Technische Universitat Munchen

Godunov/Roe Fluxes

Oliver Meister

May 20th 2015

Oliver Meister: Godunov/Roe Fluxes

, May 20th 2015 1

Technische Universitat Munchen

Integral form of Conservation Law

Start point: Integral form of a general conservation law in 1D:For any x1, x2, t1, t2 ∈ R the quantity q is conserved if:

x2∫x1

q(x , t2)− q(x , t1) dx +

t2∫t1

f (q(x2, t))− f (q(x1, t))dt = 0 (1)

where:• x , t ∈ R: space and time variables• q : R× R→ Rd : conserved quantity vector

(function over space and time)• f : Rd → Rd : flux function

Oliver Meister: Godunov/Roe Fluxes

, May 20th 2015 2

Technische Universitat Munchen

FV discretization

Finite Volumes: subdivision of domain into grid cells, eachdenoted by a range Ci := (xi− 1

2, xi+ 1

2).

Similarly, we choose a finite set of time steps (tn)n=1...N This

gives the following update rule:

xi+ 1

2∫x

i− 12

q(x , tn+1)− q(x , tn)dx =

tn+1∫tn

f (q(xi− 12, t))− f (q(xi+ 1

2, t))dt

(2)Note: No approximation has been performed. So far we havean exact solver.

Oliver Meister: Godunov/Roe Fluxes

, May 20th 2015 3

Technische Universitat Munchen

FV discretization

Discretization:Average q over space by Qn

i := 1∆x

∫Ci

q(x , tn)dx .

Average f (q) over time by F ni+ 1

2:= 1

∆t

tn+1∫tn

f (q(xi+ 12, t))dt .

⇒ Qn+1i = Qn

i −∆t∆x

(F ni+ 1

2− F n

i− 12)

Problem: Assume Qni is known, how do we get F n

i− 12, F n

i+ 12

in

order to compute Qn+1i ?

Oliver Meister: Godunov/Roe Fluxes

, May 20th 2015 4

Technische Universitat Munchen

FV discretization

Discretization:Average q over space by Qn

i := 1∆x

∫Ci

q(x , tn)dx .

Average f (q) over time by F ni+ 1

2:= 1

∆t

tn+1∫tn

f (q(xi+ 12, t))dt .

⇒ Qn+1i = Qn

i −∆t∆x

(F ni+ 1

2− F n

i− 12)

Problem: Assume Qni is known, how do we get F n

i− 12, F n

i+ 12

in

order to compute Qn+1i ?

Oliver Meister: Godunov/Roe Fluxes

, May 20th 2015 4

Technische Universitat Munchen

FV discretization

Discretization:Average q over space by Qn

i := 1∆x

∫Ci

q(x , tn)dx .

Average f (q) over time by F ni+ 1

2:= 1

∆t

tn+1∫tn

f (q(xi+ 12, t))dt .

⇒ Qn+1i = Qn

i −∆t∆x

(F ni+ 1

2− F n

i− 12)

Problem: Assume Qni is known, how do we get F n

i− 12, F n

i+ 12

in

order to compute Qn+1i ?

Oliver Meister: Godunov/Roe Fluxes

, May 20th 2015 4

Technische Universitat Munchen

An unstable method

Problem: Assume Qni is known, how do we get F n

i− 12, F n

i+ 12

in

order to compute Qn+1i ?

Use a numerical flux function

F ni+ 1

2= F(Qn

i ,Qni+1)

First attempt:

F(Qni ,Q

ni+1) :=

12

(f (Qni ) + f (Qn

i+1))

Resulting method:

Qn+1i = Qn

i −∆t

2∆x(f (Qn

i+1)− f (Qni−1))

⇒ Central difference term, unstable.

Oliver Meister: Godunov/Roe Fluxes

, May 20th 2015 5

Technische Universitat Munchen

An unstable method

Problem: Assume Qni is known, how do we get F n

i− 12, F n

i+ 12

in

order to compute Qn+1i ?

Use a numerical flux function

F ni+ 1

2= F(Qn

i ,Qni+1)

First attempt:

F(Qni ,Q

ni+1) :=

12

(f (Qni ) + f (Qn

i+1))

Resulting method:

Qn+1i = Qn

i −∆t

2∆x(f (Qn

i+1)− f (Qni−1))

⇒ Central difference term, unstable.

Oliver Meister: Godunov/Roe Fluxes

, May 20th 2015 5

Technische Universitat Munchen

An unstable method

Problem: Assume Qni is known, how do we get F n

i− 12, F n

i+ 12

in

order to compute Qn+1i ?

