numerical methods for hyperbolic balance laws with ... · in the following, numerical methods are...

Numerical methods for hyperbolic balance

laws with discontinuous flux functions and

applications in radiotherapy

Numerische Verfahren fur hyperbolische Bilanzgleichungen mit unstetiger

Flussfunktion und Anwendungen in der Strahlentherapie

von

Nadine Pawlitta

27.10.2011

Masterarbeit in Mathematik

vorgelegt der

Fakultat fur Mathematik, Informatik und Naturwissenschaften

der Rheinisch-Westfalischen Technischen Hochschule Aachen

Angefertigt bei

Prof. Dr. Martin Frank

Lehrstuhl fur Mathematik CCES

Zweitgutachter

Prof. Dr. Michael Herty

Lehrstuhl C fur Mathematik

3

Danksagung

Ich mochte mich an dieser Stelle bei all denen bedanken, die mich bei der Anfertigung

meiner Masterarbeit unterstutzt haben.

Besonderer Dank gilt Herrn Professor Martin Frank fur die Betreuung dieser Arbeit

und fur die vielen hilfreichen Anregungen.

Zudem bedanke ich mich bei den Mitarbeitern des MathCCES und meinen Freunden

und Kommilitonen fur die freundliche Atmosphare und fachliche Unterstutzung.

Nicht zuletzt mochte ich mich auch bei meiner Familie bedanken, ohne die dieses

Studium niemals moglich gewesen ware.

Contents

1 Introduction to hyperbolic conservation laws 9

1.1 Definition of conservation laws . . . . . . . . . . . . . . . . . . . . . . . . 9

1.1.1 The advection equation . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2 Theory of numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.2.1 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.3 Numerical methods for linear equations . . . . . . . . . . . . . . . . . . . 17

1.3.1 The Upwind method . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.3.2 Godunov’s method . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.3.3 The Lax-Wendroff method . . . . . . . . . . . . . . . . . . . . . . . 22

2 The advection equation with non constant velocity 27

2.1 Theoretical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.2.1 The Upwind method on a nonuniform grid . . . . . . . . . . . . . 30

2.2.2 The Upwind method on a uniform grid . . . . . . . . . . . . . . . 33

2.2.3 The Lax-Wendroff method . . . . . . . . . . . . . . . . . . . . . . . 41

2.3 Comparison and evaluation of the derived methods . . . . . . . . . . . . . 45

3 Nonlinear partial differential equations 55

3.1 Analysis of nonlinear equations . . . . . . . . . . . . . . . . . . . . . . . . 55

3.1.1 Burgers equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.2 Numerical methods for nonlinear equations . . . . . . . . . . . . . . . . . 58

3.2.1 The Lax-Friedrichs method . . . . . . . . . . . . . . . . . . . . . . 59

3.3 Nonlinear equations with spatially varying, discontinuous flux function . . 64

3.3.1 The Lax-Friedrichs method on a nonuniform grid . . . . . . . . . . 65

3.3.2 The Lax-Friedrichs method on a uniform grid . . . . . . . . . . . . 68

3.3.3 Flux functions with arbitrary, piecewise constant coefficient . . . . 71

4 Applications in radiotherapy 77

4.1 A model for dose calculation . . . . . . . . . . . . . . . . . . . . . . . . . 77

5

6 Contents

4.2 Solution of the 1D model . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.3 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.3.1 Homogeneous test case . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.3.2 One-dimensional dose calculation . . . . . . . . . . . . . . . . . . . 85

5 Conclusions and outlook 91

Introduction

Radiotherapy is a way of treating diseases like cancer by using ionizing radiation that

destroys the cells in the treated area. At the same time, the damage to the surrounding

tissue should be as small as possible. For this reason, the needed dose of radiation

has to be determined accurately. The difficulty that gave rise to this thesis is that the

radiation has to pass tissues of different density. For example, a particle that travels

through air looses much less energy than one that travels through bone. The density

can be described by a function ρ that is space dependent and can obtain values from

approximately 1, which corresponds to water, to 10−3, which is air. In this situation, the

change between the different areas and therefore densities is abrupt, so ρ is a piecewise

constant function. It can be ascertained with the data of a Computer Tomography scan

which is performed before the treatment.

The dose of radiation can then be computed by analyzing the transport of the radiation

particles. This transport can be described by a hyperbolic balance law, which is a system

of partial differential equations with a source term. In this thesis, a system of two first

order equations of the form

∂εΨ0(x, ε)− 1

ρ(x)∂xΨ1(x, ε) = 0,

∂εΨ1(x, ε)− 1

ρ(x)∂x

(Ψ0(x, ε)χ

(Ψ1(x, ε)

Ψ0(x, ε)

))= T (ε)Ψ1(x, ε)

is considered as a model for the radiative transport.

Usually, there exist several different numerical methods to solve such equations with high

accuracy, but the difference in the velocity makes them computationally too expensive

and therefore infeasible. The reason for this is the so-called CFL condition which is a

necessary condition for stability of the numerical methods. It establishes a connection

between the time step size and the spatial step size that depends on the density ρ. For

small values of ρ, as is the case when modeling air, the CFL condition demands a much

smaller time step size than spatial step size. Therefore, the existing standard methods

would solve the equation with a time step size that is only needed for particles that

travel through air and is much smaller than necessary for particles that travel through

7

8 Introduction

water.

The aim of this thesis is to find a numerical method that is better suited for this problem

by adjusting existing methods so that they no longer require such a small time step size.

Note that there does not exist sufficient theoretical knowledge of such situations. While

it will be possible to derive methods that solve the given equations, there is no theory

that provides the means to evaluate the methods analytically.

In the following, numerical methods are derived for one-dimensional systems of equations

by gradually extending methods for more simple problems. The first chapter gives a

general introduction to first-order, hyperbolic partial differential equations without such

a discontinuous coefficient and introduces basic numerical methods for the special case

of linear equations. In the second chapter, these methods are modified to solve linear

equations with the addition of a piecewise constant coefficient which will later on be the

density ρ. The results of these two chapters will be used in the third to solve nonlinear

equations, first without and then with the discontinuous coefficient. This yields the basis

for the fourth chapter. There, the numerical methods are validated and applied to dose

calculation in radiotherapy.

1 Introduction to hyperbolic conservation

laws

This chapter introduces the basics of hyperbolic partial differential equations and their

solutions and is mostly based on [11] and [12]. After establishing some theoretical facts

about such equations, the finite volume method for solving them is derived and analyzed

for convergence to the true solution. Following this, two basic methods for the special

case of a linear equation are presented.

1.1 Definition of conservation laws

Consider the initial value problem or Cauchy problem

∂

∂tu(x, t) +

∂

∂xf(u(x, t)) = 0, x ∈ R, t > 0 (1.1)

u(x, 0) = u0(x), x ∈ R

where f ∈ C1(R) and u0 ∈ L∞(R).

The equation ut + (f(u))x = 0 is the differential form of a conservation law. In order to

derive it, examine the one-dimensional example of the flow of a substance with density

u through a pipe. The mass between two points x1 and x2 at time t can be described by∫ x2

x1

u(x, t)dx.

Conservation of mass then means that the mass of the substance does not change within

the pipe but rather through flux at the boundaries of the pipe. In case this flux can be

described by the function f , it gives the integral form of the conservation law

∂

∂t

∫ x2

x1

u(x, t)dx = f (u(x1, t))− f (u(x2, t)) . (1.2)

Assuming that u and f are smooth, this results in

∂

∂t

∫ x2

x1

u(x, t)dx = −∫ x2

x1

∂

∂xf (u(x, t)) dx

9

10 1 Introduction to hyperbolic conservation laws

which can be written as∫ x2

x1

(∂

∂tu(x, t) +

∂

∂xf (u(x, t))

)dx = 0.

This yields the first introduced differential form of the conservation law.

1.1.1 The advection equation

The advection equation

∂

∂tu(x, t) + a

∂

∂xu(x, t) = 0 (1.3)

is a special case of (1.1) with a = f ′(u) since (f(u))x = f ′(u)ux = aux. It models the

advection of a substance in a fluid with velocity a.

It is easy to verify that, given an initial condition u(x, 0) = u0(x), the solution of this

equation is given by u(x, t) = u0(x− at) as

∂

∂tu0(x− at) + a

∂

∂xu0(x− at) = −au′0(x− at) + au′0(x− at) = 0.

Therefore, for arbitrary x0, the solution u(x, t) is constant along the so-called character-

istics X(t) = x0 + at of the equation because

d

dtu(X(t), t) = ut(X(t), t) +X ′(t)ux(X(t), t) = ut + aux = 0.

t

x

Figure 1.1: Characteristics for the advection equation.

Figure 1.1 shows for example some characteristics for an advection equation with positive


velocity. The value of the solution u is the same on each point of one characteristic, this

means that in practice, the initial value u0 simply shifts to the right in time, as can be

seen in figure 1.2.

0 25 500

1initial condition, time t=0

0 25 500

1time t=150

0 25 500

1time t=300

0 25 500

1time t=450

Figure 1.2: Solution of the advection equation at different time steps.

In higher dimensions, a linear, constant-coefficient equation like the advection equation

becomes a linear system

ut +Aux = 0

since the system

ut + (f(u))x = 0

can be written as

ut + f ′(u)ux = 0

if f is smooth and f ′(u) = A ∈ Rm×m is the Jacobian of f . Such a system is called

hyperbolic if the matrix A is diagonalizable with real eigenvalues λ1, . . . , λm, that is

A = R diag(λ1, . . . , λm)R−1


where R is the matrix that is composed of the corresponding eigenvectors. As a result

ut +Aux = 0 ⇔ R−1ut +R−1ARR−1ux = 0

⇔ vt + diag(λ1, . . . , λm)vx = 0

⇔ vpt + λpvpx = 0, for p = 1 . . .m

with R−1u =: v and therefore, the original system is equivalent to a system of m in-

dependent advection equations. Note that a scalar, first-order differential equation is

always hyperbolic.

1.2 Theory of numerical methods

The basic idea of the so-called finite volume methods is to divide the spatial domain

into a mesh and to approximate the integral of u over each cell in the mesh. In the

one-dimensional case, one of those cells is an interval

Ci = (xi−1/2, xi+1/2)

with size ∆xi = xi+1/2 − xi−1/2 and ∆xi = ∆x in the case of a uniform grid as pictured

in figure 1.3.

t

x

tn

tn+1

xixi-1 xi+1

Figure 1.3: Example for a uniform grid.

The approximation uni to the average value of the solution over Ci at time tn can then

be expressed as

uni ≈1

∆x

∫Ciu(x, tn)dx.


This can be used to construct a numerical method for the solution of the PDE. The

starting point is the integral form (1.2) of the conservation law

∂

∂t

∫Ciu(x, t)dx = f

(u(xi−1/2, t)

)− f

(u(xi+1/2, t)

).

Integration over time and rearrangement gives∫Ciu(x, tn+1)dx−

∫Ciu(x, tn)dx =

∫ tn+1

tn

f(u(xi−1/2, t)

)dt−

∫ tn+1

tn

f(u(xi+1/2, t)

)dt

⇔ 1

∆x

∫Ciu(x, tn+1)dx =

1

∆x

∫Ciu(x, tn)dx

− ∆t

∆x

(1

∆t

∫ tn+1

tn

f(u(xi+1/2, t)

)dt− 1

∆t

∫ tn+1

tn

f(u(xi−1/2, t)

)dt

)where ∆t = tn+1 − tn is one time step. The method should therefore have the form

un+1i = uni −

∆t

∆x(Fni+1/2 − F

ni−1/2) (1.4)

with uni approximated as before and the average flux Fni−1/2 as

Fni−1/2 ≈1

∆t

∫ tn+1

tn

f(u(xi−1/2, t)

)dt.

This can be computed by means of a numerical flux function F where it is assumed for

now that the flux Fni−1/2 depends only on the values uni−1 and uni so that

Fni−1/2 = F(uni−1, uni ). (1.5)

Thus, the method is defined by the flux and approximates the value un+1i by

un+1i = uni −

∆t

∆x

(F(uni , u

ni+1)−F(uni−1, u

ni )), (1.6)

which is the standard formulation of a finite volume method.

More generally, if the flux Fni−1/2 depends on uni−m up to uni+m−1, the method is given

by

un+1i = uni −

∆t

∆x

(F(uni−m+1, . . . , u

ni+m)−F(uni−m, . . . , u

ni+m−1)

).

Let N then describe the numerical method as

N (uni−m, . . . , uni+m) = uni −

∆t

∆x

(F(uni−m+1, . . . , u

ni+m)−F(uni−m, . . . , u

ni+m−1)

).

A method that can be written thusly is called conservative or in conservation form.

The following sections examine this condition for ∆t before specifying a method by

choosing a numerical flux function.


1.2.1 Convergence

An obvious property of the numerical method should be that the computed solution

converges to the exact one if the grid is refined, that is for ∆x,∆t→ 0. The Equivalence

Theorem of Lax and Richtmyer (see [9]) states that stability is necessary and sufficient

for convergence of a linear, consistent method. Therefore, there are two conditions that

have to be fulfilled to achieve convergence.

• Consistency deals with the error in one time step, it is given when the differential

equation is locally well approximated. For the numerical flux function F this

means that in the special case where u is constant and Fni−1/2 = f(u), F must

satisfy F(u, u) = f(u).

• Stability is the property that the error of a time step does not increase too much.

In the following, both attributes will be examined in more detail.

Consistency Consider the error caused by the application of the numerical method

after one time step. This error is called the local truncation error, it compares the true

solution at a time t + ∆t with the true solution at time t after one step of the method

and is given as

L(x, t) =1

∆t(N (u(x−m∆x, t), . . . , u(x+m∆x, t))− u(x, t+ ∆t)) . (1.7)

The numerical method is now said to be consistent if

lim∆t→0

‖L(., t)‖ = 0.

Furthermore, if the error can be written as

‖L(., t)‖ ≤ c∆tp (1.8)

for a constant c and arbitrary, smooth initial data, the method is of order p. In this

thesis, the discrete 1-norm ‖x‖1 =∑

i |xi| is used, though other norms can be used as

well, but they might give different results.

Instead of determining the order analytically, it can be estimated graphically as well. If

the ratio of ∆x and ∆t is set to be constant,

‖L(., t)‖ ≤ c∆tp ⇔ ‖L(., t)‖ ≤ c∆xp


for some other constant c. Since it will later on be important to observe the error at

every time step, consider in this thesis from now on the error

ε :=1

T

T∑t=1

∥∥∆x(u(., t∆t)−N t(u0(.))∥∥

1

where N t means that N has been applied t times. Therefore, the error ε is the weighted

sum of the difference of the exact solution and the computed one at each point in space

and time. If the method is now of order p, then ε can be estimated by ∆xp as well. For

a constant C it then holds that

ε = C∆xp ⇔ log ε = logC + p log ∆x.

For this reason, the logarithm of the error can be interpreted as a straight line with slope

p. To assess the order p simply compute the error for different values of ∆x and plot

them as a function of log ∆x.

Stability A necessary condition for stability is the so-called CFL condition, named after

Courant, Friedrichs and Lewy [4]. For the advection equation (1.3) it was shown above

that the solution is constant along the characteristics and u(x, t) = u0(x− at) for arbi-

trary x and t. In other words, the solution at a point (x, t) depends on the initial data

only at one point x− at. Because of this, the numerical solution should as well depend

on at least this point, or rather on the point on the respective characteristic that was

computed one time step before. This set of points is called the domain of dependence

D(x, t) of the PDE at the point (x, t) and simply takes the form D(x, t) = x− at for

the advection equation.

The CFL condition now states for the general case that a numerical method can only be

convergent if the domain of dependence of the PDE is a subset of the numerical domain

of dependence.

For a method of the form (1.6), which computes un+1i using a three point stencil con-

sisting of uni−1,uni and uni+1, the domain of dependence is the set

D(x, t) =

x

∣∣∣∣x− ∆x

∆tt ≤ x ≤ x+

∆x

∆tt

as can be seen in figure 1.4. There the value at the point (xi, tn+1) depends on (xi−1, tn),

(xi, tn) and (xi+1, tn) and these in turn on (xi−2, tn−1) up to (xi+2, tn−1) and so on, so

that the edges of the resulting triangle can be described by t 7→ x− ∆x∆t t and t 7→ x+ ∆x

∆t t.

Hence, for the CFL condition to be fulfilled, the domain of dependence of the given PDE

has to be situated inside this triangle.


tn

tn+1

tn-1

xixi-1 xi+1xi-2 xi+2

Figure 1.4: Three point stencil.

For the advection equation, the domain of dependence is the set D(x, t) = x− at and

therefore

x− ∆x

∆tt ≤ x− at ≤ x+

∆x

∆tt,

which can be reduced to ∣∣∣∣a∆t

∆x

∣∣∣∣ ≤ 1 (1.9)

where the number ν =∣∣a∆t

∆x

∣∣ is called the courant number.

Monotonicity and Total Variation There are other properties that are connected to

the convergence of a method and will be analyzed further here. The first is the concept of

monotonicity. It is reasonable to demand that if the solution is monotone on an interval,

the numerical solution should be monotone as well. Based on this notion, a numerical

method N is called monotone if

uni ≥ unj ⇒ N (uni ) ≥ N (unj )

and it is called monotonicity-preserving if

u0i ≥ u0

j ⇒ uni ≥ unj .

Another requirement deals with the total variation of a function. For an arbitrary

function g the total variation is defined as

TV (g) = lim supε→0

1

ε

∫ ∞−∞|g(x)− g(x− ε)|dx.


For the special case of the piecewise constant function or rather grid function un this

simplifies to

TV (un) =∞∑

i=−∞|uni − uni−1|.

The total variation can be used to measure oscillations in a function. To avoid the

appearance of oscillations which are only the result of the computations and have no

physical meaning, the total variation of the solution should not increase. A numerical

method is therefore called total variation diminishing (TVD) or more appropriately total

variation non-increasing if

TV (un+1) ≤ TV (un)

for un+1 = N (un). Furthermore, monotonicity implies TVD and TVD implies mono-

tonicity preserving, so it suffices to proof monotonicity when testing a numerical method

for all three properties.

1.3 Numerical methods for linear equations

Derive now the basic numerical methods for the solution of the advection equation and

analyze them for convergence as described above.

