Numerical Methods for (Nonlinear) Hyperbolic PDE

Georg MayMarch 2008

1 The Euler Equations

In physics there exist conservation laws for mass, momentum, and energy. Compressible fluidshave variable density, so one applies these concepts by considering conserved quantities takenper unit volume, and integrating over an arbitrary control volume Ω ⊂ R3, which we assume(without loss of generality) to be stationary. Let the velocity vector in three dimensions be givenby u = (u1, u2, u3), and denote the density (or specific mass) by ρ. For the conservation of masswe may write

d

dt

∫Ω

ρ dV = −∫∂Ω

ρu · dA , (1)

which basically states that the rate of change of mass inside a control volume is due solely to massflux across the boundary of that domain.

According to Newton’s second law of motion, the rate of change of momentum is equated withthe external forces acting on the fluid. For inviscid fluids shear forces are neglected, while normalforces are obtained by integrating the thermodynamic pressure p over the surface of the controlvolume. We neglect body forces, such as gravity. The specific momentum is given by ρui fori = 1, 2, 3. We have

d

dt

∫Ω

ρui dV = −∫∂Ω

ρuiu · dA−∫∂Ω

pni dA , i = 1, . . . , 3 . (2)

The total energy is written per unit volume as E = ρe+ 12ρu·u, where the second term is mechanical

energy and e stands for internal energy. The amont of energy inside the control volume may changedue to transport across the boundaries, but also due to work done by the forces:

d

dt

∫Ω

E dV = −∫∂Ω

(E + p)u · dA . (3)

Again viscous effects (work done by shear stresses, heat conduction) are neglected. We are thusin a position to write the conservation laws for mass, momentum, and energy for a compressibleinviscid fluid in integral form:

d

dt

∫Ω

ρ dV +∫∂Ω

ρu · dA = 0 (4)

d

dt

∫Ω

ρu dV +∫∂Ω

(ρu⊗ u + pI) · dA = 0 (5)

d

dt

∫Ω

E dV +∫∂Ω

(E + p)u · dA = 0 , (6)

where u ⊗ u = uuT is the dyadic product, and I is the unit tensor. These are nothing but theEuler equations in integral form. To obtain the differential form, apply the divergence theorem

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and note that the control volume Ω is arbitrary, which means that the integrand has to vanish.The Euler equations in differential form are written:

∂ρ

∂t+∇ · (ρu) = 0 , (7)

∂ρu∂t

+∇ · (ρu⊗ u + pI) = 0 , (8)

∂E

∂t+∇ · (u (E + p)) = 0 . (9)

Note that these equations are not closed. If we define as the vector of unknowns the five conservedvariables (ρ,m = ρ(u1, u2, u3), E)T , then all quantities in equations (7)-(9) may be expressed as afunction of these, except the pressure. To close the system one needs an equation of state, whichrelates the pressure to the other variables. For many compressible fluids and moderate operatingconditions, we assume the ideal gas law, which may be written

p = (γ − 1)E − 1

2ρu · u

, (10)

where γ = cp/cv is the ratio of specific heats at constant pressure and volume, respectively. Formany media this ratio is constant over a wide range, certainly as long as the ideal gas assumptionis valid. (For example for air one may take γ = 1.4.) The system (7)-(9) with the equation ofstate (10) are the Euler equations in strong conservation form. The significance of this terminologywill become apparent shortly. The Euler equations form a coupled system of nonlinear equationsthat are quite complicated to analyze. To see this we need to take a step back and consider thebasic properties of hyperbolic conservation laws, of which the Euler equations are a special case.

In the following we omit the vector notation that was used in this section.

2 Hyperbolic Conservation Laws

We avoid the issue of boundary conditions for the moment, and consider a Cauchy problem, i.e.find u : Rd × R+ → D ⊂ Rm such that

∂u(x, t)∂t

+d∑i=1

∂fi(u)∂xi

= 0 , u(x, 0) = φ(x) (11)

where f : D → Rm is called the flux vector. Let Ai = D1ufi be the Jacobian matrix of the ith flux

vector. Equations that can be written in the form of eq. (11) are called conservation laws.

Example 2.1. For the Euler equations (7)-(9) the state and flux vector are given by

w =

ρρu1

ρu2

ρu3

E

, fi =

ρuiρu1ui + pδ1iρu2ui + pδ2iρu3ui + pδ3i(E + p)ui

, i = 1, 2, 3 (12)

where we have denoted the state vector by w, to avoid conflict of notation with the velocity vectoru = (u1, u2, u3)T .

We have already seen the general definition of hyperbolic equations for linear second order(scalar) equations [7]. For systems of conservation laws, we need a somewhat broader definition:

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Definition 2.1. The system (11) is called hyperbolic if any combination of the Jacobian Matrices,A =

∑di=1 ηiAi, where η = (η1, . . . , ηd) ∈ Rd\0, is diagonalizable with m real eigenvalues. If

additionally the m eigenvalues are distinct, it is called strictly hyperbolic.

What are the properties of solutions to hyperbolic systems? It is useful to introduce first themost straight forward notion of a solution to the Cauchy problem (11).

Definition 2.2. We say that a function u : Rd × R+ → D is a classical solution of (11) if u is aC1 function and satisfies (11) pointwise.