Use a numerical flux function

F ni+ 1

2= F(Qn

i ,Qni+1)

First attempt:

F(Qni ,Q

ni+1) :=

12

(f (Qni ) + f (Qn

i+1))

Resulting method:

Qn+1i = Qn

i −∆t

2∆x(f (Qn

i+1)− f (Qni−1))

⇒ Central difference term, unstable.Oliver Meister: Godunov/Roe Fluxes

, May 20th 2015 5

Technische Universitat Munchen

Lax-Friedrichs Method

Problem: Assume Qni is known, how do we get F n

i− 12, F n

i+ 12

in

order to compute Qn+1i ?

Use a numerical flux function

F ni+ 1

2= F(Qn

i ,Qni+1)

Another attempt:

F(Qni ,Q

ni+1) :=

12

(f (Qni ) + f (Qn

i+1))− ∆t∆x

(Qni+1 −Qn

i )

Resulting method:

Qn+1i =

12

(Qni−1 + Qn

i+1) +∆t

2∆x(f (Qn

i+1)− f (Qni−1))

⇒ Lax-Friedrichs Method, stable but diffusive.

Oliver Meister: Godunov/Roe Fluxes

, May 20th 2015 6

Technische Universitat Munchen

Lax-Friedrichs Method

Problem: Assume Qni is known, how do we get F n

i− 12, F n

i+ 12

in

order to compute Qn+1i ?

Use a numerical flux function

F ni+ 1

2= F(Qn

i ,Qni+1)

Another attempt:

F(Qni ,Q

ni+1) :=

12

(f (Qni ) + f (Qn

i+1))− ∆t∆x

(Qni+1 −Qn

i )

Resulting method:

Qn+1i =

12

(Qni−1 + Qn

i+1) +∆t

2∆x(f (Qn

i+1)− f (Qni−1))

⇒ Lax-Friedrichs Method, stable but diffusive.

Oliver Meister: Godunov/Roe Fluxes

, May 20th 2015 6

Technische Universitat Munchen

Lax-Friedrichs Method

Problem: Assume Qni is known, how do we get F n

i− 12, F n

i+ 12

in

order to compute Qn+1i ?

Use a numerical flux function

F ni+ 1

2= F(Qn

i ,Qni+1)

Another attempt:

F(Qni ,Q

ni+1) :=

12

(f (Qni ) + f (Qn

i+1))− ∆t∆x

(Qni+1 −Qn

i )

Resulting method:

Qn+1i =

12

(Qni−1 + Qn

i+1) +∆t

2∆x(f (Qn

i+1)− f (Qni−1))

⇒ Lax-Friedrichs Method, stable but diffusive.Oliver Meister: Godunov/Roe Fluxes

, May 20th 2015 6

Technische Universitat Munchen

Upwind Method

Idea: Use Riemann problems for scalar conservation laws inorder to approximate fluxes

• Riemann problem: constant function with a singlediscontinuity at x = 0 in an infinite domain

• Finite Volumes: piecewise constant function withdiscontinuities between cell interfaces

• Find a single shock solution on the cell interfaces→ solution q on the interface is either left state ql or rightstate qr . Then f (q) = f (ql) or f (qr ).

Oliver Meister: Godunov/Roe Fluxes

, May 20th 2015 7

Technische Universitat Munchen

Upwind Method

Result:

Qn+1i =

{Qn

i −∆t∆x (f (Qn

i )− f (Qni−1)) f ′(Qn

i ) ≥ 0Qn

i −∆t∆x (f (Qn

i+1)− f (Qni )) f ′(Qn

i ) < 0

Upwind method, works if f ′ ≥ 0 or f ′ < 0. But what aboutsystems of equations and rarefaction waves?

⇒ Godunov’s method, etc.

Oliver Meister: Godunov/Roe Fluxes

, May 20th 2015 8

Technische Universitat Munchen

Upwind Method

Result:

Qn+1i =

{Qn

i −∆t∆x (f (Qn

i )− f (Qni−1)) f ′(Qn

i ) ≥ 0Qn

i −∆t∆x (f (Qn

i+1)− f (Qni )) f ′(Qn

i ) < 0

Upwind method, works if f ′ ≥ 0 or f ′ < 0. But what aboutsystems of equations and rarefaction waves?

⇒ Godunov’s method, etc.