1.3.1 The Upwind method

It is reasonable to choose the flux function Fni−1/2 according to the structure of the

solution in order to achieve better results. For the constant-coefficient advection equation

ut + aux = 0 it was shown that the solution is constant along the characteristics, that

is u(xi, tn) = u(xi + a∆t, tn+1). Hence, if a is positive, u(xi, tn+1) depends only on the

values of u at time tn to the left of xi and the same should hold for un+1i . In other words,

the flux Fni−1/2 should be determined by uni−1. Choosing

Fni−1/2 = F(uni−1, uni ) = auni−1

in (1.4) yields the first-order upwind method for the advection equation as

un+1i = uni −

a∆t

∆x(uni − uni−1). (1.10)

It is important to observe that this ansatz is only correct if u(xi, tn+1) depends solely

on u(xi−1, tn), which makes the above choice of the flux sensible. This is given by the


CFL condition for the advection equation (1.9) because u(xi, tn+1) = u(xi−a∆t, tn) and

therefore xi − a∆t ≥ xi −∆x = xi−1. Figure 1.5 first depicts the case where the CFL

condition is fulfilled because it takes more than one time step for the characteristics to

cross one spatial interval Ci. In the second part this is no longer the case and the CFL

condition is breached.

tn

tn+1

xixi-1 xi+1

tn

tn+1

xixi-1 xi+1

Figure 1.5: Two examples where the CFL condition is fulfilled and breached.

It should be noted that the upwind method for the advection equation is exact if ν =a∆t∆x = 1 because then

un+1i = uni − (uni − uni−1) = uni−1,

which complies with

u(xi, tn+1) = u(xi − a∆t, tn) = u(xi −∆x, tn) = u(xi−1, tn).

The latter is a property of the exact solution of the PDE and is given by following the

characteristics. In most cases though, ν is strictly smaller than 1, which leads to the

situation in figure 1.6.

Another possibility to derive the upwind method is to take the convex combination

of u(xi−1, tn) and u(xi, tn) to compute u(xi, tn+1) since u(xi, tn+1) = u(xi − a∆t, tn) =

u(xi−ν∆x, tn). This is the result of the evaluation of the linear interpolation polynomial

through xi−1 and xi at the point xi − a∆t. The polynomial has the form

uni +uni−1 − unixi−1 − xi

(x− xi) = uni −uni−1 − uni

∆x(x− xi),

which simplifies to

uni + ν(uni−1 − uni ) = uni − ν(uni − uni−1)

when inserting x = xi − a∆t.


tn

tn+1

xixi-1 ix -aΔt

Figure 1.6: The upwind method for ν < 1.

Properties of the upwind method The method is of first order as can be shown by

examining the local truncation error (1.7) that is given as

L(xi, tn) =1

∆t(u(xi, tn)− ν(u(xi, tn)− u(xi−1, tn))− u(xi, tn+1)) .

Replace u(xi−1, tn) and u(xi, tn+1) by their Taylor series expansion

u(xi−1, tn) = u(xi, tn)−∆xux(xi, tn) +1

2∆x2uxx(xi, tn) +O(∆x3),

u(xi, tn+1) = u(xi, tn) + ∆t ut(xi, tn) +1

2∆t2utt(xi, tn) +O(∆t3),

which yields

L(xi, tn) =1

∆t

[u(xi, tn)− ν

(u(xi, tn)− u(xi, tn) + ∆xux(xi, tn)− 1

2∆x2uxx(xi, tn)−O(∆x3)

)−u(xi, tn)−∆t ut(xi, tn)− 1

2∆t2utt(xi, tn)−O(∆t3)

]=

1

∆t

[−ν∆xux(xi, tn) +

1

2ν∆x2uxx(xi, tn) + νO(∆x3)

−∆t ut(xi, tn)− 1


]= −a ux(xi, tn) +

1

2a∆xuxx(xi, tn) +

a

∆xO(∆x3)

− ut(xi, tn)− 1

2∆tutt(xi, tn)− 1

∆tO(∆t3).

In a last step, use that for the advection equation

ut + aux = 0 ⇒ ut = −aux ⇒ utt = (−aux)t = (−aut)x = a2uxx,


so that

L(xi, tn) =1

2a∆xuxx(xi, tn)− 1

2a2∆tuxx(xi, tn) +O(∆x2,∆t2)

=1

2a(∆x− a∆t)uxx(xi, tn) +O(∆x2,∆t2).

Since the uxx term has the most influence on the local truncation error, it was proven

that the method is of first order in time and space.

This can be verified graphically as explained in section 1.2.1. Figure 1.7 shows an

example for the solution of the advection equation with velocity a = 1. The initial

condition is a part of the arc tangent on the interval [0, 1000]. Since the upwind method

then requires a boundary condition for x = 0, assume for simplicity that the value un0is given. Furthermore, the Courant number is ν = 0.9 and the step size ∆x is given as

10i/20 with i = −20, . . . , 20. The plot shows the base 10 logarithm of the error depending

on the logarithm i/20 of the step size. As can be seen, the slope and with it the order

is indeed 1.

1 0.5 0 0.5 12

1

0

1Upwind

least squares approximation(slope 1.0052)error

Figure 1.7: The logarithmic error plot of the upwind method for the solution of the

advection equation with an arc tangent as initial condition

It is easy to verify further that this method is TVD and monotonicity preserving as an


additional property since

TV (un+1) =∞∑

i=−∞

∣∣un+1i − un+1

i−1

∣∣=

∞∑i=−∞

∣∣(1− ν)(uni − uni−1) + ν(uni−1 − uni−2)∣∣

≤∞∑

i=−∞(1− ν)

∣∣uni − uni−1

∣∣+

∞∑i=−∞

ν∣∣uni−1 − uni−2

∣∣= (1− ν)TV (un) + νTV (un)

= TV (un).

1.3.2 Godunov’s method

The upwind method can be interpreted as a special case of Godunov’s method which de-

termines the values un+1i with the REA algorihm. Here, REA stands for Reconstruction,

Evolution and Averaging which is done as follows.

1. Reconstruction step: Define a piecewise polynomial u(x, tn) from the current so-

lution uni .

2. Evolution step: Use u(x, tn) as an initial condition for the given differential equa-

tion and compute its solution u(x, tn+1) after one time step ∆t.

3. Averaging step: Determine the new cell average un+1i by computing the average of

u(x, tn+1) over the interval Ci = (xi−1/2, xi+1/2), that is

un+1i =

1

∆x

∫Ciu(x, tn+1)dx.

The second step requires the solution of a so-called Riemann problem, which is the

hyperbolic equation with the discontinuous initial condition

u0(x) =

ul, for x ≤ xi+1/2,

ur, for x > xi+1/2.

The upwind method for the advection equation is the result of this method when using

the piecewise constant function

u(x, tn) = uni , for all x ∈ Ci

in the first step so that ul = uni and ur = uni+1 in the Riemann problem. Such an

approach can obviously be improved by using a more precise reconstruction. This idea

gives rise to the next section.


1.3.3 The Lax-Wendroff method

In [10], Lax and Wendroff proposed the Lax-Wendroff method, which is an improvement

of the upwind method in terms of the order of consistency. To derive the method, begin

with the Taylor series expansion of u to get

u(xi, tn+1) = u(xi, tn) + ∆tut(xi, tn) +1

2∆t2utt(xi, tn) +O(∆t3).

The time derivatives can be replaced using the advection equation (1.3) as

ut + aux = 0

⇒ ut = −aux⇒ utt = (−aux)t = (−aut)x = a2uxx.

The O(∆t) and O(∆x) terms will be omitted in the following. This yields

u(xi, tn+1) = u(xi, tn)− a∆tux(xi, tn) +1

2a2∆t2uxx(xi, tn).

In a last step, approximate the spatial derivatives with central differences

ux(xi, tn) =u(xi+1, tn)− u(xi−1, tn)

2∆x,

uxx(xi, tn) =ux(xi+1/2, tn)− ux(xi−1/2, tn)

∆x

=u(xi+1,tn)−u(xi,tn)

∆x − u(xi,tn)−u(xi−1,tn)∆x

∆x

=u(xi+1, tn)− 2u(xi, tn)− u(xi−1, tn)

∆x2.

Together, this results in

u(xi, tn+1) = u(xi, tn)− a∆t

2∆x(u(xi+1, tn)− u(xi−1, tn))

+a2∆t2

2∆x2(u(xi+1, tn)− 2u(xi, tn)− u(xi−1, tn))

so the Lax-Wendroff method for the advection equation has the form

un+1i = uni −

a∆t

2∆x

(uni+1 − uni−1

)+

1

2

(a∆t

∆x

)2 (uni+1 − 2uni + uni−1

). (1.11)

Alternatively, use a quadratic interpolation instead of a linear one as was done for

the upwind method. The data points (xi+1, uni+1), (xi, u

ni ) and (xi−1, u

ni−1) give the

interpolation polynomial

uni+1 +uni+1 − uni

∆x(x− xi+1) +

uni+1 − 2uni + uni−1

2∆x2(x− xi+1)(x− xi),


which is the same as (1.11) for x = xi − a∆t.

The method can be written in conservation form since

un+1i = uni −

a∆t

2∆x

(uni+1 − uni−1

)+

1

2

(a∆t

∆x

)2 (uni+1 − 2uni + uni−1

)= uni −

∆t

∆x

(a

2

(uni+1 − uni−1

)− a2∆t

2∆x

(uni+1 − 2uni + uni−1

))= uni −

∆t

∆x(Fni+1/2 − F

ni−1/2)

with

Fni−1/2 = F(uni−1, uni ) =

a

2

(uni + uni−1

)− a2∆t

2∆x

(uni − uni−1

).

Properties of the Lax-Wendroff method and comparison with the upwind method.

As mentioned before, the method is of second order. Analytically it can be proven by

the same approach as for the upwind method. Begin with

L(xi, tn) =1

∆t

[u(xi, tn)− ν

2(u(xi+1, tn)− u(xi−1, tn))

+ν2

2(u(xi+1, tn)− 2u(xi, tn) + u(xi−1, tn))− u(xi, tn+1)

].

Replace u(xi+1, tn), u(xi−1, tn) and u(xi, tn+1) again by their Taylor series expansion

u(xi±1, tn) = u(xi, tn)±∆xux(xi, tn) +1

2∆x2uxx(xi, tn)± 1

6∆x3uxxx(xi, tn) +O(∆x4),


2∆t2utt(xi, tn) +

1

6∆t3uttt(xi, tn) +O(∆t4).

The local truncation error then becomes

L(xi, tn) =1

∆t

[u(xi, tn)− ν

2

(2∆xux(xi, tn) +

1

3∆x3uxxx(xi, tn)

)+ν2

2

(2u(xi, tn) + ∆x2uxx(xi, tn) + 2O(∆x4)− 2u(xi, tn)

)−u(xi, tn)−∆t ut(xi, tn)− 1

2∆t2utt(xi, tn)− 1

6∆t3uttt(xi, tn)−O(∆t4)

]=

1

∆t

[−ν∆xux(xi, tn)− ν

6∆x3uxxx(xi, tn) +

ν2

2∆x2uxx(xi, tn)− ν2O(∆x4)


2∆t2utt(xi, tn)− 1

6∆t3uttt(xi, tn)−O(∆t4)

]= −aux(xi, tn)− 1

6a∆x2uxxx(xi, tn) +

a2

2∆tuxx(xi, tn)− a2∆t

∆x2O(∆x4)

− ut(xi, tn)− 1

2∆tutt(xi, tn)− 1

6∆t2uttt(xi, tn)−O(∆t3).


In a last step, write the time derivatives as space derivatives by means of the advection

equation

ut = −aux ⇒ utt = a2uxx ⇒ uttt = −a3uxxx,

to get

L(xi, tn) = −1

6a∆x2uxxx(xi, tn) +

1

6a3∆t2uxxx(xi, tn) +O(∆x3,∆t3)

=1

6a(a2∆t2 −∆x2)uxxx(xi, tn) +O(∆x3,∆t3).

The method is in fact of second order, because the coefficient of the uxxx term depends

on ∆t2 and ∆x2. Figure 1.8 proves this as well, it shows the logarithmic error plot for

the example that was used to illustrate the order of the upwind method in the previous

section.

1 0.5 0 0.5 15

4

3

2

1

0Lax Wendroff


Figure 1.8: The logarithmic error plot of the Lax-Wendroff method for the solution of

the advection equation with an arc tangent as initial condition.

However, the Lax-Wendroff method does not necessarily give much better results than

the upwind method. One disadvantage is that the Lax-Wendroff method is not TVD as

can be seen in figure 1.9.

There the upwind method smoothes the discontinuities and therewith compresses the

function while the general form is conserved. With the Lax-Wendroff method this is not

the case and oscillations are generated. Another disadvantage is that the Lax-Wendroff

method no longer has second order in such situations. The definition and the proof of

the order were based on the assumption that the solution is sufficiently smooth. This

is not given for this example and thus the theoretical order of both the upwind and

the Lax-Wendroff method is no longer reached. Figure 1.10 illustrates this problem in


0 10 20

0

0.2

0.4

0.6

0.8

1

initial condition

0 10 200.2

0

0.2

0.4

0.6

0.8

1

1.2Upwind at time t=100

0 10 200.2

0

0.2

0.4

0.6

0.8

1

1.2Lax Wendroff at time t=100

Figure 1.9: Solution of the advection equation using the upwind and the Lax-Wendroff

method.

the form of the logarithmic error plots for the above example. While the order of the

upwind method is approximately 0.6 instead of 1, the order of the Lax-Wendroff method

decreases by 1.3 to 0.7 and therefore is only marginally better than the one of the upwind

method.

2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 01.5

1

0.5

0

0.5Upwind


2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 01.5

1

0.5

0

0.5Lax Wendroff


Figure 1.10: Logarithmic error plot for the upwind and the Lax-Wendroff method.

2 The advection equation with non

constant velocity

2.1 Theoretical analysis

Consider now the advection equation as described in section 1.1.1 with the only difference

that the velocity a is no longer constant and equation (1.3) takes the form

∂

∂tu(x, t) + a(x)

∂

∂xu(x, t) = 0. (2.1)

Here a(x) is restricted to a piecewise constant function of the form

a(x) =

a1, if x ≤ α,

a2, if x ≥ α

with 0 < a1 ≤ a2 and α an arbitrary value in space where the velocity changes. This

generalization causes the characteristics to no longer have the same slope as illustrated

in figure 2.1. The slope is constant as long as the characteristics do not cross the jump

α where they kink.

t

xαFigure 2.1: Characteristics for the advection equation with piecewise constant velocity.

Like in the constant-coefficient case, the solution of the corresponding initial value prob-

lem is constant along the characteristics. For an arbitrary x0 ≥ α, they can be written

27

28 2 The advection equation with non constant velocity

as

X(t) = x0 + a2t,

so that the problem reduces to the constant-coefficient case. For x0 < α, the character-

istics can cross the position of the jump, hence

X(t) =

x0 + a1t, if x0 + a1t ≤ α ⇔ t ≤ α−x0a1

=: t∗,

x0 + a1t∗ + a2(t− t∗), else.

The result of this kink in some of the characteristics and the different slopes is that the

initial value u0 shifts to the right as before but with different speed on each side of the

jump, so that the solution dilates at the jump as shown in figure 2.2.

0 25 500

1initial condition, time t=0

0 25 500

1time t=50

0 25 500

1time t=75

0 25 500

1time t=100

Figure 2.2: Solution of the advection equation with piecewise constant velocity at differ-

ent time steps. Here a1 = 0.1, a2 = 1 and the jump lies at α = 10.

These solutions can be computed by examining the characteristics at one point. For the

example in figure 2.3 it holds that u(x1, tn) = u(x2, tn+1) and u(x5, tn) = u(x6, tn+1)

for the solutions along the characteristics that do not cross the jump and u(x3, tn) =

u(α, t∗) = u(x4, tn+1) for the one that does.


(x ,t )n+12

(x ,t )n1

(x ,t )n+14 (x ,t )n+16

(x ,t )n3 (x ,t )n5

t n+1

t n

t*

Figure 2.3: The solution to the PDE is constant along each of the three possible shapes

of the characteristics.

For the solution u(x, t) this means that if x ≤ α, one simply has to follow the charac-

teristic with slope a1 until it reaches time t = 0 where u is known through the initial

condition. The same holds for the characteristics with slope a2 if x is sufficiently bigger

than α because if x ≥ α+ a2t, the characteristic does not cross the jump. For values of

x in between, one additionally needs to determine the time t∗ where the characteristic

crosses the jump, that is

x4 = α+ a2(tn+1 − t∗) = x3 + a1(t∗ − tn) + a2(tn+1 − t∗)

⇔ t∗ =α+ a2tn+1 − x4

a2. (2.2)

Overall, this yields the solution as

u(x, t) =

u0(x− a1t), if x ≤ α,

u0

((1− a1

a2

)α+ a1

a2x− a1t

), if α < x < α+ a2t,

u0(x− a2t), if x ≥ α+ a2t

where u0

((1− a1

a2

)α+ a1

a2x− a1t

)= u0(α− a1t

∗) = u0

(α− a1

(α+a2t−x

a2

)).

It is easy to prove that this is indeed the solution of the PDE by examining the three

cases. For x ≤ α, one obtains

ut + a1ux = (u0(x− a1t))t + a1 (u0(x− a1t))x

= −a1u′0(x− a1t) + a1u

′0(x− a1t) = 0.


The case x ≥ α + a2t is analogue with a2 instead of a1. For α < x < α + a2t, inserting

the solution in the PDE results in

ut + a2ux =

(u0

((1− a1

a2

)α+

a1

a2x− a1t

))t

+ a2

(u0

((1− a1

a2

)α+

a1

a2x− a1t

))x

= −a1u′0

((1− a1

a2

)α+

a1

a2x− a1t

)+ a2

a1

a2u′0

((1− a1

a2

)α+

a1

a2x− a1t

)= 0.

2.2 Methods

In this section, different methods for solving the advection equation with piecewise con-

stant velocity a(x) =

a1, if x ≤ α,

a2, if x ≥ αwhere a1 ≤ a2 are introduced and analyzed. The

problem that arises when the velocity is no longer constant is that the CFL condition

ν = a(x)∆t∆x ≤ 1 has still to be fulfilled. Since the step size ∆t is generally constant, the

spatial step size ∆x has to be chosen complying the CFL condition for all values of a(x).