We do not know yet if such a solution always exists. To establish basic properties of hyperbolicconservation laws we consider first the simplest case

2.1 The 1D Scalar Case: Linear Equations

Consider the special case d = m = 1 with linear flux function f = cu for c ∈ R. The task is to finda classical solution u : R× R+ → R that satisfies

∂u

∂t+ c

∂u

∂x= 0 , (13)

u(x, 0) = φ(x) . (14)

It can be easily verified by direct differentation that

u(x, t) = φ(x− ct) (15)

is a solution to (13), (14). Obviously the solution is constant on the lines ξ = x − ct, which arecalled are called characteristic lines. In particular for t = 0, one has ξ = x, which allows us toidentify ξ with a starting point on the x-Axis, from which each charactaristic emanates. Formallyony may define the characteristic lines in the x−t plane by a parametrization x(t), which obeys theODE dx(t)

dt = c , x(0) = x0. It may be shown that (15) is indeed the unique solution to (13), (14).We have thus by making every possible simplification extracted a very fundamental property ofhypberbolic equations: The propagation of data along characteristic lines, which is illustrated forthe x− t plane in figure 1(a).

2.2 Linear 1D Systems

Consider a linear system of equations in one spatial dimension, i.e. find u : R × R+ → Rm suchthat

∂u

∂t+A

∂u

∂x= 0 , (16)

u(x, 0) = φ(x) , (17)

where A ∈ Rm×m is a constant matrix. According to Definition 2.1 the Jacobian of a hyperbolicsystem may be diagonalized. Let A = R−1ΛR be a similarity transformation that diagonalizes Awith eigenvalues Λ = diag(λ1, . . . , λm). Define u := Ru, and ψ(x) := Rφ(x), and re-write eq. (16)as

∂u

∂t+ Λ

∂u

∂x= 0 (18)

u(x, 0) = ψ(x) . (19)

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Note that the system is decoupled! Eq. (18) admits componentwise the same solution as eq. (13),i.e. ui(x, t) = ψi(x − λit). The solution in terms of the conserved variables is then obtained asu = R−1u.

Note, that the quantities that are transported along the characteristics are not the conservedvariables as in the scalar case, but the so-called characteristic variables u. We had already seenthat the characteristics are important for hyperbolic scalar equations when defining upwind dis-cretization in the SD (streamline diffusion) context [3]. For systems it has to be noted that one hasnow exactly m rays of characteristics emanating from a particular point, see fig. 1(b), each of whichis associated with a characteristic variable ui, and speed of propagation λi. The characteristicshave to be found by diagonalizing the Jacobian matrix1.

Remark 2.1. The distinction between the conserved variables and the characteristic variables be-comes important when considering boundary conditions. Inflow and outflow must be determinedbased on the characteristic variables in full analogy with the scalar case, where incoming charac-teristics indicate inflow boundary. It is important to note that one may not in general prescribeboundary conditions based on inflow and outflow of the conserved variables.

2.3 The Scalar 1D Case: Nonlinear Equations

Consider again the case d = 1, m = 1. However, to establish a more general concept of character-istics, and existence of a regular solution, we consider a nonlinear flux function f : D → R. We arethus looking for u : R× R+ → D such that

∂u

∂t+∂f(u)∂x

= 0 , (20)

u(x, 0) = φ(x) . (21)

Assume that u is a classical solution, which allows us to re-write the equations in non-conservativeform

∂u

∂t+ a(u)

∂u

∂x= 0 , (22)

where a(u) = f ′(u). (This form is called non-conservative, because the equation is not written asa generic conservation law, i.e. as (11)). The characteristic curve originating at x0 is defined bythe ordinary differential equation dx(t)

dt = a(u(x(t), t)) with x(0) = x0. We have not shown underwhat circumstances there is a classical solution, but even so one can state the following result:

Theorem 2.1. Assume u is a classical solution of the Cauchy problem (20), (21). Then thecharacteristics are straight lines

Proof. Since u is a classical solution, we can use the non-conservative formulation (22) to evaluatethe rate of change of the solution along the characteristic curve x(t) as

du(x(t), t)dt

=∂u(x(t), t)

∂t+ a(u(x(t), t))

∂u(x(t), t)∂x

= 0 . (23)

This means that u, is constant along the characteristics. Of course this means that a(u) is alsoconstant, which by defnition is the slope of the characteristics.

As an example consider eq. (20) for the particular case f = 12u

2 (i.e. the Burgers equation).The characteristic line originating at (x0, 0) is given by x(t) = x + φ(x0)t, and its slope is thusgiven by the initial value φ(x0).

1It remains to be seen how to handle multidimensional nonlinear systems, which in general cannot be fullydecoupled, because the Jacobian matrices Ai cannot be simultaneously diagonalized. We defer this question for themoment

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(a) Linear advection equation:Characteristics emanating from asample of points

(b) Linear Systems: Characteristicsemanating from a single point.

(c) The Burgers Equation: Charac-teristics emanating from a sample ofpoints.

Figure 1: Characteristics for one-dimensional hyperbolic equations

Fig. 1(c) illustrates the situation. It can be seen that a non-constant slope means that thecharacteristics may intersect. Obviously our notion of a classical solution, which may be determinedby transporting the initial solution along the characteristics suffers a severe crisis, because it wouldactually lead to multivalued solutions.

By considering the simplest nonlinear case (scalar, 1d) it already becomes obvious that ingeneral we cannot guarantee a unique classical solution for nonlinear hyperbolic PDE! We resortto a familiar concept: The notion of weak solutions.

