2 Conservative schemes

PW #2: Burgers equation

Given a scalar conservation law, there exist several schemes to solve the equation. But the solutions provided by these algorithms are not equivalent. For instance, it is relevant to compare accuracy on equiv- alent meshes, computational costs and time, stability or consistency conditions, and so on. This work is thus devoted to comparing some numerical schemes when solving theBurgersequation:

∂tu+∂xF(u)=0, (1a)

with:

F(u)=u²

2 . (1b)

This problem is restricted to theperiodic boundary condition casein the present work.

Let us introduce the homogeneous cartesian meshx_j = j∆xfor a given mesh size∆x. The natural framework for solving Syst. (1) consists in defining cells of whichxjis the middle. Consequently, we note x_j+1/2 =¡

j+¹₂¢

∆x such thatxj is the middle ofIj =]xj−1/2,x_j+1/2[. Likewise, the time discretization is denoted bytⁿ=n∆t,∆t>0. The approximation of the solutionuatxj andtⁿisuⁿ_j (the unknowns are located in the center of the cells).

1 Non-conservative schemes

Provided that the solutionuis smooth enough, Eq. (1a) may be rewritten as:

∂tu+u∂xu=0. (2)

On the one hand, given this new formulation, we can derive a natural scheme aiming at solving Eq. (2):

uⁿ_j⁺¹−uⁿ_j

∆t +uⁿ_j

uⁿ_j −uⁿ_j−1

∆x =0, if uⁿ_j ≥0, (3a) uⁿ⁺¹_j −uⁿ_j

∆t +uⁿ_j

uⁿ_j+1−uⁿ_j

∆x =0, if uⁿ_j <0. (3b) This scheme will be referred to as theupwind non-conservative scheme.

On the other hand, Eq. (2) may be considered as a transport equation for which the velocity field is unknown. Themethod of characteristics(MOC) provides a classical scheme for this kind of equations. It consists in tracking particules in their motion due to the velocity field (characteristic curves) in order to keep track of relevant values. More precisely, we introduce the characteristic flow as the solution to the

ODE: dX

dt =u¡

t,X(t;s,x)¢

, X(s;s,x)=x.

This flow represents the position at time t of a particule located in x at times. Under smoothness as- sumptions onu, the existence (and uniqueness) of the solutionX is guaranteed by the Cauchy-Lipschitz

The MOC relies on the following remark. Ifuis a solution to Syst. (1), then:

d dt

£u¡

t,X(t;s,x)¢¤

=0.

This means thatuis constant along characteristic curves:

u¡

t,X(t;s,x)¢

=u(s,x).

This leads to the scheme (takingt=tⁿ,s=tⁿ⁺¹andx=xj):

uⁿ_j⁺¹=uˆⁿ(ξⁿ_j), ξⁿ_j ≈X(tⁿ;tⁿ⁺¹,x_j). (4a) This scheme hence consists in two steps:

• Time step: resolution of the ODE forX on [tⁿ,tⁿ⁺¹]. A 1st-order approximation is:

ξⁿ_j =xj−uⁿ_j∆t. (4b)

• Space step: after localizingξⁿ_j in the interval [x_k,x_k+1), this step is based on a spatial interpolation using valuesuⁿ_k anduⁿ_k+1. More precisely:

uⁿ⁺¹_j =

xk+1−ξⁿ_j

∆x uⁿ_k+1+ξⁿ_j−xk

∆x uⁿ_k. (4c)

This scheme is nonlinear but unconditionally stable.

2 Conservative schemes

Among schemes solving Syst. (1), there exists a class called “conservative schemes” due to aL¹-norm con- servation principle. They read:

uⁿ⁺¹_j −uⁿ_j

∆t +

Fⁿ_j+1/2−F_jⁿ_−1/2

∆x =0, (5a)

whereF_j+1/2ⁿ is the numerical flux. This term comes from the finite volume framework. Indeed, when integrating Eq. (1a) over [tⁿ,tⁿ⁺¹]×[x_j−1/2,x_j+1/2], we get:

Uⁿ⁺¹_j −Uⁿ_j + Z tⁿ⁺¹

tⁿ

£F¡

u(t,x_j+1/2)¢

−F¡

u(t,x_j−1/2)¢¤

dt=0,

where the fluxes at the boundary of the cells appear. We get intersted in the sequel in numerical fluxes of the type:

Fⁿ_j+1/2=F(uⁿ_j,uⁿ_j+1), (5b) for every fluxFsatisfying the consistency conditionF(u,u)=F(u).

Here are several possible fluxes:

• Upwind conservative scheme:

F_upw(uⁿ_j,uⁿ_j+1)=







F(uⁿ_j) ifuⁿ_j ≥0, F(uⁿ_j+1) ifuⁿ_j <0.

• Lax-Friedrichs:

F_LF(uⁿ_j,uⁿ_j+1)=1 2

¡F(uⁿ_j)+F(uⁿ_j+1)¢

− ∆t 2∆x

¡uⁿ_j+1−uⁿ_j¢ .