Alternatively, the stencil of the method has to be adapted to a(x). The following sec-

tions derive different methods that are suited for the given situation and analyze them

theoretically. At the end of the chapter in section 2.3, they will be compared with each

other by means of exemplary computations.

2.2.1 The Upwind method on a nonuniform grid

The most simple numerical method for solving the advection equation is the upwind

method as introduced in section 1.3.1. This section studies the ansatz of using such a

simple upwind method with the modification of an adaptive spatial step size to conform

to the change in a.

Consider a piecewise constant step size

∆x =

∆x1, if x ≤ α,

∆x2 = a2a1

∆x1, if x ≥ α

where ∆x1 is the chosen value for the smaller step size. Then

∆t =ν∆x

a(x)=ν∆x1

a1=ν∆x2

a2

with ν ≤ 1, so that the CFL condition is satisfied.


Special case

To simplify matters, assume for now that the spatial grid is constructed in such a way

that the jump lies on a grid point, so that the characteristics through the grid points do

not cross the jump between time steps. This results in the situation depicted in figure

2.4.

t n+1

t nα x

Figure 2.4: The grid points and corresponding characteristics on a nonuniform grid with

courant number ν = 1.

Since the characteristics do not kink during a time step and ν = a1∆t∆x1

= a2∆t∆x2

, one can

apply the upwind method

un+1i = uni − ν(uni − uni−1)

as in section 1.3.1 on the nonuniform grid with the step sizes ∆x1 and ∆x2.

This method was constructed so that it conserves the properties of the upwind method

for the constant coefficient advection equation. The proof that the method is of first

oder and TVD remains the same as in section 1.3.1.

General case

In a next step, consider an analogous method without the constraint that the jump lies

on a grid point. There are two possible approaches to adjust the above method, which

are illustrated in figure 2.5. Before, the jump separated the area with small step size

from the one with bigger step size. Since this is no longer possible, the grid must be

constructed in such a way that the jump lies either in an interval of small step size or

in one of bigger step size.

As can be observed in the figure, the jump in an interval of smaller step size ∆x1 may

result in the characteristic crossing a grid point. Whether this happens depends on the

velocity a and the position of the jump in relation to the grid. The nearer the jump lies


t n+1

t n xα

t n+1

t n xα

Figure 2.5: The grid points and corresponding characteristics on a nonuniform grid,

where the jump is located in an interval of size ∆x2 (left) or ∆x1 (right).

to the right boundary of the interval, the higher is the possibility of the characteristic

staying in the interval. Using this approach would therefore require the method to make

a case distinction and then adjust the computations accordingly.

This can be avoided by using the other approach. Creating the grid so that the jump is in

the bigger interval of size ∆x2 ensures that no characteristic through a point (xi, tn+1)

crosses one spatial interval in one time step. This implies that the ordinary upwind

method can be used for all values except for un+1γ+1 where γ is the grid point to the left

of the jump with γ =⌊

α∆x1

⌋and α ∈ [xγ , xγ+1).

Derivation of the numerical method The most simple method for this case is to just

keep using the upwind method and to ignore that the value un+1γ+1 is then computed with

the velocity a2, which is not correct because the velocity changes at the jump. In doing

so, the properties of the method remain as before.

To improve on this, change the computation of un+1γ+1. In section 1.3.1 it was pointed

out that the upwind method can be interpreted as a linear interpolation. To compute a

value un+1i , insert the point x∗ := xi − a1∆t, where the characteristic through (xi, tn+1)

reaches tn, into the interpolation polynomial to the data points (xi−1, tn) and (xi, tn).

This can be done in this situation as well. The thus derived method is then based on

the idea of following the characteristics and is basically no upwind method anymore.

At first, compute the value x∗ for this situation. Equation (2.2) stated that the char-

acteristic through (xγ+1, tn+1) passes the jump at time t∗ = 1a2

(α − xγ+1 + a2tn+1).


Therefore,

x∗ = xγ+1 − a2(tn+1 − t∗)− a1(t∗ − tn)

= α− a1(t∗ − tn)

=

(1− a1

a2

)α+

a1

a2xγ+1 − a1∆t. (2.3)

The value of the interpolation polynomial to (xγ , tn) and (xγ+1, tn) at x∗ is

unγ+1 +unγ+1 − unγ

∆x2(x∗ − xγ+1) = unγ+1 +

(1− a1

a2

)(α− xγ+1)− a1∆t

∆x2(unγ+1 − unγ ),

which is then the value of un+1γ+1. Thus, the method has the form

un+1i =

uni +

(1−a1

a2

)(α−xi)−a1∆t

∆x2(uni − uni−1), if i = γ + 1 for α ∈ [xγ , xγ+1),

uni − ν(uni − uni−1), else.

Note that if α + a2∆t ≤ xγ+1, the characteristic through (xγ+1, tn+1) does not reach

the jump and therefore does not kink. In that case, it is not necessary to compute un+1γ+1

differently and all the properties of the upwind method are conserved as it was the case

when the grid was constructed on the jump.

Since both approaches, the ordinary upwind method and the modified method, only differ

in the computation of one value and have the same properties, they yield essentially the

same results. The small difference in un+1γ+1 has no noticeable influence on the computation

of the values adjacent to it, so that its part in the error is minimal and, because of

rounding in the computations, not noticeable.

2.2.2 The Upwind method on a uniform grid

Consider now an upwind method on a uniform grid as an alternative to the above

examined upwind method on a nonuniform grid. The idea is to construct the grid

uniformly according to the smaller velocity a1 so that for given ∆x, ∆t = ν∆xa1

with

ν ≤ 1. In this case, the CFL condition may not be fullfilled to the right of the jump

because a2∆t∆x = a2

a1ν and a2

a1≥ 1 for a1 ≤ a2. Therefore, the method computes the

solution to the right of the jump with a bigger stencil.

Under these conditions, the grid is constructed in such a way that to the left of the jump,

the interval between each of the points xi−1 and xi at time tn is only crossed by the

characteristic that passes through (xi, tn+1). To the right of the jump, this is still the

case in a way, but the characteristics pass several spatial intervals in one time step since


the slope of the characteristics to the right of the jump is s := a2a1≥ 1 times bigger than

the one to the left. Assume here that s is an integer, otherwise it has to be replaced by⌈a2a1

⌉. Furthermore, some of the characteristics now cross the jump during a time step,

which results in more than one characteristic passing the intervals around the jump at

time tn.

Special case

At first, limit the analysis to the case that the jump lies on a grid point like in the

previous section. Let β′ ≥ α be the smallest grid point for which the characteristic

through (β′, tn+1) does not reach the jump at time tn and let β = α + s ≥ β′. Denote

further α, β and β′ the indices of α, β and β′, that is xα = α, xβ = β and xβ′ = β′ where

β = α+s and β′ = α+⌈a2∆t∆x

⌉= α+dsνe. An example for this situation is illustrated in

figure 2.6. Note that β = β′ if the Courant number is sufficiently big, so that dsνe = s.

t n+1

t nα

xβ' β

Figure 2.6: The grid points and corresponding characteristics on a uniform grid.

Derivation of the numerical method As for the nonuniform grid, start with using the

upwind method and adjust the stencil accordingly. That means, compute

un+1i = uni − ν(uni − unj )

where the grid point xj has yet to be determined. As the figure suggests, there are three

different cases that have to be examined when computing the solution at (xi, tn+1).

They are distinguished from each other by the position of xi in relation to α and β.


For xi ≤ α, simply take xj = xi−1 as for the constant coefficient advection equation.

For the case xi ≥ β, note again that the slope of the corresponding characteristics is s

times bigger than for xi ≤ α, so the stencil should be s times broader, which implies

xj = xi−s. Therefore, un+1α depends on unα−1 and un+1

β on unα, so that un+1i should be

computed with unα−1 for α < xi < β. Altogether, the upwind method is

un+1i =

uni − ν(uni − uni−1), if i ≤ α,

uni − ν(uni − unα−1), if α < i < β,

uni − ν(uni − uni−s), if i ≥ β.

(2.4)

As mentioned before, s should be replaced by⌈a2a1

⌉if s is no integer. Then, ∆t could

be bigger on the interval with broader stencil, so the coefficient µ would change. Thus,

fixing the coefficient is a simplification in this case.

A distinctive feature of this method lies in the computations for α < xi < β since they

all use the value unα−1. This results in a stair-step like pattern as illustrated in figure

2.7.

Figure 2.7: Solution using the upwind method. The right figure shows a magnification

of the dashed rectangle in the left figure.

Nevertheless, the method is convergent and the discontinuities become smaller with a


smaller step size. To confirm this, examine the method for the TVD property.

TV (un+1) =∞∑

i=−∞

∣∣un+1i − un+1

i−1

∣∣=

α∑i=−∞

∣∣un+1i − un+1

i−1

∣∣+∣∣un+1α+1 − u

n+1α

∣∣+

β−1∑i=α+2

∣∣un+1i − un+1

i−1

∣∣+∣∣∣un+1β− un+1

β−1

∣∣∣+∞∑

i=β+1

∣∣un+1i − un+1

i−1

∣∣ .Compute each part separately, for the first one

α∑i=−∞

∣∣un+1i − un+1

i−1

∣∣ =

α∑i=−∞

∣∣uni − ν(uni − uni−1)− uni−1 + ν(uni−1 − uni−2)∣∣

=α∑

i=−∞

∣∣(1− ν)(uni − uni−1) + ν(uni−1 − uni−2)∣∣

≤ (1− ν)α∑

i=−∞


∣∣+ να∑

i=−∞

∣∣uni−1 − uni−2

∣∣since 0 ≤ ν ≤ 1. For the second term, it holds∣∣un+1

α+1 − un+1α

∣∣ =∣∣unα+1 − ν(unα+1 − unα−1)− unα + ν(unα − unα−1)

∣∣= (1− ν)

∣∣unα+1 − unα∣∣ .

In the third one,

β−1∑i=α+2

∣∣un+1i − un+1

i−1

∣∣ =

β−1∑i=α+2

∣∣uni − ν(uni − unα−1)− uni−1 + ν(uni−1 − unα−1)∣∣

=

β−1∑i=α+2

∣∣uni − uni−1 − ν(uni − uni−1)∣∣

≤ (1− ν)

β−1∑i=α+2


∣∣ .The fourth part is given as∣∣∣un+1

β− un+1

β−1

∣∣∣ =∣∣∣unβ − ν(unβ − u

nβ−s)− u

nβ−1 + ν(unβ−1 − u

nα−1)

∣∣∣=∣∣∣(1− ν)

(unβ − u

nβ−1

)+ ν

(unβ−s − u

nα−1

)∣∣∣≤ (1− ν)

∣∣∣unβ − unβ−1

∣∣∣+ ν∣∣∣unβ−s − unα−1

∣∣∣


and the last as

∞∑i=β+1

∣∣un+1i − un+1

i−1

∣∣ =∞∑

i=β+1

∣∣uni − ν(uni − uni−s)− uni−1 + ν(uni−1 − uni−s−1)∣∣

=∞∑

i=β+1

∣∣(1− ν)(uni − uni−1) + ν(uni−s − uni−s−1)∣∣

≤ (1− ν)∞∑

i=β+1


∣∣+ ν∞∑

i=β+1

∣∣uni−s − uni−s−1

∣∣ .Together, this yields

TV (un+1) ≤ (1− ν)

∞∑i=−∞


∣∣+ ν

α∑i=−∞


∣∣+ ν∣∣∣unβ−s − unα−1

∣∣∣+ ν∞∑

i=β+1


∣∣ ,so that the method is TVD if the last three summands add up to ν

∞∑i=−∞


∣∣.Since β = α+ s,

(∗) :=α∑

i=−∞


∣∣+∣∣∣unβ−s − unα−1

∣∣∣+∞∑

i=β+1


∣∣=

α−1∑i=−∞


∣∣+∣∣unα − unα−1

∣∣+∞∑

i=α+s+1


∣∣=

α−1∑i=−∞


∣∣+∣∣unα − unα−1

∣∣+∞∑

i=α+1


∣∣=

∞∑i=−∞


∣∣and it is guaranteed that the small discontinuities do not become oscillations.

To be able to compute a solution without this feature, try improving the method by

using the same ansatz as was done for the nonuniform grid.

Case 1. For the solution at xi ≤ α, simply take the upwind method as introduced in

section 1.3.1, that is

un+1i = uni −

a1∆t

∆x(uni − uni−1), if i ≤ α. (2.5)


Use now a linear interpolation to derive the method for xi > α as well. For the inter-

polation error to be minimal, the data points have to be chosen as close as possible to

the point where the polynomial is evaluated, that means the grid points directly to the

left and right. For xi ≤ α, this was given by taking (xi−1, tn) and (xi, tn), which is no

longer the case for xi > α.

Case 2. The characteristics through all the points (xi, tn+1) with α < xi < β′ end

between xα and xα−1 at time tn. Therefore, the method should compute the solution at

(xi, tn+1) by means of (xα, tn) and (xα−1, tn). The interpolation polynomial to this two

data points is

unα +unα − unα−1

xα − xα−1(x∗ − xα) = unα +

x∗ − α∆x

(unα − unα−1).

To determine x∗, observe that the characteristics through (xα+j , tn+1) with j = 1, . . . , β′−α− 1 pass the jump at time tn+1 − j∆t with a constant time interval ∆t. It holds that

α+ a2j∆t = xα+j ∀ j = 1, . . . , β′ − α− 1 ⇒ ∆t =∆x

a2,

so that for the characteristic through (xi, tn+1) with j = i− α

x∗ = α− a1(∆t− j∆t) = α− a1

(∆t− j∆x

a2

)= α− a1∆t+ (i− α)

∆x

s.

In the interpolation polynomial, this gives

unα +−a1∆t+ (i− α)∆x

s

∆x(unα − unα−1) = unα +

(i− αs− a1∆t

∆x

)(unα − unα−1)

so that

un+1i = unα +

(i− αs− ν)

(unα − unα−1), if α < xi < β′. (2.6)

Case 3. To the right of the jump, the exact solution at (xi, tn+1) with xi ≥ β′ depends

on (xi − a2∆t, tn). The point xi − a2∆t = xi − sν∆x lies between the grid points

xi−dsνe∆x = xi−dsνe and xi−dsνe+1, so a numerical method should compute the solution

using these two points. Using a linear interpolation through xi−dsνe and xi−dsνe+1 yields

the interpolation polynomial

uni−dsνe +uni−dsνe+1 − u

ni−dsνe

xi−dsνe+1 − xi−dsνe(x∗ − xi−dsνe) = uni−dsνe +

uni−dsνe+1 − unidsνe

∆x(x∗ − xi−dsνe).

Inserting x∗ = xi − sν∆x results in

uni−dsνe +xi − sν∆x− xi−dsνe

∆x(uni−dsνe+1 − u

ni−dsνe),


so the solution is

un+1i = uni−dsνe + (dsνe − sν)(uni−dsνe+1 − u

ni−dsνe), if i ≥ β′. (2.7)

Combining the solutions (2.5), (2.6) and (2.7) of the three cases, the method can then

be written as

un+1i =


unα +(i−αs − ν

)(unα − unα−1), if α < i < β′,

uni−dsνe + (dsνe − sν)(uni−dsνe+1 − uni−dsνe), if i ≥ β′.

(2.8)

While the approach of following the characteristics to modify the upwind method changed

the method for the nonuniform grid only at one point, the method for the uniform grid is

altered much more. This has the advantage of yielding a more accurate solution without

the additional discontinuities that where generated in the example of figure 2.7. The

proof that this method is TVD can be found in the appendix.

General case

Relax now the restriction that the jump lies on a grid point and modify the above

methods accordingly. Depending on the position of the jump in relation to the grid

points, the characteristics that cross the jump in a time step end either in the interval of

the jump or the one to the left. Figure 2.8 shows both possibilities and it can be observed

that the farther to the right of the interval the jump lies, the more characteristics stay

in the interval of the jump.

t n+1

t n xα β'

t n+1

t n xα β' ββ

Figure 2.8: The grid points and corresponding characteristics on a uniform grid.

Keep the notation of before, that is s = a2a1

. Since α is now no grid point, let γ again be

the grid point to the left of the jump, so that γ =⌊α

∆x

⌋and α ∈ [xγ , xγ+1). Furthermore,

adjust the values of β and β′ as β = γ + s+ 1 and β′ =⌈α+a2∆t

∆x

⌉=⌈α

∆x + sν⌉.


Derivation of the numerical method The simple upwind method is the same as (2.4)

with α substituted by γ, that is

un+1i =

uni − ν(uni − uni−1), if i ≤ γ,

uni − ν(uni − unγ−1), if γ < i < β,

uni − ν(uni − uni−s), if i ≥ β.

Since this method is essentially the same as method (2.4), it has the same properties

and creates the same stair-shaped discontinuities as depicted in figure 2.7. Prove again

that these small discontinuities do not oscillate for smaller Courant number by verifying

the TVD property. The first part of the proof is the same as before if α is substituted

by γ. Differences occur when proving that

(∗) :=

γ∑i=−∞


∣∣+∣∣∣unβ−s − unγ−1

∣∣∣+∞∑

i=β+1


∣∣ =∞∑

i=−∞


∣∣ ,because now γ =

⌊α

∆x

⌋and β = γ + s+ 1, so that

(∗) =

γ−1∑i=−∞


∣∣+∣∣unγ+1 − unγ−1

∣∣+∞∑

i=γ+s+2


∣∣≤

γ−1∑i=−∞


∣∣+∣∣unγ+1 − unγ

∣∣+∣∣unγ − unγ−1

∣∣+∞∑

i=γ+2


∣∣=

∞∑i=−∞


∣∣ .Consider now again the modified upwind method. For xi ≤ γ and xi ≥ β′, the value un+1

i

can be computed in the same way as derived in case 1 and 3 before, only the case γ < xi <

β′ has to be treated differently. The easiest but therefore less accurate ansatz is to make

a linear interpolation as before and use the data points (xγ−1, unγ−1) and (xγ+1, u

nγ+1) to

cover both intervals [xγ−1, xγ) and [xγ , xγ+1) where the characteristic through (xi, tn+1)

might reach the point (x∗, tn). In section 2.2.1, x∗ was already determined as

x∗ =

(1− a1

a2

)α+

a1

a2xi − a1∆t.