2.4 Weak Solutions

Let u be a classical solution to (11), and integrate the equations a smooth test function v ∈C1

0 (Rd × R+)m

−∫

R+

∫Rd

(∂u(x, t)∂t

+d∑i=1

∂fi(u)∂xi

)v dxdt = 0 (24)

⇔∫

R+

∫Rd

(u∂v

∂t+

d∑i=1

fi∂v

∂xi

)dxdt−

∫Rd

v(x, 0)φ(x) dx = 0 . (25)

We take (25) as the definition of a weak solution, and note that no differentiability is requiredfrom u anymore. We can thus look for weak solutions in a very broad class of functions.

Definition 2.3. Let φ(x) ∈ L∞loc(Rd)m. We say u ∈ L∞loc(Rd × R+)m is a weak solution (11) if usatisfies (25) for any v ∈ C1

0 (Rd × R+)m.

Remark 2.2. It is clear that a classical solution to (11) is also a weak solution. Therefore, tosimplify the nomenclature, we henceforth only refer to weak solutions to (11), meaning that foru ∈ C1 eq. (11) is satisfied pointwise, and otherwise (25) holds.

This allows us to construct a solution in cases where the theory of characteristics fails. Weanticipate that we will be able to replace the region of intersecting characteristics with a discon-tinuity, as outlined in figure 2. The information contained in the weak form of the equations isactually enough to determine the position and form of the shock. We first note the followingdefinition:

Definition 2.4. We call a function piecewise smooth in the (x, t) plane if there exist a finitenumber of orientable surfaces with normal n = (nx1 , nx2 , . . . , nxd

, nt) outside of which the solutionis smooth, and across which the solution has a jump discontinuity.

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(a) Region of intersecting characteristics removed (b) A ”shock” instead of intersecting characteristics

Figure 2: Characteristics for one-dimensional hyperbolic equations

We have the following theorem:

Theorem 2.2. For any point (x,t) on a surface of discontinuity define u± = limε→0+ u((x, t)±εn),where n is the normal associated with the surface. Let u : Rd×R+ → D ⊂ Rm be piecewise C1 (asdefined above). Then u is a weak solution of (11) if and only if u is a classical solution in regionswhere u ∈ C1, and u satisfies

(u+ − u−)nt +d∑j=1

(fj(u+)− fj(u−))nxj= 0 (26)

across jump discontinuities. Eq. (26) is known as the Rankine-Hugoniot condition.

Proof. Let u be a weak solution to (11). Consider a surface of discontinuitiy S, and a controlvolume B that includes the discontinuity, such that the only discontinuity in B is given by S ∩ B,which divides B into two subdomains B− ∪B+ = B. Since u is a weak solution, it satisfies (25) forthe control volume B with appropriately chosen test functions (i.e. with compact support on B).Using the divergence theorem in regions B− and B+, where the solution is smooth, one obtains

−∫B−

(∂u(x, t)∂t

+d∑i=1

∂fi(u)∂xi

)v dxdt+

∫S∩B

v(u−nt +d∑i=1

fi(u−) · nxi) dA (27)

−∫B+

(∂u(x, t)∂t

+d∑i=1

∂fi(u)∂xi

)v dxdt−

∫S∩B

v(u+nt +d∑i=1

fi(u+)nxi) dA = 0 . (28)

The volume integrals vanish, because the weak solution is also a strong solution in smooth regions,while the remaining terms can be rearranged to give (26), using the argument that the controlvolume B is arbitrary, so the integrand must vanish. By reversing the calculation one sees imme-diately that conversely any piecewise C1 function that satiesfies (11) in smooth regions, and (26)across discontinuities is a weak solution.

It is easy to see that the class of weak solutions is too large to make the solution to (11) unique.

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(a) Smooth Solution (b) Discontinuous Solution

Figure 3: Examples of weak solutions to (29), (30) for ul < ur.

Consider, for example, the Cauchy problem for the Burgers Equation:

∂u

∂t+

12∂u2

∂x= 0 , (29)

u(x, 0) =ul , x < x0 ,ur , x > x0 .

(30)

It may be verified that for any ul 6= ur a discontinuous solution that includes a shock is a weaksolution that satisfies the Rankine-Hugoniot condition if

u(x, t) =ul , x < x0 + st ,ur , x > x0 + st ,

(31)

where s = 12 (ul + ur). On the other hand it may be seen that for ul < ur a continuous solution

to (29), (30) can be constructed:

u(x, t) =

ul , x− x0 < ult ,x/t , ult ≤ x− x0 ≤ urt ,ur , x− x0 > urt .

(32)

Both solutions, which are illustrated in fig. (3) are prefectly acceptable weak solutions to the sameproblem! Obviously then, weak solutions are not unique. This should serve as motivation toconsider additional conditions to identify meaningful solutions.

2.5 Entropy Solutions

To motivate the concept of entropy solutions, recall that in gas dynamics only such shocks areobserved, across which the entropy increases. This is the case for compression shocks, i.e. suchshocks across which the normal velocity is decreased from supersonic to subsonic speeds. On theother hand, expansion shocks are never observed. In contrast, the entropy is a conserved quantityin smooth regions. One attempts to utilitze similar mathematical notions of entropy to narrowdown range of admissible weak solutions.

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Definition 2.5. Consider a weak solution of (11) u : Rd × R+ → D ⊂ Rm. A convex functionU : D → R is called an entropy of the system (11) if there exist d functions Fi : D → R, i = 1, . . . , d,such that

∇TU(u)Dufi(u) = ∇TFi(u) , i = 1, . . . , d , (33)

where all gradients are with respect to u. (Dufi(u) is the Jacobian matrix Ai.)