Oliver Meister: Godunov/Roe Fluxes

, May 20th 2015 8

Technische Universitat Munchen

Upwind Method

Result:

Qn+1i =

{Qn

i −∆t∆x (f (Qn

i )− f (Qni−1)) f ′(Qn

i ) ≥ 0Qn

i −∆t∆x (f (Qn

i+1)− f (Qni )) f ′(Qn

i ) < 0

Upwind method, works if f ′ ≥ 0 or f ′ < 0. But what aboutsystems of equations and rarefaction waves?

⇒ Godunov’s method, etc.

Oliver Meister: Godunov/Roe Fluxes

, May 20th 2015 8

Technische Universitat Munchen

Godunov’s method

A step further:

• Find the solution to the general Riemann problem(see last talk).

• Evaluate the solution at the cell interface. Due to theself-similarity of the Riemann solution, the result q◦ isconstant in time.

• ⇒ The numerical flux 1∆t

tn+1∫tn

f (q(x , t))dt reduces to the

simple formula f (q◦)

Oliver Meister: Godunov/Roe Fluxes

, May 20th 2015 9

Technische Universitat Munchen

Godunov’s method with Roe linearization

General Riemann problem is far too expensive, hence we try toapproximate the solution by linearization.

Goal: For each cell interface, find matrix A ∈ Rd×d such that thesystem

qt + Ai− 12qx = 0

approximates the original system

qt + f ′(q)qx = 0

by Ai− 12→ f ′(q) as Qi−1,Qi → q.

Oliver Meister: Godunov/Roe Fluxes

, May 20th 2015 10

Technische Universitat Munchen

Godunov’s method with Roe linearization

Assumption: A single shock wave with speed s connects Qiand Qi+1, so that

f (Qi)− f (Qi−1) = s · (Qi −Qi−1)

Our approximation must meet

Ai− 12(Qi −Qi+1) = s · (Qi −Qi−1) = f (Qi)− f (Qi−1)

Oliver Meister: Godunov/Roe Fluxes

, May 20th 2015 11

Technische Universitat Munchen

Godunov’s method with Roe linearization

Set:q(ξ) := Qi−1 + ξ · (Qi −Qi−1)

Then

f (Qi) − f (Qi−1) =

1∫0

df (q(ξ))

dξdξ =

1∫0

f ′(q(ξ))q′(ξ)dξ =

1∫0

f ′(q(ξ))dξ

(Qi −Qi−1)

So a suitable choice for A is1∫0

f ′(q(ξ))dξ.

Oliver Meister: Godunov/Roe Fluxes

, May 20th 2015 12

Technische Universitat Munchen

Godunov’s method with Roe linearization

Set:q(ξ) := Qi−1 + ξ · (Qi −Qi−1)

Then

f (Qi) − f (Qi−1) =

1∫0

df (q(ξ))

dξdξ =

1∫0

f ′(q(ξ))q′(ξ)dξ =

1∫0

f ′(q(ξ))dξ

(Qi −Qi−1)

So a suitable choice for A is1∫0

f ′(q(ξ))dξ.

Oliver Meister: Godunov/Roe Fluxes

, May 20th 2015 12

Technische Universitat Munchen

Godunov’s method with Roe linearization

Usually, instead of integrating over ξ a transformation z(ξ) isused to integrate a path on, because the resulting system mightnot be hyperbolic any more.

f (Qi)− f (Qi−1) =

1∫0

f ′(q(z(ξ)))dξ

(Zi − Zi−1)

where Zi = z(Qi),Zi−1 = z(Qi−1) and

Qi −Qi−1 =

1∫0

dq(z(ξ)))

dzdξ

(Zi − Zi−1)

Oliver Meister: Godunov/Roe Fluxes

, May 20th 2015 13

Technische Universitat Munchen

Roe solver for the 1D Shallow Water Equations

Nonlinear system of equations (ignoring source terms):[h

hu

]t

+

[hu

hu2 + 12gh2

]x

= 0

They can be described in quasilinear form:[h

hu

]t

+

[0 1

−(u)2 + gh 2u

] [h

hu

]x

= 0

Oliver Meister: Godunov/Roe Fluxes

, May 20th 2015 14

Technische Universitat Munchen

Roe solver for the 1D Shallow Water Equations

Choose as parameter vector z := 1√hq. Leaving out

intermediate steps we obtain:

A =

[0 1

−u2 + gh 2u

]as our Roe matrix with the arithmetic average h and the Roeaverage u. Applying Godunov’s method for the time step, wehave a full numerical scheme now.

Oliver Meister: Godunov/Roe Fluxes

, May 20th 2015 15