The interpolation polynomial is

unγ+1 +unγ+1 − unγ−1

xγ+1 − xγ−1(x∗ − xγ+1) = unγ+1 +

x∗ − xγ+1

2∆x(unγ+1 − unγ−1),


so that

un+1i = unγ+1 +

x∗ − xγ+1

2∆x(unγ+1 − unγ−1)

= unγ+1 +

(1− a1

a2

)α+ a1

a2xi − a1∆t− xγ+1

2∆x(unγ+1 − unγ−1)

= unγ+1 +1

2

((1− 1

s

)α

∆x+i

s− ν − γ − 1

)(unγ+1 − unγ−1),

which overall yields

un+1i =


unγ+1 + 12

((1− 1

s

)α

∆x + is − ν − γ − 1

)(unγ+1 − unγ−1), if γ < i < β′,


As a somewhat more efficient approach, one could as well determine beforehand, for

which i ∈ (γ, β′) the value un+1i should be computed via (xγ−1, u

nγ−1) and (xγ , u

nγ ), or

via (xγ , unγ ) and (xγ+1, u

nγ+1). This depends on x∗, if x∗ < γ, use the first pair and if

x∗ > γ, the second. For x∗ = γ, both yields the same result. These inequalities can be

solved to get

x∗ =

(1− a1

a2

)α+

a1

a2xi − a1∆t < γ

⇔ xi < sγ + sa1∆t+ (1− s)α

⇔ i < sγ + sν + (1− s) α∆x

.

The same computations as before then give

un+1i =


unγ +((

1− 1s

)α

∆x + is − ν − γ

)(unγ − unγ−1), if γ < i ≤ sγ + sν + (1− s) α

∆x ,

unγ +((

1− 1s

)α

∆x + is − ν − γ

)(unγ+1 − unγ ), if sγ + sν + (1− s) α

∆x < i < β′,


(2.9)

As for the special case above, the proof of the TVD property for this method is in the

appendix.

2.2.3 The Lax-Wendroff method

The last two sections provided different, simple methods for the solution of the advection

equation with non constant velocity and therefore ensure that a numerical solution can


be computed. In the following, these results shall be improved in the sense of gaining

a method of higher order. As before, derive the Lax-Wendroff method and adjust it to

the two cases of a nonuniform and uniform grid by following the characteristics. The

approach of using the ordinary Lax-Wendroff method will not be examined further since

it can be derived similarly to the Upwind method. Furthermore, the aim is to gain a

method that is as accurate as possible and even for the Upwind method, the modified

method yielded somewhat better results.

Nonuniform grid

Consider the situation of section 2.2.1 now without the assumption that the jump lies

on a grid point, that is α ∈ [xγ , xγ+1) with γ =⌊

α∆x1

⌋. For all i /∈ γ, γ + 1, the

stencil of the Lax-Wendroff method lies in an interval of constant step size, so that the

Lax-Wendroff method of section 1.3.3 can be used.

When computing the value un+1γ , only the different step sizes have to be considered.

Interpolate using the data points (xγ+1, tn), (xγ , t

n) and (xγ−1, tn) to get the polynomial

unγ+1 + (unγ+1 − unγ )x∗ − xγ+1

∆x2+

(unγ+1 − unγ

∆x2−unγ − unγ−1

∆x1

)(x∗ − xγ+1)(x∗ − xγ)

∆x1 + ∆x2

where x∗ = γ − a1∆t = γ − ν∆x1, so that

un+1γ = unγ+1 − (unγ+1 − unγ )

ν∆x1 + ∆x2

∆x2

+

(unγ+1 − unγ


∆x1

)(ν∆x1 + ∆x2)ν∆x1

∆x1 + ∆x2

= unγ − (unγ+1 − unγ )ν∆x1

∆x2+

(unγ+1 − unγ


∆x1

)(ν∆x1 + ∆x2)ν∆x1

∆x1 + ∆x2.

For un+1γ+1, the Lax-Wendroff method uses the data points (xγ+2, t

n), (xγ+1, tn) and

(xγ , tn), so that the step size between these points is ∆x2, but now the characteris-

tic crosses the jump. The interpolation polynomial is then the same as in section 1.3.3,

namely

uni+1 +uni+1 − uni

∆x2(x∗ − xi+1) +

uni+1 − 2uni + uni−1

2∆x22

(x∗ − xi+1)(x∗ − xi),

and x∗ =(

1− a1a2

)α+ a1

a2xi−a1∆t with i = γ+1 as in the other situations before where

the characteristic crossed the jump. Together

un+1γ+1 = unγ+2 +

x∗ − xγ+2

∆x2

(unγ+2 − unγ+1

)+

(x∗ − xγ+2)(x∗ − xγ+1)

2∆x22

(unγ+2 − 2unγ+1 + unγ

)


where

x∗ − xγ+2 = x∗ − xγ+1 −∆x2

and

(x∗ − xγ+2)(x∗ − xγ+1) = (x∗ − xγ+1)2 −∆x2(x∗ − xγ+1).

Inserting this yields

un+1γ+1 = unγ+2 −

(unγ+2 − unγ+1

)+x∗ − xγ+1

∆x2

(unγ+2 − unγ+1 −

1

2


))+

(x∗ − xγ+1)2

2∆x22


)= unγ+1 +

x∗ − xγ+1

2∆x2

(unγ+2 − unγ

)+

(x∗ − xγ+1)2

2∆x22


)= unγ+1 +

(1− a1

a2

)(α− xγ+1)− ν∆x1

2∆x2

(unγ+2 − unγ

)+

((1− a1

a2

)(α− xγ+1)− ν∆x1

)2

2∆x22


).

Overall, the Lax-Wendroff method for a nonuniform grid is then defined as

un+1i =

uni − (uni+1 − uni )ν∆x1∆x2

+(uni+1−uni

∆x2− uni −uni−1

∆x1

)(ν∆x1+∆x2)ν∆x1

∆x1+∆x2, if i = γ,

uni +

(1−a1

a2

)(α−xi)−ν∆x1

2∆x2

(uni+1 − uni−1

)+

((1−a1

a2

)(α−xi)−ν∆x1

)2

2∆x22


), if i = γ + 1,

uni − ν2

(uni+1 − uni−1

)+ν2

2


), else.

Uniform grid

The difference between the upwind method and the Lax-Wendroff method on a uniform

grid is the higher order interpolation. Since the step size ∆x is now constant, the

interpolation polynomial for three arbitrary data points (xi−1, tn), (xi, t

n) and (xi+1, tn)


has the form

uni+1 + (uni+1 − uni )x∗ − xi+1

∆x+ (uni+1 − 2uni + uni−1)

(x∗ − xi+1)(x∗ − xi)2∆x2

=uni+1 + (uni+1 − uni )x∗ − xi −∆x

∆x+ (uni+1 − 2uni + uni−1)

(x∗ − xi)2 −∆x(x∗ − xi)2∆x2

=uni +

((uni+1 − uni )− 1

2(uni+1 − 2uni + uni−1)

)x∗ − xi

∆x+ (uni+1 − 2uni + uni−1)

(x∗ − xi)2

2∆x2

=uni + (uni+1 − uni−1)x∗ − xi

2∆x+ (uni+1 − 2uni + uni−1)

(x∗ − xi)2

2∆x2.

According to the cases examined in section 2.2.2, the corresponding data points and x∗

have to be inserted. Note again that α ∈ [xγ , xγ+1), γ =⌊

α∆x1

⌋and β′ =

⌈α+a2∆t

∆x

⌉=⌈

α∆x + sν

⌉with s = a2

a1.

The first case, i ≤ γ, is not affected by the jump, so that the ordinary Lax-Wendroff

method with (xi−1, tn), (xi, t

n), (xi+1, tn) and x∗ = xi − a1∆t can be used with

un+1i = uni −

ν

2

(uni+1 − uni−1

)+ν2

2


).

The second case, γ < i < β′, was subdivided into the two cases where the intervals

[xγ−1, xγ ] and [xγ , xγ+1] were differentiated. This is no longer necessary when using

three data points for the interpolation as both intervals are covered by one interpolation

polynomial. Thus, choose (xγ−1, tn), (xγ , t

n) and (xγ+1, tn), while the point x∗ remains

x∗ =(

1− a1a2

)α+ a1

a2xi − a1∆t as before. The result is then

un+1i = unγ + (unγ+1 − unγ−1)

x∗ − xγ2∆x

+ (unγ+1 − 2unγ + unγ−1)(x∗ − xγ)2

2∆x2

= unγ +1

2

((1− a1

a2

)α

∆x+i

s− ν − γ

)(unγ+1 − unγ−1)

+1

2

((1− a1

a2

)α

∆x+i

s− ν − γ

)2

(unγ+1 − 2unγ + unγ−1).

In the third case, i ≥ β′, the higher velocity had to be considered, so the data points

(xi−dsνe, tn) and (xi−dsνe+1, t

n) and x∗ = xi− a2∆t = xi− sν∆x were used. Now adding

(xi−dsνe−1, tn) yields

un+1i = uni−dsνe + (uni−dsνe+1 − u

ni−dsνe−1)

x∗ − xi−dsνe2∆x

+ (uni−dsνe+1 − 2uni−dsνe + uni−dsνe−1)(x∗ − xi−dsνe)2

2∆x2

= uni−dsνe +1

2(dsνe − sν)(uni−dsνe+1 − u

ni−dsνe−1)

+1

2(dsνe − sν)2(uni−dsνe+1 − 2uni−dsνe + uni−dsνe−1).


Altogether, the thus obtained Lax-Wendroff method on a uniform grid is

un+1i =

uni − ν2

(uni+1 − uni−1

)+ ν2

2


), if i ≤ γ,

unγ + 12

((1− a1

a2

)α

∆x + is − ν − γ

)(unγ+1 − unγ−1)

+12

((1− a1

a2

)α

∆x + is − ν − γ

)2(unγ+1 − 2unγ + unγ−1), if γ < i < β′,

uni−dsνe + 12(dsνe − sν)(uni−dsνe+1 − u

ni−dsνe−1)

+12(dsνe − sν)2(uni−dsνe+1 − 2uni−dsνe + uni−dsνe−1), if i ≥ β′.

It was explained in section 1.3.3 that the Lax-Wendroff method for the constant coeffi-

cient advection equation generates oscillations if the initial condition is not sufficiently

smooth, therefore it is not TVD. The same holds then for the above methods since

they are essentially the same method, only on a different grid with different advection

velocities. For this reason, the methods will not be examined analytically but rather

computationally, which is the content of the next section.

2.3 Comparison and evaluation of the derived methods

The aim of this section is to gain a better understanding of the influence of the initial

condition and the jump in advection velocity on the numerical solution. Theoretically,

the Lax-Wendroff method is of higher order than the upwind method and should there-

fore yield better results. Practically, this might not be the case as soon as the solution

is no longer differentiable, which is nearly always the case at the jump in velocity. Bear

in mind that the exact solution of the advection equation is simply the initial condition

shifted along the spatial axis. If now a non constant part of the initial condition crosses

the jump, the difference in velocity results in the creation of a kink in the solution. An

example for this can be found in figure 2.2 of section 2.1.

Consider now three different situations.

• A sufficiently smooth initial condition without extrema where the jump creates a

minimal kink in the solution.

• A smooth initial condition with extrema where the jump creates a minimal kink

in the solution.

• An initial condition with extrema where the jump creates a much sharper kink in

the solution.


Because of such a kink in the solution, one would expect the order of the upwind and

the Lax-Wendroff method to reduce from the first to the third case. Furthermore, the

order of the Lax-Wendroff method should decrease more than that of the upwind method

because of the extrema.

To verify this assumption, examine an example for each situation stated above. The

Courant number is given as ν = 0.9, the velocity as a1 = 0.1 and a2 = 1, the spatial

interval is [0, 1000] and the temporal one [0, T ] with T =(αa1

+ 1000−αa2

)/2. The initial

conditions are then defined in the following way.

1. A part of the arctan, u0(x) = arctan (0.01(x− 100))+ π2 , with the jump at x = 800,

where the function appears nearly constant. This is the same example as was used

to illustrate the order of the upwind and Lax-Wendroff method without a jump in

section 1.3.1 and 1.3.3.

2. A sine wave, u0(x) = sin(

2πx1000

), with the jump at the maximum x = 250, so that

mostly the part with almost zero slope passes the jump.

3. The same sine wave, u0(x) = sin(

2πx1000

), with the jump at x = 500.

The solution to all three examples is depicted in figure 2.9. It is computed with both

the upwind and the Lax-Wendroff method on a uniform and a nonuniform grid. As a

boundary condition at x = 0, assume for simplicity that the initial conditions above are

valid for x ≤ 0 and that the corresponding value un0 is known at all times. Furthermore,

for the Lax-Wendroff method, compute the value at the right boundary with the upwind

method instead. The following figures show the logarithmic error plots with step size

10i/20 for i = −20, . . . , 20.

Consider first how the error is computed. It should foremost be noted that the computed

order of the method is the slope of the least squares approximation to the error values.

If the error values lie almost on a straight line, this slope is a good approximation to the

order. If there are greater fluctuations in the set of all error values, the approximation

might be much less exact. A small difference between the computed slopes therefore

does not necessarily mean that one method is of higher order than the other.

Another factor that influences the error values and the computed order is the definition

of the grid. In case the jump lies on a grid point, the grid is constructed around the

jump, that means that the jump is fixed as a grid point and the other grid points are

constructed to the left and right of it with the corresponding step size up to the boundary

of the spatial interval. This approach is generally not used in praxis. Usually, the first

grid point is the left boundary of the spatial interval and the others are defined according


(1) 0 200 400 600 800 1000

1

2

3

time t=0

0 200 400 600 800 1000

1

2

3

time t=100

0 200 400 600 800 1000

1

2

3

time t=300

(2) 0 200 400 600 800 10001

0

1

time t=0

0 200 400 600 800 10001

0

1

time t=30

0 200 400 600 800 10001

0

1

time t=60

(3) 0 200 400 600 800 10001

0

1

time t=0

0 200 400 600 800 10001

0

1

time t=10

0 200 400 600 800 10001

0

1

time t=100

Figure 2.9: Solution of the three examples at different times.

to the step size. Whether the jump then lies on a grid point or not depends on the step

size. One result of this construction of the grid is that the grid points do not always

coincide and the error is computed at different points of the spatial interval, which leads

to the observed fluctuations. The second result is that the ratio of the number of grid

points and the step size is not constant. Take the given interval [0, 1000] with step size

∆x = 100. If the jump were to lie at x = 500, there would be the eleven grid points


0, 100, . . . , 1000, whereas there would only be the ten grid points 50, 150, . . . , 950 if the

jump would lie at x = 550. Therefore, the number of grid points can vary, which is not

considered in the computation of the error and is another reason why the fluctuations

occur that can be seen in the figures. This effect is worse if the grid is constructed so

that the jump lies on a grid point because then not only the distance of the last grid

point to the boundary of the interval, but also the first grid point influences the number

of grid points. Figure 2.10 illustrates this for the first example by comparing the uniform

and nonuniform modified upwind method for different definitions of the grid.

1 0.5 0 0.5 1

3

2

1

0

1

example 1, uniform modified Upwind, jump on grid point


1 0.5 0 0.5 1

3

2

1

0

1

example 1, nonuniform modified Upwind, jump on grid point


1 0.5 0 0.5 1

3

2

1

0

1

example 1, uniform modified Upwind, jump between grid points


1 0.5 0 0.5 1

3

2

1

0

1

example 1, nonuniform modified Upwind, jump between grid points


Figure 2.10: Error plots for example 1 with the modified Upwind method.

Note that all of the methods can be adjusted to the more general case of an advection

equation with arbitrary, piecewise constant velocity. This would require an additional

distinction by cases for each jump and the grid could no longer be constructed so that all


jumps lie on a grid point. Furthermore, the case where the velocity decreases at a jump

would need to be examined as well. The approach is analogue, but somewhat easier,

since at most one characteristic through a grid point crosses the jump in a time step.

This fact is verified in figure 2.11. In the uniform case, the method has to be constructed

so that the border between bigger and smaller stencil lies on the grid point to the right

of the jump. For the nonuniform case, the ordinary upwind method can be used where

the grid has to be constructed as in the figure.

t n+1

t n x

t n+1

t n x

Figure 2.11: The grid points and corresponding characteristics on a uniform and nonuni-

form grid for a decreasing advection velocity.

Thus, there are two reasons why the methods for which the jump lies between grid points

should be preferred and are used solely in the following. First, the greater oscillations in

the error values and second, the fact that the grid has to be constructed independently

of the jump to generalize the method for more than one jump.

Consider now the error plots in the three figures 2.12, 2.13 and 2.14. It should be noted

that the slope for the upwind method has approximately the value one in all cases al-

though a kink arises in the solution due to the jump. For the example in figure 1.10,

the discontinuity in the initial condition led to the slope of the upwind method falling

to 0.6. The only difference between these two situations is that the kink is now only

a discontinuity in the derivative, whereas it was a discontinuity in the solution itself in

the other example. This leads to the assumption that the upwind method yields good

results as long as there are no discontinuities in the initial condition.

Observe as well that for a nonuniform grid, there is no difference between the upwind

and the modified upwind method as mentioned when deriving the two. On a uniform

grid on the other hand, the modified method yields much better results since the order

is slightly higher, the error for each step size is smaller and there are less oscillations in

the error values. With the solution for example 1 being nearly smooth, this is hardly

noticeable but for example 2 and 3 it can be seen clearly. The reason lies mostly in the


1 0.5 0 0.5 1

3

2

1

0

1

example 1, uniform Upwind


1 0.5 0 0.5 1

3

2

1

0

1

example 1, nonuniform Upwind


1 0.5 0 0.5 1

3

2

1

0

1

example 1, uniform modified Upwind


1 0.5 0 0.5 1

3

2

1

0

1

example 1, nonuniform modified Upwind


1 0.5 0 0.5 1

3

2

1

0

1

example 1, uniform modfied Lax Wendroff


1 0.5 0 0.5 1

3

2

1

0

1

example 1, nonuniform modified Lax Wendroff


Figure 2.12: Error plots for example 1.


1 0.5 0 0.5 1

2

1

0

1



1 0.5 0 0.5 1

2

1

0

1



1 0.5 0 0.5 1

2

1

0

1



1 0.5 0 0.5 1

2

1

0

1



1 0.5 0 0.5 1

2

1

0

1

example 2, uniform modified Lax Wendroff


1 0.5 0 0.5 1

2

1

0

1





1 0.5 0 0.5 1

2

1

0

1



1 0.5 0 0.5 1

2

1

0

1



1 0.5 0 0.5 1

2

1

0

1



1 0.5 0 0.5 1

2

1

0

1



1 0.5 0 0.5 1

2

1

0

1

example 3, uniform modified Lax Wendroff


1 0.5 0 0.5 1

2

1

0

1





somewhat lesser accuracy around the jump as depicted in figure 2.7, which corresponds

to the third example.