Note that finding an entropy function is in general not trivial, as eq. (33) presents a systemof m × d partial differential equations for the (d + 1) unknowns U and Fi. In the scalar case,however, any convex function is an entropy, upon taking for the flux Fi any primitive of thefunction U ′(u)f ′i(u).

Equipped with an entropy function we may define the entropy solution as follows:

Definition 2.6. A weak solution u to (11) is called an entropy solution if u satisfies for all entropyfunctions U(u) of (11)∫

R+

∫Rd

(U(u)

∂v

∂t+

d∑i=1

Fi∂v

∂xi

)dxdt−

∫Rd

v(x, 0)U(φ(x)) dx = 0 , ∀v ∈ C10(Rd×R+) , v ≥ 0 .

(34)

It is straightforward to show that u is an entropy solution to (11) if

• u is a classical solution that satisfies (11) pointwise in regions where u ∈ C1,

• u satisfies the Rankine-Hugoniot shock conditions (26),

• u satisfies the entropy jump conditions,

nt (U(u+)− U(u−)) +d∑i=1

nxi(Fi(u+)− Fi(u−)) ≤ 0 . (35)

Note in particular that in smooth regions where we can use classical solutions, the entropy satisfiesa conservation law pointwise, i.e. ∂tU +

∑i ∂xiFi = 0.

Example 2.2. The function U = −ρs is an entropy for the Euler equations, where s is pro-portional to the thermodynamic entropy function, i.e. s ∝ log(p/ργ). It is known from inviscidfluid mechanics that the etnropy is a conserved quantity in smooth regions (it is constant alongstreamlines), but increases across compression shocks.

Example 2.3. For the 1D Burgers equations one may choose any convex function, for examleU(u) = u2. From the entropy flux conditions we thus find Fi(u) = 2

3u3. After introducing the

parametrization (t, ξ(t)) for the surface of discontinuity we write for the shock speed s = dξdt .

We have for the shock normal n = (nx, nt)T = (1,−s)T , and from the Rankine-Hugoniot shockconditions we find s = 1

2 (ul + ur). The entropy shock conditions then give

s(u2+ − u2

−)− 23(u3

+ − u3−)≥ 0 , (36)

⇔ 12

(u+ + u−)(u2+ − u2

−)− 23(u3

+ − u3−)≥ 0 , (37)

⇔ 16

(u− − u+)3 ≥ 0 . (38)

This means that we allow shocks only for u− > u+, i.e. we allow only discontinuous solutionswhenever characteristics intersect as a consequence of a compression as in figure 2(b) while con-structing a smooth solution as in (32) for diverging characteristics, i.e. ”expansions”, see fig. 3(a).We would reject the solution (31) depicted in fig. 3(b).

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It may be shown that entropy solutions are unique in the scalar case. For nonlinear systemsthis has not yet been proven [5], although it is conjectured. However, the complete mathematicalframework of entropy solutions for multidimensional nonlinear systems is beyond the scope ofthis introductory text. It should be mentioned, however, that such solutions can be found byconsidering ”nearly” hyperbolic systems with a small amount of viscosity

∂uε(x, t)∂t

+d∑i=1

∂fi(uε)∂xi

= ε∇2uε . (39)

It may be shown that if uε → u as ε→ 0+, then u is an entropy solution to (11). For our purposesit is sufficient to define the entropy solution as the solution that is obtained from eq. (39) in thelimit as ε→ 0.

3 Finite-Volume Methods for Scalar Nonlinear Conserva-tion Laws

We derive the classical finite volume method as a special case of the Discontinuous GalerkinMethod, which has already been discussed for the linear advection-diffusion equation [3]. Weconsider scalar nonlinear equations, i.e. find u : Rd × R+ → R such that

∂u

∂t+

d∑i=1

∂fi(u)∂xi

= 0 , u(x, 0) = φ(x) . (40)

Generally we seek entropy weak solutions, for which we assume φ ∈ L∞(Rd).Consider a tessalation Th consisting of shape-regular polyhedral elements T , which are used as

control volumes to define the weak form the equations in the usual way. Define the scalar productsover the control volumes, ∫

T

uv = (u, v)T ,

∫∂T

uv = 〈u, v〉∂T , (41)

and consider the weak form of (40) obtained by integrating against a smooth test function v overa mesh element T , i.e.

(∂tu, v)T −d∑i=1

(fi(u), ∂xiv)T +

d∑i=1

〈fi(u)ni, v〉∂T = 0 . (42)

The DG approximation is then to find uh ∈ Vkh =uh ∈ L2

loc(Rd) : uh|T ∈ Pk(T )

such that

(∂tuh, vh)T −d∑i=1

(fi(uh), ∂xivh)T +

⟨h, vh

⟩∂T

= 0 , ∀vh ∈ Vkh , (43)

where h is a numerical flux function that approximates f(u) · n on ∂T . We write the DG approxi-mation in generic form as

N (uh, vh) = 0 , ∀vh ∈ Vkh . (44)

The operator N is nonlinear, which complicates the analysis of the method enormously, sincethe standard methods of showing boundedness, coercivity (or inf-sup stability) for bilinear formsare not applicable. For the approximation to nonlinear conservation laws, the numerical flux is

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the essential ingredient, as the flux alone determines the physics of the problem. The analysisof numerical schemes for hyperbolic conservation laws has historically focused very much on thenumerical flux. The extension of this formulation to systems of equations is straightforward, in thatvector-valued test functions may be used in a very similar manner. The numerical flux functionsand the analysis, however, are more involved for the systems case. We discuss the extension tononlinear systems with emphasis on numerical flux functions for the Euler Equations in a latersection.

