Numerical Methods for PDEs: Godunov's Method & Slope Limiting, Slides of Mathematical Methods for Numerical Analysis and Optimization

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Numerical Methods for Partial

Differential Equations

Today

• Flux functions for finite-volume methods
• Higher order reconstruction
• Limiters

cont

• We define the i’th cell average of q as:
• i.e. the average value of q in the interval
• Assume the line is divided into uniform cells of

width dx then on each cell:

1/ 2

1/ 2 1/ 2 1/ 2

i

i

x

x i i i

qdx

q x x

−

• −

= −

I i = [ xi −1/ 2 ,xi+1/ 2]

{ (^ (^ 1/ 2 ))^ (^ (^ 1/ 2 ))}

i i ,^ i ,

d

q f q x t f q x t

dt dx

cont

xi − 1 x i^ xi^ + 1

xi − 3/ 2 xi −1/ 2 xi +1/ 2 xi +3/ 2

i. The x axis is subdivided into cells (also known as elements)

ii. The cells are centered at the x with integer sub indices

iii. The cell boundaries are at the midpoints between the cell

centers.

Exact Time-Integration

A traditional first step is to integrate the

evolution equation over one time step:

{ (^ (^ ))^ (^ (^ ))}

( (^ )) ( (^ ))

( (^ )) ( (^ ))

1

1 1

1/ 2 1/ 2

1 1/ 2 1/ 2

1 1/ 2 1/ 2

n

n

n n

n n

i i i

t n n i i i i t t

t t n n i i i i t t t t

d

q f q x t f q x t

dt dx

q q f q x t f q x t dt

dx

q q f q x t dt f q x t dt

dx dx

• −

• − =

• − = =

∫

∫ ∫

Godunov’s Method

The mean evolution equation without approximation:

Godunov’s approach:

1. At the start of each time step we only have cell average values for q

2. Use the cell averages to reconstruct a polynomial approximation to q defined on each cell

3. Form the exact right hand side time integrals, but for the new propagation problem with the reconstructed solution as initial condition

4. Compute the cell averages for the new time level.

( (^ )) ( (^ ))

1 1 1 1 1/ 2 1/ 2

1 , ,

n n

n n

t t n n i i i i t t t t

q q f q x t dt f q x t dt dx

− + = =

  = + (^)  −   

∫ ∫

1) Local Reconstruction cont

• At each cell we will establish an approximation

to the slope.

• The reconstructed solution will pass through

the cell average at the cell center and have the

computed slope (shown as blue lines):

xi − 1 x i xi + 1

( , ) ( )

n n n qi x t = qi + x − xi σi

1

n

qi −

n

qi

1

n

qi +

2. Advect Reconstructed Solution Exactly

• For the simple case of the advection equation it is obvious how to compute the flux integral.
• We know how to analytically evolve the reconstructed solution in time, starting at t^n
• The solution shifts to the positive x direction with speed u
• Thus we can use a method of characteristics argument to evaluate the flux integral (for the reconstructed solution as it propagates through the end points of each cell).
• In the more general case, a fairly standard approach is to linearize the flux function about some appropriate mean state, decompose the solution into characteristic variables of the linearized system and perform the same kind of linear wave propogation analysis as we are about to describe. Docsity.com

2) Time Step Restriction

• If the stencil for the slopes only includes the i’th

and i+1’th cells then the time step must satisfy u*dt

<=dx

• Otherwise we would have to use the reconstruction

from the i-1’th cell.

xi − 1 xi + 1

( ) ( )

n n n qi x = qi + x − xi σi

n

t

n 1

t

dt

u*dt

dx

2) Advect Exactly -- inflow

• In the space-time plane we can plot the

characteristics for the advection equation.

• For the inflow – we backtrack into the i-1’th cell

xi − 1 x i xi + 1

1 (^ )^1 (^1 ) 1

n n n qi (^) − x = qi (^) − + x − xi − σi−

( )

( (^ ))

1/ 2

1 1/ 2

n i i n n i i

q x t

q x u t t

−

− −

n

t

n 1

t

2) Advect Exactly cont

• Given a piecewise linear reconstructed solution we know how to

exactly evaluate each of the integrals:

• We back tracked along the characteristics to find the value of the

qtilde field at

( (^ ))

( (^ (^ )))

( (^ (^ ) ) )

1

1

1

1/ 2

1/ 2

1/ 2

2 2

n n n n n n t

i t t t n n i i t t t n n i i i t t

n n n i

n

i

i

i

x u

f q x t dt

f q x u t t dt

u q x dt

dx

t

u dt

ud

t

tq u dt

σ

σ σ

=

=

=

∫

∫

∫

( )

1

1/ 2 ,^ for^ ,

n n

xi t t t t

+ ∈ ^  Docsity.com

3) Update Averages (p

• Repeating for the inflow integral (using the neighbor cellLeVeque)

reconstruction), we can express approximations for the right hand integrals in terms of cell averages (and the as yet unspecified slopes):

( (^ )) ( (^ ))

( )

1 1 1 1/ 2 1/ 2

2 2 1 1 1

2 2

2 2 1

1 , ,

1 2 2

1 2 2

1 2 2

n n

n n

t t n n i i i i t t t t

n n n n i i i i

n n n i i i

n n n n i i i i

q q f q x t dt f q x t dt dx

dx u dt q udtq u dt dx

dx u dt udtq u dt dx

dx u dt q udt q q u dt dx

− + = =

− − −

−

  = + (^)  −   

  = + (^)  + −   

  − (^)  + −   

  = − − + (^)  − (^)  −  

∫ ∫

( 1 )

n

σ i−

     (^) Docsity.com

Accuracy

• We can establish method order accuracy by

Taylor series expansions.

• The latter 3 methods (Fromm, Beam-Warning,

Lax-Wendroff) are 2 nd^ order accurate for

sufficiently smooth solutions.

• This is one order better that the straight

upwind version (assuming zero slope).

Lax-Wendroff Reconstruction (down wind slope)

xi − 1 x i xi + 1

1

n

qi −

n

qi

1

n

qi +

• We consider a piecewise constant solution
• The blue line is the reconstructed version of the approx. solution
• The red dots are the original cell averages,