The modified Lax-Wendroff method, while still of higher order than the ordinary and

modified upwind method, reaches at most the order 1.355 in example 1 where the jump

resulted in only a small kink in the solution. For this example, the difference between the

uniform and nonuniform Lax-Wendroff method is rather conspicuous as it is too high to

be a result of the fluctuations in the error terms. In fact, the nonuniform methods yield

in all cases nearly the same or a smaller slope than the corresponding uniform methods.

The reason for this is the bigger step size ∆x2, the solution to the right of the jump is

computed with a bigger step size and is less accurate. While the error is computed by

weighting the error of one point in time and space with the corresponding step size so

that the difference in ∆x is considered, the lesser accuracy of the solution is neglected.

For this reason, the error is additionally smaller for the uniform methods, even if the

slope is nearly the same. An example for this are the solutions of the upwind method in

figures 2.13 and 2.14 where the jump lies between grid points.

The only advantage of the nonuniform method over the uniform one is that the bigger

step size needs fewer computations and is computationally less expensive. The nonuni-

form method should be preferred in a situation where the initial condition is sufficiently

smooth, so that the lesser accuracy on the area with bigger step size does not result in

a much higher error.

Regarding the choice between upwind and Lax-Wendroff method, consider the advan-

tage of the higher order against the disadvantage of the higher computational complexity.

Compare the error plots for the uniform Lax-Wendroff method for the three examples.

The sharper the kink that was created by the jump, the lesser became the order of the

method. Since there are very few cases where the initial condition would be shaped so

that only a small or no kink would arise with time, the upwind methods seems to be

better suited for this situation.

3 Nonlinear partial differential equations

This chapter is based on the theory that was presented in section 1.2 and elaborates on

the nonlinear case as described in [11] and [12]. Until now, only the advection equation

and numerical methods for its solution were examined. Use the acquired results now to

study the solution of other, nonlinear equations and derive corresponding methods.

3.1 Analysis of nonlinear equations

Consider the conservation law

∂

∂tu(x, t) +

∂

∂xf(u(x, t)) = 0 (3.1)

where f(u) is now a nonlinear function of u. The so-called quasilinear form of this

equation is given by differentiating f , so that

∂

∂tu(x, t) + f ′(u(x, t))

∂

∂xu(x, t) = 0

if f is smooth. Use this form to get a better idea of the solution of the equation. As

was done for the advection equation in section 1.1.1, determine the characteristics X(t)

of the equation under the assumption that u(x, t) is smooth. It holds

d

dtu(X(t), t) = ut(X(t), t) +X ′(t)ux(X(t), t) = 0

⇔ X ′(t) = f ′ (u(X(t), t)) ,

so that u(x, t) is constant along the characteristics. Furthermore, because X ′(t) =

f ′ (u(X(t), t)) = f ′(const) = const, X(t) is a straight line. While this agrees with the

properties of the advection equation, the crucial difference lies in the slope of the char-

acteristics. For the advection equation, all characteristics had slope a. For an arbitrary

equation they have a constant slope f ′(u(X(t), t)) which is different for each character-

istic though. For this reason, the characteristics may cross each other over time, so that

shock formations arise and the solution can no longer be traced by the characteristics

as was done for the advection equation. This will later be examined further.

55

56 3 Nonlinear partial differential equations

At first, note that the solution can still be determined analytically under certain assump-

tions. Consider for this a smooth initial condition u(x, 0) = u0(x) and a sufficiently small

time interval [0, T ] in which the characteristics do not collide and the solution remains

smooth. The characteristic X(t) through a point (x, 0) has the slope

f ′(u(X(t), t)) = f ′(u(x, 0)) = f ′(u0(x)),

so that the characteristic itself is given by

X(t) = x+ f ′(u0(x))t.

For the solution this means that

u(x, t) = u0(x) with x = x+ f ′(u0(x))t (3.2)

for every (x, t) with t ∈ [0, T ]. If T is chosen so that the characteristics do cross, then

(3.2) has no unique solution and the solution of the differential equation can no longer

be computed by means of the characteristics.

There are basically three ways in which the initial condition can change over time. The

first possibility is that the characteristics do not cross on a certain area, but rather

fan out. This results in a so-called rarefaction wave, that means that the value of the

solution in the area is rarefied. The other possibility is the exact opposite where the

value is compressed by the characteristics converging, which generates a compression

wave. The merging or crossing of the characteristics creates shock waves across which

the solution becomes discontinuous.

The problems that therefore arise when dealing with nonlinear equations, is that there

is possibly more than one solution and that these solutions are no longer smooth, so

that they do not fulfill the differential equation at discontinuities. This motivates the

definition of weak solutions, which only fulfill the integral form of the conservation law

and not the differential one. Under the assumption that the solution u is smooth, start

with multiplying the differential equation with a test function φ(x, t) ∈ C10 , which is a

continuous function with compact support, and integrate over time and space, so that∫ ∞0

∫ ∞−∞

(ut + f(u)x)φ(x, t) dx dt = 0.

Now use integration by parts to transfer the derivatives to φ

0 =

∫ ∞0

∫ ∞−∞

(ut + f(u)x)φ(x, t) dx dt

= −∫ ∞

0

∫ ∞−∞

(uφt + f(u)φx) dx dt+

∫ ∞−∞

[uφ]∞0 dx+

∫ ∞0

[f(u)φ]∞−∞dt.


Since φ has compact support, it is zero at infinity, so it holds∫ ∞0

∫ ∞−∞

(uφt + f(u)φx) dx dt = −∫ ∞−∞

u(x, 0)φ(x, 0) dx.

The function u that satisfies this equation is then called a weak solution of the differential

equation.

This weak solution can be used to determine the speed s with which a shock propagates

with time. It can be computed by examining the Riemann problem on a sufficiently

small interval around the shock, so that the initial condition for the problem is given by

ul and ur where ul is the value to the left of the shock and ur the one to the right. The

Rankine-Hugoniot condition gives the shock speed as

s =f(ul)− f(ur)

ul − ur. (3.3)

The problem with the concept of weak solutions is that they are not necessarily unique

while one would assume that the differential equation describes a physical situation that

is unique. To check which of the possible weak solutions is the right one in the sense of

its physical meaning, consider different so-called entropy conditions. One example is the

entropy condition by Lax, which states that for convex f , the shock speed must fulfill

f ′(ul) > s > f ′(ur).

There exist several more approaches to finding weak solutions and proving that they are

physically admissible; they can for example be found in [12] and will not be elaborated

here further.

3.1.1 Burgers equation

Exemplify the above theory by means of the inviscid Burgers equation

ut +

(1

2u2

)x

= 0

with f(u) = 12u

2. It can be written in the quasilinear form

ut + uux = 0

and has the characteristics

X(t) = x+ u0(x)t.


Therefore, if u0 is monotonically increasing, the characteristics fan out and never cross.

In this case, the solution of the equation is unique at all times and is given implicitly by

u(x, t) = u0(x− u0(x)t).

Figure 3.1(a) shows this case for the example of u0(x) = arctan(x). On the other hand, if

u0 is monotonically decreasing on an arbitrary small interval, the characteristics at this

interval cross at some point and only a weak solution of the equation can be found. Such

a situation can be illustrated by considering u0(x) = − arctan(x) as in figure 3.1(b).

(a) (b)

Figure 3.1: The initial condition and some corresponding characteristics for

(a) u0(x) = arctan(x) and (b) u0(x) = − arctan(x).

Both examples will be used later on in this chapter to examine the numerical method

which will be introduced in the following sections.

3.2 Numerical methods for nonlinear equations

The derivation of numerical methods for nonlinear differential equations is more difficult

than for the linear case, because in most cases, there exists no unique solution and the

method should converge to the right weak solution.

A standard example where this is not the case is given in [11] as the Riemann problem

u0i =

1, for i < 0,

0, for i ≥ 0


for Burger’s equation. Adjusting the upwind method of section 1.3.1 to the nonlinear

case yields

un+1i = uni −

∆t

∆xuni (uni − uni−1),

which results in uni = u0i for all i and n, so that the solution converges to u0, which is

no weak solution.

A numerical method for a nonlinear equation should hence have the properties described

in section 1.2.1 and compute the right weak solution, so it should fulfill some entropy

condition. Note that the CFL condition is still a necessary condition for stability, but it

is now given by ∣∣∣∣∆t

∆xf ′(u)

∣∣∣∣ ≤ 1 for the solution u of (3.1), (3.4)

since information propagates with the slope of the characteristics, which is f ′.

For systems of equations, f ′(u) is replaced by the biggest eigenvalue λmax of the Jacobian

f ′. The reason for this is that the equation

ut + f ′(u)ux = 0

can be written as

vpt + λpvpx = 0, for p = 1 . . .m

with f ′ = R diag(λ1, . . . , λm)R−1 and R−1u =: v as described in section 1.1.1.

Since f ′ depends on u and is not necessarily monotone, the value un+1i can either depend

on a value to the right or to the left. For this reason, an upwind method as used up

to now does not seem to be the best approach. The next section introduces a better

alternative for this situation.

3.2.1 The Lax-Friedrichs method

According to the above explanation, the stencil of a method for solving nonlinear equa-

tions should be centered. Under this condition, the easiest choice is to use the two

points directly to the left and right. Thus, the Lax-Friedrichs method is derived by first

choosing the numerical flux as

Fni−1/2 =1

2(f(uni−1) + f(uni ))


and then replacing uni by the average 12(uni+1 + uni−1), so that

un+1i =

1

2(uni+1 + uni−1)− ∆t

2∆x(f(uni ) + f(uni+1)− f(uni−1)− f(uni ))

=1

2(uni+1 + uni−1)− ∆t

2∆x(f(uni+1)− f(uni−1)) (3.5)

and the flux is then given by

Fni−1/2 = F(uni−1, uni ) =

1

2

(f(uni−1) + f(uni )

)− ∆x

2∆t(uni − uni−1).

In order to examine the order of the method, consider again the local truncation error for

the special case of the advection equation, as was done for the upwind and Lax-Wendroff

method in chapter 1. For the Lax-Friedrichs method, the error is defined as

L(xi, tn) =1

∆t

[1

2(u(xi+1, tn) + u(xi−1, tn))

− ∆t

2∆x(au(xi+1, tn)− au(xi−1, tn))− u(xi, tn+1)

].

Apply Taylor series expansion to replace u(xi+1, tn), u(xi−1, tn) and u(xi, tn+1) by

u(xi±1, tn) = u(xi, tn)±∆xux(xi, tn) +1

2∆x2uxx(xi, tn) +O(∆x3),


2∆t2utt(xi, tn) +O(∆t3)

in the local truncation error so that

L(xi, tn) =1

∆t

[1

2

(2u(xi, tn) + ∆x2uxx(xi, tn) + 2O(∆x3)

)− a∆t

2∆x(2∆xux(xi, tn))

−u(xi, tn)−∆t ut(xi, tn)− 1


]=

1

∆t

[1

2∆x2uxx(xi, tn) +O(∆x3)− a∆t ux(xi, tn)



]=

∆x2

2∆tuxx(xi, tn) +

1

∆tO(∆x3)− aux(xi, tn)

− ut(xi, tn)− 1

2∆t utt(xi, tn)−O(∆t2).

Now, use that

ut = −aux ⇒ utt = a2uxx


for the advection equation so that the first order derivatives cancel each other out. Then,

the error has the form

L(xi, tn) =∆x2

2∆tuxx(xi, tn)− 1

2a2∆t uxx(xi, tn) +O(∆x2,∆t2)

=1

2∆t

(∆x2

∆t2− a2

)uxx(xi, tn) +O(∆x2,∆t2).

At this point, the Lax-Friedrichs method differs from the upwind and Lax-Wendroff

method that where of first and second order respectively. A reduction of either the

spatial or time step size resulted in a more accurate solution as long as the step sizes

fulfilled the CFL condition. For the Lax-Friedrichs method, this is no longer the case.

If the spatial step size is constant,

lim∆t→0

∆t

(∆x2

∆t2− a2

)= lim

∆t→0

(∆x2

∆t−∆ta2

)=∞

and the method diverges for decreasing ∆t. Therefore, the relation between ∆t and ∆x

has to be fixed so that ∆x∆t is constant. In that case, the above limit changes to

lim∆t→0

∆t

(∆x2

∆t2− a2

)= lim

∆t→0C∆t = 0

for some constant C and the method converges with first order accuracy. The step

size ∆t that results in the most accurate solution can be determined by minimizing

∆t(

∆x2

∆t2− a2

). The CFL condition states that ∆t ≤ ∆x

a so that

∆x2

∆t−∆ta2 ≥ ∆x2

∆xa

− ∆x

aa2 = 0.

Therefore, the time step size should be chosen as big as the CFL condition allows.

As a further property, the Lax-Friedrichs method is TVD, which can be proved as follows.

TV (un+1) =∑i∈Z

∣∣un+1i − un+1

i−1

∣∣=∑i∈Z

∣∣∣∣12(uni+1 + uni−1)− ∆t

2∆x(f(uni+1)− f(uni−1))

−1

2(uni + uni−2) +

∆t

2∆x(f(uni )− f(uni−2))

∣∣∣∣=∑i∈Z

∣∣∣∣12(uni+1 − uni −

∆t

∆x(f(uni+1)− f(uni ))

+uni−1 − uni−2 +∆t

∆x(f(uni−1)− f(uni−2))

)∣∣∣∣ .


The mean value theorem states

f(uni+1)− f(uni ) = f ′(ξi+1/2)(uni+1 − uni ), for ξi+1/2 ∈ [uni , uni+1],

so that

TV (un+1) =∑i∈Z

∣∣∣∣(1

2−

∆tf ′(ξi+1/2)

2∆x

)(uni+1 − uni )

+

(1

2+

∆tf ′(ξi−2+1/2)

2∆x

)(uni−1 − uni−2)

∣∣∣∣ .Because of the CFL condition (3.4),∣∣∣∣∆tf ′(ξj+1/2)

2∆x

∣∣∣∣ ≤ 1

2for arbitrary j ∈ Z

and therefore

TV (un+1) ≤∑i∈Z

(1

2−

∆tf ′(ξi+1/2)

2∆x

) ∣∣uni+1 − uni∣∣

+∑i∈Z

(1

2+

∆tf ′(ξi−2+1/2)

2∆x

) ∣∣uni−1 − uni−2

∣∣=∑i∈Z

(1

2−

∆tf ′(ξi−1+1/2)

2∆x

) ∣∣uni − uni−1

∣∣+∑i∈Z

(1

2+

∆tf ′(ξi−1+1/2)

2∆x

) ∣∣uni − uni−1

∣∣=∑i∈Z


∣∣ = TV (un).

Consider again the example of Burgers equation with the initial condition u0(x) =

± arctan(x). Figure 3.1 suggests that for u0(x) = arctan(x) a rarefaction wave occurs.

This is indeed the case as figure 3.2 shows.

The initial condition u0(x) = − arctan(x) should, according to the characteristics, result

in a compression wave that becomes a shock with time. This shock then propagates

with the speed s given by the Rankine-Hugoniot condition in (3.3), so that

s =f(ul)− f(ur)

ul − ur=

12u

2l −

12u

2r

ul − ur=

1

2(ul + ur) = 0

since ul = −ur for the initial condition. The solution should therefore converge to

u(x, t) =

π2 , for x < 0,

−π2 , for x > 0,


0100 1002

0

2

0 1001002

0

2

0 1001002

0

2

Figure 3.2: The solution to Burgers equation with initial condition u0(x) = arctan(x) at

different time steps.

0 1001002

0

2

0 1001002

0

2

0 1001002

0

2

Figure 3.3: The solution to Burgers equation with initial condition u0(x) = − arctan(x)

at different time steps.

which is displayed in figure 3.3.

Analogously, if the initial condition where of the form u0(x) = − arctan(x) + c for some

constant c, the shock would propagate with speed c. A problem that should be noted

is that in cases like this, the step size ∆x has to be sufficiently small. Take for example

the initial condition on the interval [−100, 100] with step size ∆x = 10. The result of

the low accuracy is that the solution after one time step is too different from the real

solution and thus evolves differently.


3.3 Nonlinear equations with spatially varying, discontinuous

flux function

Extend the analysis now to equations like (3.1) with the addition of a discontinuous,

space dependent coefficient

a(x) =

a1, if x ≤ α,

a2, if x ≥ α

in the flux where it is assumed that 0 < a1 ≤ a2. The differential equation then has the

form

∂

∂tu(x, t) + a(x)

∂

∂xf(u(x, t)) = 0, (3.6)

or alternatively the quasilinear form

∂

∂tu(x, t) + a(x)f ′(u(x, t))

∂

∂xu(x, t) = 0.

Such equations are difficult to handle since some of the existing theory for nonlinear

equations does not apply. As a matter of fact, this situation is not yet completely

understood and there are many unanswered questions. For example, it is not possible to

use the concept of entropy solutions and thus to guarantee convergence to the right weak

solution as before. For this reason, the theoretical aspects are from now on neglected.

However, it is still possible to compute the characteristics as in the autonomous case in

section 3.1, as long as they do not cross the jump where they kink. The slope is then

given as

X ′(t) = a(X(t))f ′ (u(X(t), t)) ,

so that the characteristic through the point (x, 0) is

X(t) = x+ a(x)f ′(u0(x))t

for x ≤ α and X(t) ≤ α so that a(x) = a1, or for x ≥ α and X(t) ≥ α so that a(x) = a2.

If the characteristic does cross the jump, distinguish the two cases x ≤ α and x ≥ α. For

x ≤ α, the first part of the characteristic has a slope with the factor a1 and the second

part with a2. Therefore

X(t) = x+ a1f′(u0(x))t∗ + a2f

′(u0(x))(t− t∗)


with

x+ a1f′(u0(x))t∗ = α ⇔ t∗ =

α− xa1f ′(u0(x))

.

The case x ≥ α is the same for interchanged values of a, that is

X(t) = x+ a2f′(u0(x))t∗ + a1f

′(u0(x))(t− t∗)

with

t∗ =α− x

a2f ′(u0(x)).