Consider the special case V0h =

uh ∈ L2

loc(Ω) : uh|T ∈ P0(T )

. This means the approximationis piecewise constant. In this case one may write the DG basis in explicit form as

uh =∑i

uiχ(Ti) , (45)

where χ(Ti) is the characteristic function defined on the element Ti:

χ(Ti) =

1 , x ∈ Ti0 , otherwise . (46)

The second term in (43) vanishes for piecewise constant test functions. If the integrals are approx-imated with a one-point integration rule, we may write for a mesh element Ti:

|Ti|duidt

+∑

eik∈∂Ti

h(ui, uk;nik)Sik = 0 , (47)

where eik is the edge separating element Ti from Tk, and nik and Sik are the outward pointingnormal and face area associated with ∂Ti, respectively. This is the classical finite-volume schemein semi-discrete form, i.e. eq. (47) is a system of nonlinear ordinary differential equations (ODE).

Remark 3.1. Historically finite-volume schemes have been derived directly from integral conser-vation laws, independently of Discontinuous Galerkin Schemes, by approximating the unknownfunction by volume avarages for each mesh element. This has given rise to the name. FiniteVolume schemes are only identical to DG schemes for the case p = 0.

The simplest fully discrete finite-volume scheme is obtained by using a forward Euler discretiza-tion in time, i.e. d

dtui ≈un+1

i −uni

∆t , where uni denotes the numerical approximation of the cell avarageat time tn, which leads to

un+1i = uni −

∆t|Ti|

∑eik∈∂Ti

h(uni , unk ;nik)Sik = 0 , (48)

where for simplicity we assume a constant time step tn = n∆t.In praparation to analyzing the finite-volume schemes we note two basic consistency properties

that we henceforth assume the numerical flux functions satisfy.

Definition 3.1. A numerical flux function h is called conservative if it satisfies

h(u, v;n) = −h(v, u;−n) . (49)

Definition 3.1 ensures that flux contributions from interior faces cancel in the summation ofeq. (48) over all mesh elements. This is a discrete analog of the property of the integral conservationlaws, which require that the rate of change for arbitrary control volumes be given by the flux overthe boundary.

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Definition 3.2. We call a numerical flux function h consistent if it satisfies

h(u, u;n) = f(u) · n (50)

Definition 3.2 expresses the obvious consistency requirement, that the numerical flux be identi-cal to the actual flux function of the conservation law for smooth input. (In particular this is truefor the smooth exact solution!) In the following we always assume that the numerical flux func-tions are conservative and consistent. A third property is crucial for the analysis of finite-volumeschemes: The monotonicity property. This was inspired by the classical analysis of schemes fornonlinear 1D conservation laws. i.e. (20), (21), due to Harten et al. [6]. Note first the followingdefinition for monotonicity

Definition 3.3. Consider a 1D finite-volume discretization of (20), (21) on a uniformly spacedmesh in time and space such that

un+1i = H(ui+k, . . . , ui, . . . , ui−k) (51)

= uni −∆t∆x

(hi+ 1

2− hi− 1

2

), (52)

where hi+ 12

= h(ui+k, . . . , ui, . . . ui−k+1) is a consistent numerical flux. The scheme is calledmonotone if

∂Hi

∂ui+l≥ 0 , ∀|l| ≤ k (53)

The importance of monotone schemes is summarized in the following theorem

Theorem 3.1. Consider a 1-D monotone scheme as in Definition 3.3. Then uni converges bound-edly to u(x, t) as ∆x→ 0 and ∆t→ 0 with ∆t/∆x = const, where the limit function u(x, t) is anentropy satisfying weak solution of (20), (21).

The monotonicity condition (53) motivates the following definition for the semi-discrete schemein higher dimensions

Definition 3.4. We call a finite volume scheme of the form (47) monotone if the numericalflux function h(u, v;n) is conservative and consistent, in addition is Lipschitz continuous in bothinput arguments, and is nondecreasing / nonincreasing in the first and second input argument,respectively. In the case of differentiability we have

∂h

∂u≥ 0 ,

∂h

∂v≤ 0 . (54)

Equivalently we sometimes say that the scheme employs a monotone numerical flux function toemphasize the fact that monotonicity thus defined is a property of the space discretization, i.e. thesemidiscrete ODE.

