Advection Non-trivial solutions to the advection equation (right-moving)

x

fv

t

f (1)

give a moving pulse, whose direction is determined by the sign of v. Positive v = right-moving.

f(x)

x

Figure 1

For arbitrary functions f, the solution to the advection equation is f(u), where u = (x - vt).

Proof:

t

uf

x

ufv

vdu

df

t

u

du

df

t

uf

du

df

x

u

du

df

x

uf

)()()(

)()(

)(

This solution works for all cases. yields a right-moving wave, and a left-moving

wave.

Consider Figure 1, where time advances with each graph from left to right. Notice that each

point on the curve is simply being dragged forward. If we mark two x values and their

corresponding f(x) values, the movement of the pulse along characteristic lines (lines along

which the function does not change) becomes apparent. Figure 3 is a contour plot for Figure 2

as it advances with time.

Figure 2 Figure 3

xb xa xb xa

t

x

t

Derivative Approximations

Forward Time Centered Space (FTCS) (Garcia 2000)

Consider a set of grid points with known x and f(x) values. We will say this set of data points

exists at time level t = , and denotes x at the ith place. The distance between points is equal

to Δ , and the duration between time levels is Δ .

What if we want to find information at another time level? (Figure 4)

?

Figure 4 known

We assume this data obeys the advection equation and estimate the function values at time

level t = using a forward time centered space approximation (FTCS).

Recall equation (1).

x

fv

t

f

The partial derivative on the right side of this equation can be approximated as

The partial derivative on the left as

Equating the two with the appropriate constant v yields

Solving for gives

The approximation of the space derivative is second order accurate by virtue of being a

centered derivative approximation. The approximation of the time derivative is only first order

accurate since it is a forward derivative approximation.

The computational molecule, which shows what points are used to approximate the value at

the desired point, for FTCS is

Figure 5

FTCS is unconditionally unstable. If allowed to run for long enough, an FTCS algorithm will

always blow up – infinities will start to show up in later time steps.

In most numerical methods for evolving hyperbolic partial differential equations, the stability of

the system is determined by the Courant factor. The courant factor is , or the ratio of

physical wave speed to the speed at which the algorithm propagates information.

Consider a physical light cone.

Figure 6

Only events within in the light cone can influence or be influenced by an observer at the origin.

For our purposes, the physical light cone of the wave is determined by the physical speed of the

wave. The numerical light cone is determined by the spacing of the grid on which function

x

t

values are known. For example, looking at the computational molecule for FTCS, you can see

that numerical light cone.

Figure 7

Computationally, the only information affecting the approximation of the function value at the

gray point is the information at the three black points. Consider a physical light cone mapped

onto Figure 7.

Figure 8

Here, the physical light cone is smaller than the numerical light cone because . Stable

approximations will have a numerical light cone bigger than a physical light cone, or a Courant

factor . Keeping with the physical light cone analogy (Figure 6), the computational system

can be understood heuristically as taking into account all possible physical influences.

However, recall that FTCS is unconditionally unstable, and will eventually blow up even when

this condition is satisfied.

To get an idea of the problems with FTCS, consider a simple situation such as when a Courant

factor is . Here, we would expect that this algorithm would essentially drag the pulse

along characteristic lines. This is not true for FTCS. Recall equation (2).

For a Courant factor , the algorithm becomes

Since we have provided the algorithm with a very simple system and are looking at a symmetric

situation that it should be able to handle easily, we should expect a rather clean-looking answer

like

In the case of equation (4), the wave is simply pulled along the characteristic line, which is what

should happen computationally when numerical and physical speeds are equal; however, this is

clearly not the case for FTCS. Although this is not a rigorous explanation for the fact that FTCS

is unconditionally unstable, it does show some of the issues with using FTCS, even when solving

very simple systems.

Lax (Garcia 2000)

Instead of using a known data point at and then adding the time derivative multiplied by

, the Lax method uses an average of the two points adjacent to at time level The

lax method uses the average of points to find the point at the next time step.

Figure 11

The Lax method includes a diffusion term, which acts as though it takes energy away from the

wave, preventing it from blowing up to infinity. As is expected, the Lax method becomes

unstable with large Courant factors, but it will also become unstable with very small Courant

factors. In the latter condition, the pulse dissipates rather quickly.

Centered Time Centered Space (Garcia 2000)

The Centered Time Centered Space method uses two initial time levels of data in order to get

centered derivatives that are second-order accurate in both space and time

Figure 9

From before

The derivative approximations are

And

n

Average

Solving equation (5) yields

Note that the computational molecule for CTCS is

Figure 10

If , we can assume within a reasonable uncertainty that = . Therefore, equation

(6) reduces to . This becomes exact if initial data ensures that .

Lax-Wendroff (Garcia 2000)

This method finds second derivatives, which it then uses to get to the next time step. Figure 12

provides a graphical representation of estimating second derivatives. The derivative is being

approximated at the center black point.

Figure 12

Consider a Taylor series expansion

The FTCS effectively truncates the Taylor series estimation at the term with the first-order

partial differential. Lax-Wendroff carries the estimation one term further in the expansion.

An alternative way of using the Lax-Wendroff method takes a lax step to an intermediate time

level and then “leaps” across that intermediate step to the desired value.

Figure 12

For each step in Figure 12:

1.

2.

3.

4.

5.

6.

Lax-Wendroff is essentially FTCS plus a diffusion term. Although the difference between FTCS

and Lax-Wendroff is basically a matter of an additional term, Lax-Wendroff is stable for the

Courant condition and is second-order accurate in both time and space. This method

does suffer from a dissipation problem similar to the Lax method. However, Lax-Wendroff is not

affected as greatly by this issue as Lax is.