For the advection equation, the change of the characteristics had a great impact on the

solution of the equation in the sense that it contained a kink that was not part of the

initial condition. For the nonlinear case, the kink in the characteristics creates such a

kink in the solution as well, but this is of less importance now if a problem where a

shock is generated by the crossing of characteristics is considered. Figure 3.4 shows the

same example of figure 3.1 with the addition of the coefficient a(x). For example (a), the

kink in the characteristics creates a kink in the solution while the solution would remain

smooth otherwise. In example (b), the jump does not make a big difference, apart from

the fact that the characteristics converge sooner.

The influence of the factor a on the shape of the characteristics has to be considered for

the CFL condition as well. Therefore, take the CFL condition (3.4) and add the factor

a, yielding ∣∣∣∣∆t

∆xaf ′(u)

∣∣∣∣ ≤ 1 for the solution u of (3.6). (3.7)

How to choose the time and spatial grid specifically depends on the choice between a

uniform and nonuniform grid as in chapter 2. The adjustment of the Lax-Friedrichs

method for these two cases is the content of the next section. Note that the approach

of modifying the method by following the characteristics, as was done in section 2.2, is

now no longer possible, because the slope of the characteristic through a point (xi, tn+1)

is no longer known.

3.3.1 The Lax-Friedrichs method on a nonuniform grid

Use the results of section 2.2.1 to derive the Lax-Friedrichs method on a nonuniform

grid. Define a piecewise constant step size

∆x =

∆x1, if x ≤ α,

∆x2 = a2a1

∆x1, if x ≥ α.


(a) (b)

Figure 3.4: The initial condition and some corresponding characteristics for the modified

Burger’s equation ut + a(x)uux = 0 with a1 = 0.5 and a2 = 1 and the initial

condition (a) u0(x) = arctan(x) and (b) u0(x) = − arctan(x). The vertical

line marks the jump across which the characteristics kink.

The CFL condition then determines ∆t with∣∣∣∣ ∆t

∆xiai max f ′

∣∣∣∣ =∆t

∆xiai max

∣∣f ′∣∣ =: ν ≤ 1 for i = 1, 2,

so that

∆t =ν∆x1

a1 max |f ′|=

ν∆x2

a2 max |f ′|.

Now one can simply use the Lax-Friedrichs method (3.5) with the addition of the coef-

ficient a(x). Denote again γ =⌊

α∆x1

⌋with α ∈ [xγ , xγ+1) the grid point where the step

size changes. To the left of the jump with x ≤ xγ−1,

un+1i =

1

2(uni+1 + uni−1)− a1∆t

2∆x1(f(uni+1)− f(uni−1)).

Analogously, to the right with x ≥ xγ+1, it holds that

un+1i =

1

2(uni+1 + uni−1)− a2∆t

2∆x2(f(uni+1)− f(uni−1)).


At the border between the step sizes for xi = xγ , the points xi−1 and xi+1 have a different

distance to xi, so that

un+1γ =

1

2(unγ+1 + unγ−1)− a1∆t

2∆x1f(unγ−1) +

a2∆t

2∆x2f(unγ+1)

=1

2(unγ+1 + unγ−1)− a1∆t

2∆x1

(f(unγ−1)− f(unγ+1)

).

For µ = a1∆t∆x1

= a2∆t∆x2

, the Lax-Friedrichs method is then given as

un+1i =

1

2(uni+1 + uni−1)− µ

2(f(uni+1)− f(uni−1)).

Note that there are two possible situations in which this method is less accurate, because

the wrong value for a was chosen. First, consider the grid point xγ . If the slope of the

characteristic through (xγ , tn+1) is negative, the characteristic crosses the jump in the

time interval [tn, tn+1] and the factor a2 should be replaced with the average of a on the

interval [xγ , xγ+1]. Analogously, if the slope of the characteristic through (xγ+1, tn+1)

is positive, the characteristic crosses the jump as well and the same effect arises. By

ignoring this, the method becomes much simpler, as was the case for the upwind method

for the advection equation.

To get an idea whether the computed solution is sensible, consider again Burger’s equa-

tion with the addition of the piecewise constant coefficient a

ut + a(x)uux = 0.

The nonuniform Lax-Friedrichs method is then

un+1i =

1

2(uni+1 + uni−1)− µ

4((uni+1)2 − (uni−1)2).

Figure 3.5 shows the computed solution of this equation at different times with the

initial condition u0(x) = arctan(x − 100) and a(x) =

0.1, if x ≤ α,

1, if x ≥ α,with the jump

at α = 100. This example is chosen because it yields a somewhat nice solution in the

sense that it contains none of the problems that arise for nonlinear equations. Because

u0(α) = 0, the characteristics do not cross the jump and because the initial condition is

strictly monotonic increasing, the characteristics fan out and do not coincide. A similar

situation was illustrated in figure 3.4(a). As expected, the part to the left of the jump

propagates further left and the part to the right further right. Because of the different

values of a, this happens much faster at the right, so that a kink in the solution arises

at the position of the jump.


0 100 2002

0

2

0 100 2002

0

2

0 100 2002

0

2

Figure 3.5: Solution of the nonuniform Lax-Friedrichs method.

3.3.2 The Lax-Friedrichs method on a uniform grid

After successfully constructing a method for a nonuniform grid, do the same for a uni-

form grid. As for the upwind method in section 2.2.2, this is more difficult than on

a nonuniform grid. The reason for this is that the characteristics to the right of the

jump pass more than one spatial interval in a time step because of the higher velocity

a2. For the upwind method, the result was that all the values un+1i for γ < i < β were

computed by means of the value unγ−1, which generated small discontinuities in the form

of a stair-step like pattern. It will be seen that the same happens for the Lax-Friedrichs

method on a uniform grid.

Start with defining the uniform grid. Like before, choose the time step size for given

spatial step size according to the CFL condition for the smaller value of a. That is for

fixed ∆x, set

∆t =ν∆x

a1 max |f ′|.

The solution to the left of the jump can then be computed with the ordinary Lax-

Friedrichs method while the solution to the right requires a broader stencil. The approach

to deriving the method is the same as for the upwind method, that is fix the method as

un+1i =

1

2(unj + unk)− µ

2(f(unj )− f(unk))

for µ = a1∆t∆x and determine the indices j and k accordingly. Note that as described

in section 2.2.2, defining the method with constant coefficient µ is a simplification if

s is no integer and thus replaced by⌈a2a1

⌉. For γ =

⌊α

∆x

⌋with α ∈ [xγ , xγ+1) and

β = γ + a2a1

= γ + s, distinguish the cases i < γ, γ ≤ i < β and i ≥ β.


As mentioned before, for i < γ, it suffices to define j = i+ 1 and k = i− 1, which is the

stencil of the ordinary Lax-Friedrichs method, so that

un+1i =

1

2(uni+1 + uni−1)− µ

2(f(uni+1)− f(uni−1)).

Unlike for the upwind method, consider i < γ instead of i ≤ γ because the stencil of the

Lax-Friedrichs method at the point xγ contains the jump.

For i ≥ β, the factor a2 is s times bigger than a1, so that the stencil should be s times

bigger as well. Therefore, take j = i+ s and k = i− s, which yields

un+1i =

1

2(uni+s + uni−s)−

µ

2(f(uni+s)− f(uni−s)).

The case γ ≤ i < β is more difficult. Note that according to the two cases before,

un+1γ−1 =

1

2(unγ + unγ−2)− µ

2(f(unγ )− f(unγ−2)) ⇒ j = γ, k = γ − 2,

un+1β

=1

2(unβ+s + unβ−s)−

µ

2(f(unβ+s)− f(unβ−s)) ⇒ j = γ + 2s, k = γ,

so that it should hold that j ∈ [γ, γ + 2s] and k ∈ [γ − 2, γ]. This is indeed reason-

able, because the characteristics through the points (xi, tn+1) reach at most the interval

(xγ−1, xγ ] at time tn. The reason for this is the same as for the uniform upwind method,

since the different slope f ′ is already included in the step size ∆t. Furthermore, for nega-

tive f ′, the characteristics through (xi, tn+1) cross at most s spatial intervals. Together,

the stencil should be chosen so that k = γ − 1 and j = i+ s, which gives

un+1i =

1

2(uni+s + unγ−1)− µ

2(f(uni+s)− f(unγ−1)).

Altogether, the uniform Lax-Friedrichs method has the form

un+1i =

12(uni+1 + uni−1)− µ

2 (f(uni+1)− f(uni−1)), if i < γ,


2 (f(uni+s)− f(unγ−1)), if γ ≤ i < β,


µ2 (f(uni+s)− f(uni−s)), if i ≥ β.

The fact that the solution is computed by means of unγ−1 for all γ ≤ i < β results in the

already announced discontinuities, that were observed for the uniform upwind method.

To illustrate this, consider again the example of the previous section. For the differential

equation

ut + a(x)uux = 0


the method is given as

un+1i =

12(uni+1 + uni−1)− µ

4 ((uni+1)2 − (uni−1)2), if i < γ,


4 ((uni+s)2 − (unγ−1)2), if γ ≤ i < β,


µ4 ((uni+s)

2 − (uni−s)2), if i ≥ β.

Examining the problem for u0(x) = arctan(x − 100) and a(x) =

0.1, if x ≤ α,

1, if x ≥ α,with

the jump at α = 100 yields the solution that is presented in figure 3.6.

0 100 2002

0

2

0 2001002

0

2

0 100 2002

0

2

Figure 3.6: Solution of the uniform Lax-Friedrichs method.

As for the nonuniform method, the computed solution has the expected form. Addition-

ally, a stair-step like pattern arises directly to the right of the jump, which is illustrated

in figure 3.7 and could already be observed for the upwind method in chapter 2.

95 100 1250.25

1.25

10095 1250.25

1.25

10095 1250.25

1.25

Figure 3.7: Detail of the solution of the uniform Lax-Friedrichs method at the right of

the jump.


The method on the uniform grid is at a disadvantage, because it requires additional

boundary conditions. The method on the nonuniform grid can be used for the solution

at all grid points except for the first and the last one. On the uniform grid, it is not

possible to compute the solution at the last s grid points, since it would require the

value of the solution outside the spatial interval. There exist different approaches to

defining boundary conditions, some can for example be found in [12], but they will not

be elaborated on here.

To simplify matters, the values at points outside of the interval are replaced by the value

at the right boundary. Whether this definition of the method yields acceptable results

depends on the shape of the solution at all times. Consider again the example of the

arctan up to the time step, that is illustrated in the third part of figure 3.6. At all times,

the straight line π2 is an asymptote of the solution. Therefore, the value at the right

boundary is approximately the same as the values further to the right, if the spatial

interval were longer and they were computed as well. For this reason, the approach can

be used in this situation for both the uniform and the nonuniform method.

3.3.3 Flux functions with arbitrary, piecewise constant coefficient

Up to now, only equations with one jump were considered, where the jump separated

an area with smaller coefficient a1 from an area with bigger coefficient a2. Expand this

now to the more general case with the coefficients given as

a(x) =

a1, if x ≤ α1,

a2, if α1 ≤ x ≤ α2,

a3, if x ≥ α2

for a1 ≤ a2 and a2 ≥ a3. Since this example contains both the transition from smaller

to bigger and from bigger to smaller coefficient, the results of this section can be used

in all situations with arbitrary, piecewise constant coefficient.

Start with the Lax-Friedrichs method on a nonuniform grid. Here, the addition of the

second jump does not require a change of the method, as figure 3.8 suggests. The domain

of dependence of the PDE at a grid point contains all possible characteristics through

this grid point and the CFL condition specifies the step size. Together, this guarantees

that the domain of dependence of a point (xi, tn+1) is a subset of the interval [xi−1, xi+1],

so that no distinction of cases is necessary. For this reason, the Lax-Friedrichs method

un+1i =

1

2(uni+1 + uni−1)− µ

2(f(uni+1)− f(uni−1))


with µ = a1∆t∆x1

for

∆x =

∆x1, if x ≤ α1,

∆x2 = a2a1

∆x1, if α1 ≤ x ≤ α2,

∆x3 = a3a1

∆x1, if x ≥ α2,

and ∆t =ν∆x1

a1 max |f ′|

can be used as before.

t n+1

t n xα 1 α 2

Figure 3.8: The nonuniform grid for two jumps and the domain of dependence at different

grid points.

Note that in this situation, the step size ∆x1 and thus ∆x2 should be chosen small

enough in relation to the distance between the jumps. Otherwise, there exists only one

interval of size ∆x2. This is illustrated in figure 3.9. As can be seen, a part of the

bigger interval contains no characteristics. Therefore, it would be more accurate to use

a smaller ∆x2 and to add more grid points as the dashed lines in the figure suggest.

Since this is difficult to implement, the step size has to be chosen so that the area that

contains no domain of dependence is relatively small.

t n+1

t n xα 1 α 2

Figure 3.9: The example of figure 3.8 if the distance between the jumps is small in

relation to the step size.

Examine the method for a uniform grid next. In contrast to the nonuniform case, this is


much more complicated. The situation for the first jump at α1 is the same as in section

3.3.2. For the second jump at α2, observe in figure 3.10 that principally the same holds,

only reversed. Assume that a1 ≤ a3 ≤ a2 and define the step size according to a1, so

x

t n+1

t n

α 1 α 2

Figure 3.10: The uniform grid for two jumps and the domain of dependence at different

grid points.

that for given ∆x,

∆t =ν∆x

a1 max |f ′|.

Then define s2 = a2a1

and s3 = a3a1

as the factors by which the coefficients a2 and a3

are bigger than a1 and assume further that they are integer or replace them by the

next bigger integer. Keep in mind that si describes the maximal number of spatial

intervals that a characteristic passes in a time step, so one could as well define s1 = 1.

Furthermore, let γ1 =⌊α1∆x

⌋and γ2 =

⌈α2∆x

⌉be the grid points to the left of α1 and to

the right of α2 respectively. Lastly, set β1 = γ1 +s2, β2 = γ2−s2 and β3 = γ2 +s3. Now,

consider all the possible cases. The first three were already examined in the previous

section, so only the notation needs to be adjusted.

• i < γ1: un+1i = 1

2(uni+s1 + uni−s1)− µ2 (f(uni+s1)− f(uni−s1))

• γ1 ≤ i < β1: un+1i = 1

2(uni+s2 + unγ1−s1)− µ2 (f(uni+s2)− f(unγ1−s1))

• β1 ≤ i ≤ β2: un+1i = 1


The fourth case corresponds to the second when interchanging the left and right side

and correcting the indices.

• β2 < i ≤ γ2: un+1i = 1

2(unγ2+s3 + uni−s2)− µ2 (f(unγ2+s3)− f(uni−s2))

The fifth case is similar, with the only difference that a characteristic with negative slope

is not bounded by xγ2+s3 .


• γ2 < i < β3: un+1i = 1


For the last case, like the first and third, the corresponding stencil does not contain a

jump, so the stencil is given by s3.

• β3 ≤ i: un+1i = 1


Altogether, define the method as

un+1i =

12(uni+s1 + uni−s1)− µ

2 (f(uni+s1)− f(uni−s1)), if i < γ1,

12(uni+s2 + unγ1−s1)− µ

2 (f(uni+s2)− f(unγ1−s1)), if γ1 ≤ i < β1,


2 (f(uni+s2)− f(uni−s2)), if β1 ≤ i ≤ β2,

12(unγ2+s3 + uni−s2)− µ

2 (f(unγ2+s3)− f(uni−s2)), if β2 < i ≤ γ2,


2 (f(uni+s3)− f(uni−s2)), if γ2 < i < β3,


2 (f(uni+s3)− f(uni−s3)), if β3 ≤ i.

Obviously, this can be quite confusing, which only worsens for more jumps. On the other

hand, the uniform grid yields a more accurate solution. If for example a2 a1, a3, the

nonuniform method would define a much bigger step size between the two jumps, which

could yield a solution that is too inexact in this area.

As was the case for the nonuniform method, the uniform method is based on the as-

sumption that the distance between the jumps is relatively big, so that γ1 ≤ β1 ≤ β2 ≤γ2 ≤ β3. If the distance is smaller, a characteristic could pass both jumps in one time

step as depicted in figure 3.11. In this case, the above scheme is no longer valid. An

x

t n+1

t n

α 1 α 2

Figure 3.11: The example of figure 3.10 if the distance between the jumps is small in

relation to the step size.


alternative scheme could be

un+1i =


2 (f(uni+s1)− f(uni−s1)), if i < γ1,

12(unγ2+s3 + unγ1−s1)− µ

2 (f(unγ2+s3)− f(unγ1−s1)), if γ1 ≤ i < β3,


2 (f(uni+s3)− f(uni−s3)), if β3 ≤ i.

For this distinction between the possible situations to not be necessary, the step size

should be chosen small enough as required for the nonuniform method.

In the next chapter, the two modified versions of the Lax-Friedrichs method will be used

for the computation of the solution of some test cases, including the calculation of the

dose in radiotherapy.

4 Applications in radiotherapy

4.1 A model for dose calculation

In this chapter, the derived methods will be applied to the radiation transport problem.

In [2] and [5], the particle transport is described by

∂ε(SM (x, ε)Ψ0(x, ε)

)−∇xΨ1(x, ε) = 0,

∂ε(SM (x, ε)Ψ1(x, ε)

)−∇xΨ2(x, ε) = Ttot(x, ε)Ψ

1(x, ε). (4.1)

Here, Ψ(x, ε,Ω) is the fluence in phase space with the space variable x ∈ Rd, the energy

ε > 0 of a particle and the flight direction Ω in the unit sphere S2. Note that the energy

ε assumes the role of the time variable of the previous chapters. Ψ0, Ψ1 and Ψ2 define

the moments of Ψ in angle, that is

Ψ0(x, ε) =

∫S2

Ψ(x, ε,Ω) dΩ,

Ψ1(x, ε) =

∫S2

ΩΨ(x, ε,Ω) dΩ,

Ψ2(x, ε) =

∫S2

(Ω⊗ Ω)Ψ(x, ε,Ω) dΩ,

so that Ψ0 is a scalar, Ψ1 a vector and Ψ2 a tensor. Furthermore, SM (x, ε) is the stopping

power that describes the energy loss of a particle and Ttot(x, ε) is the transport coefficient

that models angular deviations. Because both are influenced by the density ρ(x) of the

tissue, it is assumed that they can be written as

SM (x, ε) = ρ(x)S(ε), Ttot(x, ε) = ρ(x)T (ε)

for some functions S(ε) and T (ε). As mentioned in the introduction, the density ρ is a

piecewise constant function whose values differ by a high factor. For example, air has

approximately density 10−3 and water density 1. The solution of (4.1) then yields the

sought after dose

D(x) =

∫ ∞0

S(ε)Ψ0(x, ε) dε,

77

78 4 Applications in radiotherapy

which is the amount of energy that reaches a point x.