It is easily seen that monotone flux functions are a necessary condition for the scheme (51),when written in the form (52) to satisfy the monotonicity condition (53). Equally easily it isseen however, that the discrete form (52) needs an additional constraint, which we introduce asa time step restriction. Before going to the fully discrete scheme, we note the consequence of the

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monotinicity definition can be seen by writing scheme (47) as follows:

duidt

= − 1|Ti|

∑eik∈∂Ti

h(ui, uk;n)Sik

= − 1|Ti|

∑eik∈∂Ti

h(ui, uk;n)− f(ui) · nuk − ui

(uk − ui)Sik

= − 1|Ti|

∑eik∈∂Ti

∂h

∂uk(ui, uik)(uk − ui)Sik

=1|Ti|

∑eik∈∂Ti

Cik(uh)(uk − ui) , (55)

where uik ∈ [ui, uk] 2. Definition 3.4 ensures that all the operations are well defined, and further-more that Cik ≥ 0. This has the immediate consequence that the ODE (47) has the property thatlocal maxima/minima cannot increase/decrease. This is an essential step in showing stability inL∞ norm for the discrete scheme for which we augment the monotoniciy condition with a time-stepconstraint. One has the following theorem

Theorem 3.2. Assume a discrete finite volume scheme of the form (48) with monotone fluxes.Assume that the time step restriction

1− ∆t|Ti|

∑eik∈∂Ti

Cik(unh) ≥ 0 (56)

holds for each time increment [tn, tn+1], where the coefficients Cik are defined in (55). Then thescheme is L∞-stable, and the following estimate holds

infx∈Rd

u(x, 0) ≤ uni ≤ supx∈Rd

u(x, 0) , ∀(Ti, tn) . (57)

Proof. Using the constraint (56) in the discrete form of (55) shows that in each time increment[tn, tn+1] a local maximum principle is verified. Upon applying the argument recursively up to theinitial conditions, one immediately finds (57), which asserts L∞ stability.

Very few ingredients are thus needed in order to establish nonlinear stability results of the finitevolume scheme on general triangulations. Unfortunately, due to the nonlinearity of the equations,establishing convergence to the entropy weak solution and pertaining convergence rates is muchmore difficult. It has to be stated that no general equivalence theorem between consistency andstability on the one hand, and convergence on the other hand applies in the nonlinear case. Whilethe details are beyond the scope of these lectures (see [1, 4] for more in-depth analysis), it turns outthat for monotone finite volume schemes convergence to the entropy solution can be established,as stated by the following theorem:

Theorem 3.3. Consider the finite-volume approximation (48) using monotone fluxes and assumethe time step restriction (56) holds. Assume a sequence of solutions (um)m∈N obtained by letting∆x and ∆t tend to zero for m → 0. Then um converges strongly to the entropy weak solutionof (40) in Lploc(Rd × R+) for all p ∈ [1,∞).

2Note that differentiability of the flux is assumed. Using the monotonicity and the Lipschitz constant in placeof the derivate the statement also holds for non-differentiable fluxes.

12

Unfortunately, the error estimates for the scheme are (yet) more complicated. In addition tothe assumptions made thus far, one needs an extra assumption, which can be stated that the initialdata u(x, 0) = φ(x) has bounded total variation. Formally the space of functions with boundedtotal variation is defined as follows

Definition 3.5. The space of functions with bounded total variation is defined as

BV =g ∈ L1(Rd) : |g|BV <∞

, (58)

where

|g|BV = sup∫

Rd

g∇ · φdx : φ ∈ C10 (Rd)d , ||φ||∞ ≤ 1

. (59)

Note that this definition in the weak form ensures that BV is defined for non-differentiablefunctions. For smooth functions this definition ensures that any linear combination (with coeffi-cients bounded by 1) of the partial derivatives stays bounded. Thus we see that in practice thisadditional requirement on u(x, 0) does not impose unreasonable restrictions.

Theorem 3.4. Assume u(x, 0) ∈ BV (Rd)∩L∞(Rd), and let uh be the approximation given by thefinite volume scheme (48) with monotone fluxes and time step restriction (56). Then there existsC ≥ 0 such that

||u− uh||L1(K) ≤ Ch1/4, (60)

for any K ⊂ Rd × R+, where u is the entropy weak solution to (40). For the case d = 1 one canimprove the result in that

||u− uh||L1(K) ≤ Ch1/2 (61)

holds.

4 Numerical Flux Functions

For hyperbolic equations in general, and for the Euler equations in particular, the concept of up-winding plays an important role in defining numerical flux functions. Upwinding may be looselyinterpreted as adding a directional bias to spatial discretization, so as to respect the flow of infor-mation determined by the characteristics. To motivate this recall some basic concepts from thediscretization of the one-dimensional linear advection equation:

4.1 Discretization of the Linear Advection Equation

Consider the linear advection equation (13). We may write a conservative discretization of theform (52) as

un+1i = uni −

∆t∆x

(hi+ 1

2− hi− 1

2

), (62)

with hi+ 12

a consistent and conservative approximation of the flux f(u) = cu. For c ∈ R one maychoose

hi+ 12

=

cui c > 0cui+1 c < 0 . (63)

Using f(u) = cu the numerical flux may be written equivalently as

hi+ 12

=12(f(uni+1) + f(uni )

)− |c|

2(uni+1 − uni

)(64)

=c

2(uni+1 + uni

)− |c|

2(uni+1 − uni

). (65)

13

(a) Discretization for c > 0 (b) Discretization for c < 0

Figure 4: Stencil for stable discretization of the linear advection equation, as in eqns. (62) and (63).The dashed lines indicate the characteristic lines, as determined by x(t) = x0 + ct.

Consider the case c > 0. Then the discrete scheme may be written

un+1i = (1− C)uni + Cuni−1 , (66)

where C = c∆t∆x . From definition 3.3 we infer that the scheme is monotone, provided all the

coefficients are positive, which is ensured as long as 0 ≤ C ≤ 1. This is sufficient for stability asseen previously. One may also show, using e.g. Neumann analysis, that it is necessary. We callsuch effective time-step restrictions a CFL condition, following the classical analysis by Courant,Friedrichs, and Levy [2].