1 2

Black dots = initial data

3 4 5

6

Iterated Crank-Nicholson (Teukolsky 2008)

The iterated Crank-Nicholson method makes a guess at the values of the next time step and

then improves the guess by averaging. This method is stable for 4n+2 and 4n+3 iterations

where

n = 0,1,2,3,… However, doing more than two corrections is not really helpful.

Consider the advection equation (1) without the velocity factor.

x

f

t

f

FTCS gives

Backwards differencing (which is unconditionally stable) gives

Crank-Nicholson averages these two methods.

The first iteration of Crank-Nicholson calculates an intermediate variable using FTCS:

Another intermediate variable is found by averaging

Then the time step is made using in an FTCS approximation

A two-step iteration is done the same way. After equations (9) and (10):

The time step is then made as in equation (12).

As shown by Teukolsky, 2008, Crank-Nicholson is stable for courant factors , or .

Upwind Method (Garcia 2000)

This method exploits the directionality of the wave by using only points that would trail the

wave (physically) in the calculations. One benefit of this method is that we need only provide a

boundary condition for the side of the computational domain away from which the wave

travels. The computational molecule for this method is

Figure 13

And its numerical light cone is

Figure 14

The algorithm for a right-moving pulse, obeying the advection equation is

At the left boundary, some known value is provided (e.g., constant at 0) at each time step.

Providing known values as boundaries is known as Drichlet boundary conditions. A right

boundary condition is not necessary as the wave will simply move off the edge of the

computational world – outgoing boundary condition. For a left-moving pulse, we would simply

need to mirror the points used (e.g., becomes the left-most point for a given set rather than

the right-most).

Beam-Warming (Dullemond and Kuiper 2008)

Whereas the upwind method only used two points at the previous time level (Figure 13), the

Beam-Warming method uses three.

Figure 15

The partial differential equation (PDE) for the Beam-Warming method looks something like

The second derivative term is a diffusion term, which acts as if it is taking energy away from the

pulse, preventing it from blowing up. The right-moving algorithm for this method is

Again, these points can simply be mirrored for a left-moving algorithm. Also, note that this

system requires boundary conditions for the two left-most points, whereas upwind only

needed a boundary condition for the one, left-most point.

Box Method

The Box method uses two points at the previous time step and one point on the current time

step in order to approximate a point at the current time step.

Figure 16

The PDE for the Box method is

The algorithm is

Recall that

n-1 i

So, the algorithm can be reduced to

Note the numerical light cone for this system

Figure 17

Because the light cone encompasses all previous points at all previous time levels, this system

should be stable even for very large Courant factors.

Wave Equation The simplest one-dimensional wave equation is

2

22

2

2

x

fv

t

f (19).

This equation has the solution f(u), where u = x + vt or x – vt.

Proof:

Show that f(u) where u = x + vt is a solution to equation 2.

1st partial

2nd partial

1st partial

2nd partial

The proof for u = x – vt is similar. Also, it has already been shown that characteristics have

velocities .

2

2

2

22

2

2

2

2

2

2

2

2

2

2

2

x

f

t

fv

du

fd

x

f

x

u

du

df

du

d

du

df

x

du

df

x

f

du

df

x

f

x

f

xx

f

vdu

fd

t

f

vt

u

du

df

du

d

du

df

t

vdu

df

t

u

du

df

t

f

Crank Nicholson and the Wave Equation

Recall the wave equation (19).

2

22

2

2

x

fv

t

f

In two-variable form, this equation is.

We can discretize this equation using Crank-Nicholson. However, we need two pieces of initial

data. Either and at one time level or at two time levels. For discretization, I will assume

the first situation, with known values at time level t = .

Note that here

Note that here

Boundary Conditions for the Wave Equation

From the section on approximating numerical derivative, the Upwind, Beam-Warming, and Box

methods are particularly useful for establishing boundary conditions.

When a wave is evolved forward in time numerically, the system doing the evolution loses

information at the boundaries of the domain over which the wave is being evolved.

Figure 18

Consider the wave equation (19)

2

22

2

2

x

fv

t

f

This equation has as a solution

The F term is the right-moving part of the wave and the G term is the left-moving part. These

two parts are independently both solutions of the advection equation (1).

Therefore, we can use the aforementioned methods for boundary conditions by forcing the

wave equation to obey the appropriate advection equation (left-moving or right-moving) at the

boundaries.

Numerical Domain

Time advances

Information lost at boundaries

Wave equation in Spherical Coordinates

The three dimensional wave equation in spherical coordinates is

The Laplacian in spherical coordinates is

If f=f(r,t)

The Laplacian reduces to

So

Note that at the origin, the term will be zero, because is 0 at the origin.

Boundary Conditions for the Wave Equation in 3-D

Outgoing Boundary

Recall that solutions to the wave equation (19) have left and right moving parts that obey the

advection equation. See equation (21), copied here.

However, the 3-D wave equation in spherical coordinates (24) has an additional term.

How can we use our out-going boundary conditions (Upwind, Beam-Warming, and Box) with a

3-D wave equation?

Let

That is to say,

Also, let

Where

Recalling that

Find that makes this work.

Now all we have to do is solve the ODE

Now,

Considering the previous boundary conditions, such as the box method, all that needs to

change is .

For example, the Box method algorithm (18) becomes

In short, obeys a simpler wave equation.

Boundary at the Origin

Find

At the origin

Consider the Taylor series

So, we can see that

Exploiting the fact that the wave is symmetric about the origin, the second derivative at the

origin can easily be approximated as

Figure 19

Also, by considering the Laplacian in Cartesian coordinates

Due to the spherical symmetry about the origin (r=0),

Therefore,

Discretizing the Wave Equation

CFL

At r = 0

At outer boundary

Where

Crank-Nicholson

Let

At r = 0

At outer boundary

Where