Since the system (4.1) consists of two equations with three unknowns, some further

assumptions are needed to close the system. In [2], an entropy minimization strategy is

used and the system is transformed to the so-called M1 model

∂ε(ρ(x)S(ε)Ψ0(x, ε)

)−∇xΨ1(x, ε) = 0,


)−∇xDe

(Ψ1(x, ε)

Ψ0(x, ε)

)Ψ0(x, ε) = ρ(x)T (ε)Ψ1(x, ε), (4.2)

where

De(f) =1− χ(f)

2Id +

3χ(f)− 1

2

f

‖f‖⊗ f

‖f‖is the Eddington tensor with the Eddington factor

χ(f) =3 + 4‖f‖2

5 + 2√

4− 3‖f‖2.

Now, only an initial condition is needed to solve the system of equations. Denote εmax

as the highest possible energy of a particle and consider the energy interval (0, εmax).

The initial condition is given as

Ψ0(x, εmax) = δ > 0, Ψ1(x, εmax) = 0

and (4.2) is solved for ε from εmax to 0.

4.2 Solution of the 1D model

Restrict the following analysis to the one-dimensional case d = 1. The system (4.2) then

reduces to


)− ∂xΨ1(x, ε) = 0,


)− ∂x

(Ψ0(x, ε)χ

(Ψ1(x, ε)

Ψ0(x, ε)

))= ρ(x)T (ε)Ψ1(x, ε). (4.3)

To be able to use the Lax-Friedrichs method as derived in section 3.3.3, this system needs

to be transformed so that the energy derivative only contains Ψ0 and Ψ1 respectively. In

addition, only homogeneous systems with the right side equal to zero were examined up

to now, but the equations in this chapter all contain a source term ρTΨ1 in the second

equation which needs to be regarded as well.

Following the approach in [2], use a change of variables to transform the system. Denote

u(x, ε) =

(Ψ0(x, ε)

Ψ1(x, ε)

).


In a first step, multiply (4.3) with S(ε) to obtain

S∂ε(ρSΨ0

)− ∂xSΨ1 = 0,

S∂ε(ρSΨ1

)− ∂x

(SΨ0 χ

(SΨ1

SΨ0

))= ρTSΨ1,

since S(ε)∂xf(u(x, ε)) = ∂x (S(ε)f(u(x, ε))). By defining

u(x, ε) := S(ε)u(x, ε),

this is the same as

S∂ε

(ρΨ0

)− ∂xΨ1 = 0,

S∂ε

(ρΨ1

)− ∂x

(Ψ0 χ

(Ψ1

Ψ0

))= ρT Ψ1.

Secondly, it holds that ∂ε (ρ(x)u(x, ε)) = ρ(x)∂ε (u(x, ε)) and dividing the system by

ρ(x) > 0 yields

S∂εΨ0 − 1

ρ(x)∂xΨ1 = 0,

S∂εΨ1 − 1

ρ(x)∂x

(Ψ0 χ

(Ψ1

Ψ0

))= T Ψ1.

In a last step, let

ε : R+ → R+, ε 7→∫ ε

0

1

S(t)dt

be a change of variables. This is valid because S(ε) > 0 so that ε is monotonically

increasing. Then define further

u(x, ε) := u(x, ε),

so that S(ε)∂εu(x, ε) = ∂εu(x, ε) and ∂xf(u(x, ε)) = ∂xf(u(x, ε)), and the system can be

written as

∂εΨ0(x, ε)− 1

ρ(x)∂xΨ1(x, ε) = 0,

∂εΨ1(x, ε)− 1

ρ(x)∂x

(Ψ0(x, ε)χ

(Ψ1(x, ε)

Ψ0(x, ε)

))= T (ε)Ψ1(x, ε). (4.4)

For the homogeneous case, (4.3) has the form

∂ε (ρSu)− ∂xf (u) = 0 (4.5)


with the flux function

f(u) =

(Ψ1

Ψ0 χ(Ψ1/Ψ0

)) .It was just shown that this is equivalent to

∂εu−1

ρ(x)∂xf(u) = 0,

which conforms to (3.6) with a(x) = 1ρ(x) as introduced in section 3.3. Therefore, (4.5)

can be solved with the methods that were derived in the previous chapters.

On a nonuniform grid, the Lax-Friedrichs method for this equation is

uni =1

2(un+1i+1 + un+1

i−1 )− µ

2(f(un+1

i+1 )− f(un+1i−1 )). (4.6)

Here, the solution is computed for ε from εmax to 0, so that εn = εn+1 − ∆ε and the

temporal index, or in this case the one describing the energy, is counted backwards.

Furthermore, the step size ∆x(x) is chosen depending on the density ρ(x) and the

coefficient µ is now given by µ = ∆ε2ρ(x)∆x(x) . More precisely, the scheme has the form

(Ψ0)ni =1

2

((Ψ0)n+1

i+1 + (Ψ0)n+1i−1

)− µ

2

((Ψ1)n+1

i+1 − (Ψ1)n+1i−1

),

(Ψ1)ni =1

2

((Ψ1)n+1

i+1 + (Ψ1)n+1i−1

)− µ

2

((Ψ0)n+1

i+1 χ

((Ψ1)n+1

i+1

(Ψ0)n+1i+1

)− (Ψ0)n+1

i−1 χ

((Ψ1)n+1

i−1

(Ψ0)n+1i−1

)).

To apply this method in the inhomogeneous case, a discretization of the source term

is necessary. This gives rise to some difficulties when computing the solution because

the source term depends on the solution Ψ1 that is multiplied with the possibly large

transport coefficient. Such a problem is called stiff and some numerical methods for

solving it require a very small step size ∆ε in order to be stable.

According to [12], one possible approach to solving this equation without reducing the

step size is to split the problem. In a first step, the solution to the homogeneous system

is computed as described before. In a second step, the ordinary differential equation

∂εu(x, ε) =

(0

T (ε)Ψ1(x, ε)

)

is solved. This equation contains a further difficulty since the transport coefficient de-

pends on ε while the numerical method uses the variable ε. A constant step size ∆ε

seldom implies a constant step size ∆ε, so it is not easy to compute εn from εn. A possi-

ble solution to this problem will be presented later on when computing some examples.


For the solution of the ODE, the most simple approach is to just discretize the derivative

implicitly

∂εu(x, ε) =u(x, ε+ ∆ε)− u(x, ε)

∆ε⇒ u(x, ε) = u(x, ε+ ∆ε)−∆ε

(0

T (ε)Ψ1(x, ε)

).

Combining these two steps yields the scheme

uni =1

2(un+1i+1 + un+1

i−1 )− µ

2

(f(un+1

i+1 )− f(un+1i−1 )

)−∆εΣn

i (4.7)

with Σni =

(0

(Ψ1)ni Tn

)where Tn = T (εn) is the discretization of the transport coeffi-

cient. Hence,

(Ψ0)ni =1

2

((Ψ0)n+1

i+1 + (Ψ0)n+1i−1

)− µ

2

((Ψ1)n+1

i+1 − (Ψ1)n+1i−1

),

(Ψ1)ni =1

1 + ∆ε Tn

[1

2

((Ψ1)n+1

i+1 + (Ψ1)n+1i−1

)−µ

2

((Ψ0)n+1

i+1 χ

((Ψ1)n+1

i+1

(Ψ0)n+1i+1

)− (Ψ0)n+1

i−1 χ

((Ψ1)n+1

i−1

(Ψ0)n+1i−1

))].

For the uniform grid, the scheme can be written similarly, but since it depends on the

number of discontinuities in ρ and requires several distinctions of the different stencils

that are used, it will not be written down explicitly.

For the computation of the dose, use that

D(x) =

∫ ∞0

S(ε)Ψ0(x, ε) dε

=

∫ εmax

0S(ε)Ψ0(x, ε) dε

=

N−1∑n=0

∫ εn

εn+1

S(ε)Ψ0(x, ε) dε

=N−1∑n=0

εn − εn+1

2

(S(εn)Ψ0(x, εn) + S(εn+1)Ψ0(x, εn+1)

)=

N−1∑n=0

εn − εn+1

2

(Ψ0(x, εn) + Ψ0(x, εn+1)

)with ε0 = εmax, εN = 0 and εn = ε(εn) and with the trapezoidal rule. To compute this,

determine each summand after the corresponding step of the above scheme (4.7) and

add them together. As for the discretization of the source term, the values for εn need

to be known to be able to do so.


4.3 Numerical experiments

In this section, the previous results are validated by means of some test cases.

4.3.1 Homogeneous test case

At first, consider the simple example of the Riemann problem suggested in [2]. The

initial condition is given as

Ψ0(x, εmax) =

0.5, if x < 0.5

3, if x > 0.5, Ψ1(x, εmax) = 0

for x ∈ [0, 1], the transport coefficient as T (ε) = 0 and the stopping power as S(ε) = 1

for ε ∈ [0, 0.2]. Therefore, the differential equation contains no source term and

u(x, ε) = u(x, ε) = u(x, ε)

with ε = ε, so that the transformations of the previous section become redundant.

Furthermore, the density is defined as

ρ(x) =

1, if x < 0.3,

0.01, if 0.3 < x < 0.5,

1, if x > 0.5.

Figure 4.1 shows the solution u(x, 0) = u(x, 0) of the M1 model for εmax = 0.2 that was

computed with the Lax-Friedrichs method with ∆ε = 1310−3 and with the spatial step

sizes ∆x = 10−3 for the uniform grid and ∆x1 = 10−3, ∆x2 = 10−1 and ∆x3 = 10−3 for

the nonuniform grid. As introduced in chapter 3, these step sizes are required by the

CFL condition

∆ε ≤ ∆x(x)1ρ1|λmax|

= ∆x(x),

where |λmax| = 1 is an estimation of the biggest eigenvalue of f ′ that was computed with

prior knowledge of the structure of the solution of the Riemann problem.

Note again that both methods were constructed under the assumption that a sufficient

number of grid points lies between the two jumps in density. In this case, the step sizes

∆x and ∆x2 for the uniform and nonuniform method have to be small enough in relation

to the width 0.2 of the interval (0.3, 0.5) where ρ(x) = 0.01.

As can be seen, both methods yield essentially the same results. They differ only slightly

in the areas with nonzero slope as illustrated in the magnification in figure 4.2. The


0 0.2 0.4 0.6 0.8 11

0.5

0

0.5

1

1.5

2

2.5

3

3.5uniform

0

1

0 0.2 0.4 0.6 0.8 11

0.5

0

0.5

1

1.5

2

2.5

3

3.5nonuniform

0

1

0 0.2 0.4 0.6 0.8 10.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

nonuniform 0 uniform

0

Figure 4.1: Solution of the Riemann problem using the uniform method and the nonuni-

form method and their difference.

0.58 0.6 0.62

1.8

2

2.2

2.4

2.6

2.8

3uniform

0

1

0.58 0.6 0.62

1.8

2

2.2

2.4

2.6

2.8

3nonuniform

0

1

0.58 0.6 0.62

0

1

2

3

4

5

6

7

8

9

x 10 3

nonuniform 0 uniform

0

Figure 4.2: Magnification of the solutions in figure 4.1.

more important difference of the two methods is the number of grid points. While there

are 1000 grid points for the uniform method, there are only 802 for the nonuniform

one. Because 600 steps in energy are needed for εmax = 0.2, the consequence of this

difference is that for the uniform method, 6 · 105 values u(xi, εn) need to be computed

and for the nonuniform method only 4.812 · 105. Therefore, the nonuniform method is

computationally less expensive for the given step size.

To check whether the computed solution is correct, consider again system (4.4). A


further change of variables with

x : R+ → R+, x 7→∫ x

0ρ(t)dt

and

u(x, ε) := u(x, ε)

transforms the system to

∂εΨ0(x, ε)− ∂xΨ1(x, ε) = 0,

∂εΨ1(x, ε)− ∂x

(Ψ0(x, ε)χ

(Ψ1(x, ε)

Ψ0(x, ε)

))= T (ε)Ψ1(x, ε). (4.8)

This is a system without the discontinuous coefficient ρ and can be solved easily and

accurately. Since it is written in the tilde variables, the initial conditions have to be

transformed accordingly. Because x(0.5) = 0.302, they are given as

Ψ0(x, εmax) =

0.5, if x < 0.302

3, if x > 0.302, Ψ1(x, εmax) = 0.

Then, the solution u(x, ε) of this Riemann problem should fulfill

u(x, ε) = u(x, ε).

Figure 4.3 shows the solution Ψ0(x, 0) and its transformation Ψ0(x, 0) from the x-plane

to the x-plane. Since the thus constructed Ψ0 and the previously computed Ψ0 in figure

4.1 match, the derived methods are indeed valid for the homogeneous case.

Figure 4.4 shows the normalized dose D(x) for this example.


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.5

0

0.5

1

1.5

2

2.5

3

3.5

!0

!0

Figure 4.3: Solution of the Riemann problem in the x-plane and its transformation to

the x-plane.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

normalized dose

Figure 4.4: Normalized dose for the homogeneous test case.

4.3.2 One-dimensional dose calculation

Examine now an example that was used in [7] and is better suited for dose calculation.

There, a part of the body is irradiated such that the radiation enters the body at the left

boundary. When interpreting the energy as time, this means that an initial condition

should be zero everywhere except at the left boundary. With increasing time, the initial

radiation propagates to the right and further into the tissue. Here, x ∈ [xmin, xmax] =


[0, 9], ε ∈ [εmin, εmax] = [0.01, 10] and the initial condition is given as

Ψ0(x, εmax) =

105 exp(−200(εmax − 9.9)2), if x = xmin

0, else,

Ψ1(x, εmax) = 0.95 ·Ψ0(x, εmax)

and for smaller energies ε, a boundary condition of the form

Ψ0(xmin, ε) = 105 exp(−200(ε− 9.9)2), Ψ1(xmin, ε) = 0.95 ·Ψ0(xmin, ε),

Ψ0(xmax, ε) = 0, Ψ1(xmax, ε) = 0

is used. Furthermore, consider an area x ∈ [0, 9] that is composed of water, air and

again water, so that

ρ(x) =

1, if 0 ≤ x ≤ 4,

10−3, if 4 < x < 6,

1, if 6 ≤ x ≤ 9.

The stopping power and the transport coefficient are given in form of a table that is the

result of previous computations that are based on cross section data as explained in [5].

The needed values are then determined with a linear interpolation of the values in the

table.

The same idea is used for the computation of the grid points εn. In a first step, construct

a set of grid points εi that covers the examined energies. After that, compute for each εi

the corresponding ε(εi) and save both sets. It is then possible to use a linear interpolation

with the nodes εi and values εi in order to compute other values ε from ε(ε). Obviously,

this approach is not ideal since the accuracy of the interpolation depends on the position

of the nodes and the interpolation has to be done in each time step.

The experiments in the previous section showed that the derived methods yield an

accurate solution of the M1 model without a source term and with the stopping power

S(ε) = 1. To first test whether the scheme is valid for an inhomogeneous equation with

arbitrary stopping power, consider the case ρ ≡ 1 so that the area [0, 9] is composed of

only water. Figure 4.5 shows the results of the computed dose for different spatial step

sizes. Since the biggest eigenvalue of f is now approximately 3, the energy step size is

chosen as ∆ε = 13∆x because of the CFL condition.

To be able to interpret these results, compare the dose with the lowest step size ∆x =

10−2.5 to the results of some other methods. The first benchmark is the Monte Carlo


0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

dx = 10 1

dx = 10 1.5

dx = 10 2

dx = 10 2.5

Figure 4.5: Normalized dose for εmax = 10 on water.

code PENELOPE (see [13]) which is a probabilistic method that is one of the standard

approaches to dose calculation. The other benchmark is as well a common Monte Carlo

code by the name GEANT4 (see e.g. [1]). The dose for each of the three methods is

depicted in figure 4.6.

0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

PENELOPEGEANT4Lax Friedrichs for dx = 10 2.5

Figure 4.6: Comparison of the computed dose with the results of GEANT4 and

PENELOPE.

It was explained in [5] that the difference at the left boundary is a result of the M1 model

which does not capture all the physical effects and thus, does not yield exactly the same


results as the Monte Carlo methods. Apart from that, the derived method yields good

results. As figure 4.5 suggests, the method converges for a refined grid, as the descent of

the dose to the right of the interval steepens. This can be verified by plotting the `1-error

of each dose with ∆x = 10−18i−1, i = 0, . . . , 11, in comparison to the most accurate dose

for ∆x = 10−2.5 as was done in figure 4.7. There, a linear interpolation was used to

expand each dose onto the grid with step size ∆x = 10−2.5 so that the values at each

grid point can be compared.

2.25 2 1.75 1.5 1.25 15.3

5.8

6.3

6.8

Figure 4.7: Logarithmic error plot for the computed doses.

Thus, it was shown that the method is able to correctly handle the transport coefficient

and the transformation to the ε variables because of the stopping power.

In a last step, compute the dose with the discontinuous coefficient ρ in order to prove

that the method is able to compute the correct dose for the application. To do so, the

nonuniform method is used for the M1 model with the spatial step sizes ∆x1 = ∆x3 =

10−3 and ∆x2 = 100 and with the energy step size ∆ε = 1310−3. Figure 4.8 shows the

solution u(x, ε) after different amounts of steps in energy ε. Since the radiation that

passes the air cavity does not loose much energy because of the low density, one would

expect the dose to behave similar to figure 4.6 with the difference, that it is shifted from

the left to the right border of the air cavity like the solution in figure 4.8.