Recall that for the scalar case the characteristic directions are given by x = f ′(u). For the linearadvection equation (13) this means that that the characteristic lines are given by x(t) = x0 + ct.These define a domain of dependence, meaning that each point (x, t) can only be influenced bya region spanned by the triangle (x, t), (x, 0), (x0 = x − ct, 0). A graphical interpretation of theCFL condition can thus be given in terms of the characteristics, see fig. 4. The CFL conditionimplies that the numerical discretization stencil, i.e. the numerical domain of dependence, mustinclude the physical domain of dependence, given by the characteristics. This concept explainsmany discretization techniques employed for the Euler equations. Before considering the Eulerequations, we extend the concept to linear systems of equations.

4.2 Numerical Fluxes for Linear 1D Systems

Consider a linear system, i.e. find u : R×R+ → Rm, such that eq. (16) holds. Recall that a linear1D hyperbolic systems can be globaly decoupled. It is easy to see that by defining the numericalflux as

hi+ 12

=12(f(uni+1) + f(uni )

)− |A|

2(uni+1 − uni

), (67)

where f(u) = Au, and

|A| = R−1|Λ|R , Λ = diag(|λ1|, . . . , |λi|, . . . , |λm|) , (68)

the system may be reduced to m independent scalar equations

w(j),n+1i = w

(j),ni − ∆t

∆x

(h

(j)

i+ 12− h(j)

i− 12

), j = 1, . . . ,m , (69)

14

where

h(j)

i+ 12

=λj2

(w

(j),ni+1 + w

(j),ni

)− |λj |

2

(w

(j),ni+1 − w

(j),ni

). (70)

This is identical to (65), with c replaced by λj . Note that the upwinding is thus applied to thecharacteristic variables, where the characteristics are defined by the Eigenvalues of the Jacobian, i.e.the jth characteristic curve is given by x(j) = λj . (see fig. 1(b) for an illustration ofm characteristicsemanating from a single point.) Eeach characteristic field is thus upwinded individually. Stabilityfor the system thus follows directly from stability for the scalar case, if we take

∆t ≤ ∆xmaxj=1,...,m |λj |

. (71)

This means that the time step is taken such that all characterstics are included in the domain ofdependence. In particular it is seen that discontinuous initial data leads to discontinuities in eachcharacteristic field j, which will be transported with the ”speed” λj , according to the theory ofcharacteristics. Thus the system produces a fan of discontinuities with jumps across each of thecharacteristic lines.

4.3 Numerical Fluxes for the Euler Equations

Unfortunately multidimensional systems cannot be simultaneously diagonalized, so that in generalthe reduction to scalar equations is impossible. However, by definition of hyperbolic systems, anylinear combination of the Jacobian may be diagonalized. In section 3 we have established that onlythe projection of the fluxes, i.e. f(u) ·n, into faces of polyhedral control volumes, are of interest forboth finite-volume and DG methods. This allows locally a diagonalization. Since discretizing thenormal flux is essentially a one-dimensional problem we consider a one-dimensional model problemto illustrate the approach

Recall the flux functions for the Euler Equations, (12). We consider the full 3D Euler statevector, however, applied to the one-dimensional case d = 1. This is equivalent to assuming thatan interface in a control volume is perpendicular to the x-direction. For this case we have for theflux Jacobian Duf(u)

A =

0 1 0 0 0

γq2 − u2 (3− γ)u −γv −γw γ−uv v u 0 0uw w 0 u 0

(γq2 − h)u h− γu2 −γuv −γuw γu

, (72)

where γ = γ − 1, q2 = 12 (u2 + v2 + w2), and h = (E + p)/ρ is the total enthalpy. It may be

shown that under restrictions consistent with the physics of the problem (p > 0, ρ > 0) the Eulerequations are always hyperbolic. The eigenstructure may be written 3

Λ = diag (u− c, u, u, u, u+ c) , (73)

K =

1 1 0 0 1

u− c u 0 0 u+ cv v 1 0 vw w 0 1 w

h− uc q2 v w h+ uc

, K−1 =γ

2c2

h+ c(u− c) −(u+ c) −v −w 1−2h+ 4cc 2u 2v 2w −2−2vcc 0 2cc 0 0−2wcc 0 0 2cc 0

h− c(u+ c) −u+ c −v −w 1

,

(74)3Note that there are some degress of freedom in choosing eigenvectors, as there are repeated eigenvalues.

15

where c = c/γ. We wish to produce a numerical flux function as in (67), so as to locally decouplethe system. A suitable and consistent linearization of the Flux Jacobian, as a function of thetwo input values A = A(ui, ui+1), is needed, such that the system can be diagonalized. This istantamount to locally approximating the equations with the linear system

∂u

∂t+ A

∂u

∂x= 0 , where A = A(ui, ui+1) . (75)

Note that in principle any consistent linearization may be used. Simple approximations that cometo mind are A(ui, ui+1) = 1

2 (A(ui) + A(ui+1)), or A(ui, ui+1) = A( 12 (ui + ui+1)). Following Roe,

a particular linearization can be obtained, which was derived to satisfy the following properties

1. the linearized system is hyperbolic

2. A(u, u) = A(u)

3. for any ui, uk we have A(ui, uk)(ui − uk) = f(ui)− f(uk)

Proposition 4.1. Property 3 implies that the shock speed of an isolated discontinuity is an eigen-value of the linearized matrix A, and the state difference is an eigenvector.