• Richtmyer-Lax-Wendroff:

F_LW(uⁿ_j,uⁿ_j+1)=F(u^n+1/2_j+1/2), whereu^n+1/2_j+1/2 =1 2

¡uⁿ_j+1+uⁿ_j¢

− ∆t 2∆x

¡F(uⁿ_j+1)−F(uⁿ_j)¢ .

• MacCormack:

F_McC(uⁿ_j,uⁿ_j+1)=1 2

hF(uⁿ_j+1)+F(u^n+1/2_j )i

, whereu^n+1/2_j =uⁿ_j−∆t

∆x

¡F(uⁿ_j+1)−F(uⁿ_j)¢ .

• Godunov: the idea of the Godunov scheme relies on a piecewise constant approximation over cells.

More precisely, we consider at timetⁿthe function ˆuⁿsuch that ˆuⁿ(tⁿ,x)=Uⁿ_j ifx∈I_j.U_jⁿdenotes the mean value of the approached solution ˆuⁿ at timetⁿ over the cellIj. To progress in time, we introduce Riemann problems at each interfacex_j+1/2. Letwbe the solution to the Riemann model problem:

∂tw+∂xF(w)=0, w(0,x;uL,uR)=







uL ifx<0, uR ifx>0.

Integrating Eq. (1a) over [tⁿ,tⁿ⁺¹]×Ij, the Godunov scheme writes as (5a) with:

F(uⁿ_j,uⁿ_j+1)= 1

∆t Z tⁿ⁺¹

tⁿ F¡ ˆ

uⁿ(t,xj+1/2)¢

dt=F¡ ˆ

uⁿ(tⁿ⁺¹,xj+1/2)¢ ,

for ˆuⁿ(tⁿ⁺¹,x_j+1/2)=w(∆t, 0;Uⁿ_j,Uⁿ_j+1). For theBurgersequation, the solution to the Riemann prob- lem is a similarity solution and readsw(t,x;uL,uR)=wˆ³x

t;uL,uR

´ where:

w(ξ;ˆ u_L,u_R)=











uL ifξ≤uL, ξ ifuL<ξ<uR, uR ifξ≥uR, ifuL≤uRand:

ˆ

w(ξ;uL,uR)=







uL ifξ≤σ, u_R ifξ>σ,

3 Kinetic scheme

A scheme may be obtained by considering Syst. (1) as the limit of the following kinetic equation when ε→0:

∂tf +a(v)∂xf =1 ε

¡χu(v)−f¢

, (6a)

with:

a(v)=F⁰(v). (6b)

The functionχu(v) is defined by:

χu(v)=











1 if 0<v<u,

−1 ifu<v<0, 0 otherwise,

(7)

and the unknows of Syst. (1) and Syst. (6) are linked by the relationship:

u(x,t)= Z

Rf(x,v,t)d v. (8)

We use a spliting scheme to approach the solution of Syst. (6):

• Step 1: Linear transport step

∂tf +a(v)∂xf =0. (9)

We consider here an upwind scheme:

fⁿ⁺

1 2

j (v)=f_jⁿ(v)−∆t

∆x

ha⁺(v)³

f_jⁿ(v)−f_j−1ⁿ (v)´

−a⁻(v)³

f_j+1ⁿ (v)−f_jⁿ(v)´i

(10) wherea⁺(v)=max(a(v), 0) anda⁻(v)=max(−a(v), 0).

• Step 2: Collision step

∂tf =1 ε

¡χu−f¢

. (11)

Whenε→0, this step reduces to f =χuand thus, the second step of the scheme consists in solving:

f_jⁿ⁺¹(v)=χ

uⁿ⁺

1 2 j

(v), (12)

whereuⁿ⁺

1 2

j =

Z

Rfⁿ⁺

1 2

j (v)d v. Noting thatuⁿ⁺¹_j =

Z

Rf_jⁿ⁺¹(v)d v = Z

Rχ

uⁿ⁺

12 j

(v)d v=uⁿ⁺

1 2

j , we can now return to variables of interest, that isuⁿ_j. Integrating Eq. (10) with respect tov∈Ryields an equation of the form (5a) where the fluxF_jⁿ_+1/2 is defined by:

F_j+1/2ⁿ = Z

Ra⁺(v)χuⁿ_j−a⁻(v)χuⁿ_j+1d v.

4 Directions

You may consider the periodic domain [0, 2] and three initial conditions for theBurgersequation:

• u⁰₁(x)=x;

• u⁰₂(x)=1[1/4,3/4](x);

• u⁰₃(x)=exp

µ−(x−1/2)² x(1−x)

¶

1(0,1)(x).

You may implement 1 non-conservative scheme, 3 conservative schemes and the kinetic scheme. In each case, try to highlight theCFL condition.

Eventually, compare the solutions to Syst. (1) and the ones to:

∂tv+∂xFˆ(v)=0,

withv=u²and ˆF(v)=²₃v^3/2. These equations are equivalent in the smooth case.

You may deliver your results electronically at latest onFebruary 6, 2011.

([email protected]@inria.fr)