Starting at the energy ε(εmax) = ε(10) = 5.1878, the step size ∆ε = 1310−3 requires 15562

steps to reach the energy 4.6667 ·10−4 > 2.5171 ·10−4 = ε(0.01). This is the advantage of

the derived methods over the ordinary Lax-Friedrichs method. Here, the CFL condition


0 2 4 6 8

0

4

8

12

16x 104 solution after 500 steps

0

1

0 2 4 6 8

0

1.5

3

4.5

6x 104 solution after 4000 steps

0

1

0 2 4 6 8

0

0.625

1.25

1.875

2.5x 104 solution after 10000 steps

0

1

0 2 4 6 8

0

20000

10000

5000

15000

solution after 12000 steps

0

1

0 2 4 6 8

0

3000

6000

9000

12000solution after 14000 steps

0

1

0 2 4 6 8

0

2000

4000

6000

8000solution after 15562 steps

0

1

Figure 4.8: Solution u(x, ε) of the M1 model at different energies.

demands a step size ∆ε that is smaller than ∆x or ∆x1 respectively. For the ordinary

method, the CFL condition states that for all x

∆ε ≤ ∆x1

ρ(x) |λmax|=

1

3ρ(x)∆x

so that ∆ε ≤ 1310−3∆x. This means that the step size for the ordinary method is given

by the minimal value of ρ and for the modified method by the maximal value of ρ. For

the dose calculation this results in a computation time for the ordinary method that


is approximately a thousand times higher than for the modified method. Indeed, the

ordinary Lax-Friedrichs method would require the computation of about 1.4 ·1011 values

u(x, ε) while the modified method on a uniform grid needs about 1.4 · 108 values and on

a nonuniform grid 1.1 · 108. Generally, it holds that the more different the values of the

coefficient ρ, the better are the modified methods in comparison to the ordinary ones.

0 2 4 6 80

0.2

0.4

0.6

0.8

1

PENELOPEmodified Lax Friedrichs

Figure 4.9: Comparison of the computed dose with the results of PENELOPE.

The corresponding dose to figure 4.8 and the dose computed with PENELOPE are

displayed in figure 4.9. The difference near the left boundary was already mentioned to

be a result of the M1 model. On the interval (4, 6), the density is very small so that the

particles get transported much faster. This is the reason why the dose remains nearly

constant on this interval. Furthermore, it explains the great difference between the two

doses, since their value on the interval is influenced by their value at x = 4 and the dose

of the M1 model is higher there. This was already the case in the example in figure 4.6,

now the dose is essentially just shifted to the right from x = 4 to x = 6 as expected.

Altogether, the observed differences are results of the M1 model for which the method

yields an accurate solution.

5 Conclusions and outlook

In this thesis, the Lax-Friedrichs method was successfully modified in order to solve

one-dimensional systems of hyperbolic balance laws with a discontinuous flux function.

The aim was to derive a method that does not require a very fine grid like common

methods do because of the CFL condition. This was done by gradually generalizing eas-

ier problems and validating them by means of some examples, starting with the upwind

method for the scalar advection equation and ending with the modified Lax-Friedrichs

method for dose calculation in radiotherapy. There are two versions of the final method.

The first was constructed for a uniform spatial grid where the stencil of the method was

adjusted to the piecewise constant coefficient in the flux. The other version was defined

on a nonuniform grid which was chosen according to the coefficient. Both methods were

then applied to some examples including the dose calculation for radiotherapy.

It can be observed that it depends on the given situation whether the uniform or the

nonuniform method should be preferred. In chapter 2.3, the results suggested that the

uniform upwind method is better than the nonuniform one since it yielded a greater

order and smaller error. However, these observations were valid for arbitrary examples

and situations where the coefficient a(x) had values between 0.1 and 1. If the difference

between the values of a(x) is much bigger and if the initial condition is confined to a

special structure, then the results change. In the application, an initial condition would

define some values at the left boundary from where the radiation originates and it would

be zero everywhere else. This was the case in the example of section 4.3.2. Before,

the disadvantage of the nonuniform method was the lesser accuracy in areas of bigger

coefficient a(x). Now, there is only a limited interval of energies for which the nonzero

part of the initial condition passes through the spatial interval with bigger a(x) and

which would therefore require a smaller step size. In comparison to the uniform method,

the resulting inaccuracy is only a minor disadvantage that is made up for by the much

lesser computing time and the much simpler form of the scheme. The same conclusion

was reached for the example in section 4.3.1. For this reason, the nonuniform method is

better suited for the application in dose calculation.

Nevertheless, both methods have a great advantage over a standard method. For the

91

92 5 Conclusions and outlook

example of the one-dimensional dose calculation in section 4.3.2, the computation time

of the modified methods was proven to be approximately a thousand times smaller than

for an ordinary method. The reason for this was the restriction of the energy step size

due to the CFL condition. It was explained that the difference in computation time

depends on the quotient of the maximum and minimum value of the coefficient ρ, which

is especially high for dose calculation.

The derived methods can be applied to higher dimensional problems as well. For exam-

ple, a two-dimensional conservation law can be written as

∂

∂tu(x, t) +

∂

∂x(1)f(u(x, t)) +

∂

∂x(2)g(u(x, t)) = 0,

where x =

(x(1

x(2)

)and f and g describe the flux in the direction of x(1) and x(2)

respectively. As suggested in [12], the x-plane can be subdivided into rectangles

Cij = [x(1)i−1/2, x

(1)i+1/2]× [x

(2)j−1/2, x

(2)j+1/2]

with ∆x(1) = x(1)i+1/2 − x

(1)i−1/2 and ∆x(2) = x

(2)j+1/2 − x

(2)j−1/2. The approximation to the

average value of the solution over Cij then has the form

unij ≈1

∆x(1)∆x(2)

∫ x(2)j+1/2

x(2)j−1/2

∫ x(1)i+1/2

x(1)i−1/2

u(x, tn) dx(1)dx(2)

and it can be computed by means of the numerical method

un+1ij = unij −

∆t

∆x(1)

(Fni+1/2,j − F

ni−1/2,j

)− ∆t

∆x(2)

(Gni,j+1/2 −G

ni,j−1/2

)where Fn and Gn are the average flux functions of f and g. Now, the same approach as

for the one-dimensional case can be used by considering each spatial dimension separately

and the derived Lax-Friedrichs method can be generalized to higher dimensions.

In the application, the system 4.4 needs to be solved and the change of variables was

only done for the energy which is always one-dimensional. Therefore, the solution of the

system and hence the dose can be computed as described for the one-dimensional case.

For further studies, note that the aim of this thesis was to foremost derive a method that

is suited for the given situation. For this reason, only simple methods were considered

and modified. As a possible improvement, a replacement of the Lax-Friedrichs method

could be examined with the purpose of constructing a method of higher order. In

addition, the method uses some approximations that could be improved upon, like the

discretization of the source term or the computation of ε from ε.

Appendix

Show that the modified upwind method on a uniform grid of section 2.2.2 is TVD. To

do so, use the same approach as for the simple upwind method and analyze the total

variation for each case in the definition of the method separately.

Start with the proof for the special case where the jump lay on a grid point. There, the

method was defined as

un+1i =


unα +(i−αs − ν

)(unα − unα−1), if α < i < β′,


For the total variation it holds

TV (un+1) =∞∑

i=−∞

∣∣un+1i − un+1

i−1

∣∣=

α∑i=−∞

∣∣un+1i − un+1

i−1

∣∣+∣∣un+1α+1 − u

n+1α

∣∣+

β′−1∑i=α+2

∣∣un+1i − un+1

i−1

∣∣+∣∣∣un+1β′− un+1

β′−1

∣∣∣+∞∑

i=β′+1

∣∣un+1i − un+1

i−1

∣∣For the first summand,

α∑i=−∞

∣∣un+1i − un+1

i−1

∣∣ =α∑

i=−∞

∣∣(1− ν)uni + νuni−1 − (1− ν)uni−1 − νuni−2

∣∣=

α∑i=−∞

∣∣(1− ν)(uni − uni−1) + ν(uni−1 − uni−2)∣∣

≤ (1− ν)α∑

i=−∞


∣∣+ να∑

i=−∞


∣∣ ,where the last inequality holds, because 0 ≤ ν ≤ 1. The second term is given as∣∣un+1

α+1 − un+1α

∣∣ =

∣∣∣∣unα +

(1

s− ν)

(unα − unα−1)− unα + ν(unα − unα−1)

∣∣∣∣=

1

s

∣∣unα − unα−1

∣∣ .

93

94 Appendix

Because β′ = α+ dsνe, the third summand can be written as

β′−1∑i=α+2

∣∣un+1i − un+1

i−1

∣∣=

β′−1∑i=α+2

∣∣∣∣unα +

(i− αs− ν)

(unα − unα−1)− unα −(i− 1− α

s− ν)

(unα − unα−1)

∣∣∣∣=

β′−1∑i=α+2

∣∣∣∣1s (unα − unα−1)

∣∣∣∣ = (β′ − α− 2)

∣∣∣∣1s (unα − unα−1)

∣∣∣∣ =dsνe − 2

s


∣∣ .The fourth part resembles the second, as it is defined by two different cases as∣∣∣un+1

β′− un+1

β′−1

∣∣∣=

∣∣∣∣unβ′−dsνe + (dsνe − sν)(unβ′−dsνe+1 − unβ′−dsνe)− u

nα −

(β′ − 1− α

s− ν)

(unα − unα−1)

∣∣∣∣=

∣∣∣∣unα + (dsνe − sν)(unα+1 − unα)− unα −(dsνe − 1

s− ν)

(unα − unα−1)

∣∣∣∣≤ (dsνe − sν)

∣∣unα+1 − unα∣∣+

(ν − dsνe − 1

s

) ∣∣unα − unα−1

∣∣ .Obviously, (dsνe − sν) ≥ 0 and

(ν − dsνe−1

s

)≥ 0, since(

ν − dsνe − 1

s

)≥ 0 ⇔ ν ≥ dsνe − 1

s⇔ sν ≥ dsνe − 1 = bsνc

is always satisfied. Finally, the fifth term is

∞∑i=β′+1

∣∣un+1i − un+1

i−1

∣∣=

∞∑i=α+dsνe+1

∣∣∣uni−dsνe + (dsνe − sν)(uni−dsνe+1 − uni−dsνe)

−uni−1−dsνe − (dsνe − sν)(uni−dsνe − uni−1−dsνe)

∣∣∣=

∞∑i=α+1

∣∣uni + (dsνe − sν)(uni+1 − uni )− uni−1 − (dsνe − sν)(uni − uni−1)∣∣

=

∞∑i=α+1

∣∣uni − uni−1 + (dsνe − sν)(uni+1 − uni ) + (sν − bsνc − 1)(uni − uni−1)∣∣

≤ (dsνe − sν)∞∑

i=α+1

∣∣uni+1 − uni∣∣+ (sν − bsνc)

∞∑i=α+1


∣∣ .

Appendix 95

Altogether, the method is TVD because it holds, that

TV (un+1)

≤ (1− ν)α∑

i=−∞


∣∣+ να∑

i=−∞


∣∣+dsνe − 1

s


∣∣+ (dsνe − sν)

∣∣unα+1 − unα∣∣+

(ν − dsνe − 1

s

) ∣∣unα − unα−1


∞∑i=α+1


∞∑i=α+1


∣∣= (1− ν)

α∑i=−∞


∣∣+ να−1∑i=−∞


∣∣+ ν∣∣unα − unα−1


∞∑i=α


∞∑i=α+1


∣∣= (1− ν)

α∑i=−∞


∣∣+ να∑

i=−∞



∞∑i=α+1


∣∣+ (sν − bsνc)∞∑

i=α+1


∣∣=

α∑i=−∞


∣∣+ (dsνe − bsνc)∞∑

i=α+1


∣∣ = TV (un)

Now, prove analogously that the method is still TVD if the jump lies between grid

points. Denote t := sγ + sν + (1− s) α∆x , then the method can be written as

un+1i =


unγ + i−ts (unγ − unγ−1), if γ < i ≤ t,

unγ + i−ts (unγ+1 − unγ ), if t < i < β′,


Therefore, the total variation is composed of seven different summands

TV (un+1) =

γ∑i=−∞

∣∣un+1i − un+1

i−1

∣∣︸︷︷︸

(1)

+∣∣∣un+1γ+1 − u

n+1γ

∣∣∣︸︷︷︸(2)

+

t∑i=γ+2

∣∣un+1i − un+1

i−1

∣∣︸︷︷︸

(3)

+∣∣un+1t+1 − u

n+1t

∣∣︸︷︷︸(4)

+

β′−1∑i=t+2

∣∣un+1i − un+1

i−1

∣∣︸︷︷︸

(5)

+∣∣∣un+1β′− un+1

β′−1

∣∣∣︸︷︷︸(6)

+

∞∑i=β′+1

∣∣un+1i − un+1

i−1

∣∣︸︷︷︸

(7)

.

96 Appendix

For these, it holds that

(1) =

γ∑i=−∞

∣∣uni − ν(uni − uni−1)− uni−1 + ν(uni−1 − uni−2)∣∣

≤ (1− ν)

γ∑i=−∞


∣∣+ ν

γ∑i=−∞


∣∣ ,

(2) =

∣∣∣∣unγ +γ + 1− t

s(unγ − unγ−1)− unγ + ν(unγ − unγ−1)

∣∣∣∣≤(γ + 1− t

s+ ν

)︸︷︷︸

=(1− 1s )(

α∆x−γ)+ 1

s≥0

∣∣unγ − unγ−1

∣∣ ,

(3) =t∑

i=γ+2

∣∣∣∣unγ +i− ts

(unγ − unγ−1)− unγ −i− 1− t

s(unγ − unγ−1)

∣∣∣∣=

t∑i=γ+2

∣∣∣∣1s (unγ − unγ−1)

∣∣∣∣ =t− γ − 1

s

∣∣unγ − unγ−1

∣∣ ,

(4) =

∣∣∣∣unγ +t+ 1− t

s(unγ+1 − unγ )− unγ −

t− ts

(unγ − unγ−1)

∣∣∣∣ =1

s

∣∣unγ+1 − unγ∣∣ ,

(5) =

β′−1∑i=t+2

∣∣∣∣unγ +i− ts

(unγ+1 − unγ )− unγ −i− 1− t

s(unγ+1 − unγ )

∣∣∣∣=

β′−1∑i=t+2

∣∣∣∣1s (unγ+1 − unγ )

∣∣∣∣ =β′ − t− 2

s

∣∣unγ+1 − unγ∣∣ ,

(6) =

∣∣∣∣unβ′−dsνe + (dsνe − sν)(unβ′−dsνe+1 − unβ′−dsνe)− u

nγ −

β′ − 1− ts

(unγ+1 − unγ )

∣∣∣∣≤∣∣∣unβ′−dsνe − unγ ∣∣∣+ (dsνe − sν)

∣∣∣unβ′−dsνe+1 − unβ′−dsνe

∣∣∣+

∣∣∣∣ β′ − 1− ts

∣∣∣∣ ∣∣unγ+1 − unγ∣∣ ,

Appendix 97

(7) =∞∑

i=β′+1

∣∣∣uni−dsνe + (dsνe − sν)(uni−dsνe+1 − uni−dsνe)

−uni−dsνe−1 − (dsνe − sν)(uni−dsνe − uni−dsνe−1)

∣∣∣=

∞∑i=β′+1

∣∣∣(dsνe − sν)(uni−dsνe+1 − uni−dsνe)− (dsνe − sν − 1)(uni−dsνe − u

ni−dsνe−1)

∣∣∣=

∞∑i=β′+1

∣∣∣(dsνe − sν)(uni−dsνe+1 − uni−dsνe)− (bsνc − sν)(uni−dsνe − u

ni−dsνe−1)

∣∣∣≤ (dsνe − sν)

∞∑i=β′+1

∣∣∣uni−dsνe+1 − uni−dsνe

∣∣∣+ (sν − bsνc)∞∑

i=β′+1

∣∣∣uni−dsνe − uni−dsνe−1

∣∣∣ .Overall, it holds for the total variation

TV (un+1) = (1) + (2) + (3) + (4) + (5) + (6) + (7)

≤ (1− ν)

γ∑i=−∞


∣∣+ ν

γ+1∑i=−∞


∣∣+

(β′ − t− 1

s+

∣∣∣∣ β′ − t− 1

s

∣∣∣∣) ∣∣unγ+1 − unγ∣∣+∣∣∣unβ′−dsνe − unγ ∣∣∣

+ (dsνe − sν)∞∑i=β′

∣∣∣uni−dsνe+1 − uni−dsνe

∣∣∣+ (sν − bsνc)∞∑

i=β′+1

∣∣∣uni−dsνe − uni−dsνe−1

∣∣∣=

γ∑i=−∞


∣∣+∞∑

i=β′−dsνe+1


∣∣+∣∣∣unβ′−dsνe − unγ ∣∣∣

+

(β′ − t− 1

s+

∣∣∣∣ β′ − t− 1

s

∣∣∣∣) ∣∣unγ+1 − unγ∣∣

=

b α∆xc∑i=−∞


∣∣+∞∑

i=d α∆x+sνe−dsνe+1


∣∣+∣∣∣und α∆x+sνe−dsνe − u

nb α∆xc

∣∣∣+

(β′ − t− 1

s+

∣∣∣∣ β′ − t− 1

s

∣∣∣∣) ∣∣∣unb α∆xc+1− unb α∆xc

∣∣∣=

b α∆xc∑i=−∞


∣∣+

∞∑i=b α∆x+sνc−bsνc+1


∣∣+∣∣∣unb α∆x+sνc−bsνc − u

nb α∆xc

∣∣∣+

(β′ − t− 1

s+

∣∣∣∣ β′ − t− 1

s

∣∣∣∣) ∣∣∣unb α∆xc+1− unb α∆xc

∣∣∣ .

98 Appendix

Since⌊α

∆x + sν⌋− bsνc =

⌊α

∆x + sν − bsνc⌋∈ ⌊α

∆x

⌋,⌊α

∆x

⌋+ 1, the above equates to

TV (un) if the last term is zero. This is the case if

β′ − t− 1

s≤ 0 ⇔ β′ ≤ t+ 1

⇔⌈ α

∆x+ sν

⌉≤ s

⌊ α

∆x

⌋+ sν + (1− s) α

∆x+ 1

⇔⌊ α

∆x+ sν

⌋≤ α

∆x+ sν + s

⌊ α

∆x

⌋− 1

s

α

∆x,

and therefore, if 1sα

∆x is small enough in relation to the other terms, which is the case in

all relevant examples.

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Erklarung

Hiermit versichere ich, dass ich die vorliegende Arbeit selbstandig und ohne Benutzung

anderer als der angegebenen Quellen und Hilfsmittel angefertigt habe. Alle Zitate sind

als solche kenntlich gemacht.

Aachen, den 27.10.2011

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