Proof. Parametrize the surface of discontinuity as in example 2.3, yielding a shock speed S, andcombine the Rankine-Hugoniot condition with property 3. This yields

f(ui)− f(uk) = S(ui − uk) = A(ui, uk)(ui − uk) (76)

The second equation shows that S is an eigenvalue of the matrix A, and the state difference is aneigenvector.

The significance of this property is that isolated discontinuities that are connected by the RankineHugoniot condition are recognized as such by the linearized equations. Recall that the solution forthe linear Cauchy problem is given by solving the decoupled scalar equations in the characteristicvariables and transforming back to the conserved variables, i.e. u = Kw. In particular for initialdata

u(x, 0) =

ul , x < 0ur , x > 0 (77)

we have for t ≥ 0 that the initial jump degenerates into a fan of smaller jumps, because eachcharacteristic follows its own wavespeed λj . (See also fig. 1(b)). For the total jump, expressed inthe characteristic variables, one has

ul − ur = ∆u = K∆w =m∑j=1

∆w(j)kj , (78)

where kj are the eigenvectors. If ∆u, however, is an eigenvector of A, up to a constant we have∆u = αk(j) for some j. Since the eigenvectors are linearly independent, this means that ∆w(i) = 0for i 6= j. This means that the linearization recognizes the jump as an isolated discontinuity withcorrect shock speed S.

Remark 4.1. Note that property 3 also means that the numerical flux reduces to pure upwindingfor supersonic flow, for which all the eigenvalues have the same sign. This means

hi+ 12

=

f(ui) , λi > 0 ,∀if(ui+1) , λi < 0 ,∀i (79)

16

The particular construction due to Roe uses a parameter vector q to construct the matrix A. Itcan be shown by a straightforward computation that every element of u, F is a quadratic functionof a parameter vector q, which is given by

q =√ρ(1, u, v, w, h)T . (80)

The usefulness of this observation is revealed by noting that for arbitrary scalar r and s we have∆(rs) = r∆s + s∆r, where the overbar denotes a simple arithmetic average. This make it veryeasy to calculate ∆F = C∆q, and ∆u = B∆q. It then follows that for the Roe matrix ARoe with∆F = ARoe∆u we have ARoe = (CB−1), where

B =

2q1 0 0 0 0q2 q1 0 0 0q3 0 q1 0 0q4 0 0 q1 0q5γ

γ−1γ q2

γ−1γ q3

γ−1γ q4

q1γ

, (81)

C =

q2 q1 0 0 0

γ−1γ q5

γ+1γ q2

γ−1γ q3

γ−1γ q4

γ−1γ q1

0 q3 q2 0 00 q4 0 q2 00 q5 0 0 q2

(82)

It may be shown that the linearized matrix thus obtained is equivalent to evaluating the fluxJacobian with averaged values

φ =√ρiφi +

√ρkφk√

ρi +√ρk

, (83)

where φ is any of the quantities u, v, w, h. The speed of sound may then by calcuated as

c =

√(γ − 1)(h− 1

2(u2 + v2 + w2)) . (84)

The eigenvalues are given byΛ = diag (u− c, u, u, u, u+ c) . (85)

For d > 1 one replaces the Jacobian A in eq. (72) with An =∑Aini. The Roe-linearization

of An can be accomplished using the same averages as before, i.e. (83) (the parameter vector (80)works for all flux jacobians Ai), producing the Roe matrix ARoe(uk, ui;n). This means that thenumerical flux function at a face of a polyhedral control volume, separating cells i and k may bewritten

h(ui, uk;n) =12

(f(ui) + f(uk)) · n− 12|ARoe(uk, ui;n)| (uk − ui) , (86)

where the eigenvalues of the matrix ARoe are given by

Λ = diag (un − c, un, un, un, un + c) , (87)

where un = u · n is the normal velocity.

References

[1] Timothy Barth and Mario Ohlberger. Finite volume methods: Foundation and analysis. InErwin Stein, Rene de Borst, and Thomas J. R. Hughes, editors, Encyclopedia of ComputationalMechanics, Part 1: Fundamentals, chapter 15. Wiley, 2004.

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[2] R. Courant, K. Friedrichs, and H. Lewy. Uber die partiellen Differenzengleichungen der math-ematischen Physik. Math. Ann., 100(1):32–74, 1928.

[3] H. Egger. Finite element methods for convection diffusion reaction problems. Advanced FEMLecture Notes, http://www.mathcces.rwth-aachen.de/doku.php/teaching/afem08, 2008.

[4] Robert Eymard, Thierry Gallouet, and Raphaele Herbin. Finite volume methods. In P. G. Cia-rlet and J. L. Lions, editors, Handbook of Numerical Analysis, volume 7, pages 713–1020. NorthHolland, updated manuscript, http://www.cmi.univ-mrs.fr/ herbin/PUBLI/bookevol.pdf, Oc-tober 2006.

[5] Edwige Godlewski and Pierre-Arnaud Raviart. Numerical Approximation of Hyperbolic Systemsof Conservation Laws. Springer Verlag, 1996.

[6] A. Harten, J. M. Hyman, and P. D. Lax. On finite-difference approximations and entropyconditions for shocks. Comm. Pure Appl. Math., 29:297–322, 1976.

[7] J. Schoberl. Numerical methods for PDE. Lecture Notes, RWTH Aachen,http://www.mathcces.rwth-aachen.de/doku.php/teaching/afem08, 2007.

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