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An in-depth explanation of godunov's method for solving partial differential equations (pdes) using finite volume methods. Topics covered include flux functions, higher order reconstruction, limiters, and exact time-integration. The document also discusses the importance of accurately computing pointwise values of the flux function at cell boundaries and the use of godunov's approach to form exact right-hand side time integrals.
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1/ 2
1/ 2 1/ 2 1/ 2
i
i
x
x i i i
qdx
q x x
−
= −
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.
{ (^ (^ ))^ (^ (^ ))}
( (^ )) ( (^ ))
( (^ )) ( (^ ))
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
−
− =
− = =
∫
∫ ∫
The mean evolution equation without approximation:
Godunov’s approach:
At the start of each time step we only have cell average values for q
Use the cell averages to reconstruct a polynomial approximation to q defined on each cell
Form the exact right hand side time integrals, but for the new propagation problem with the reconstructed solution as initial condition
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
− + = =
= + (^) −
∫ ∫
( , ) ( )
n n n qi x t = qi + x − xi σi
1
n
n
1
n
( ) ( )
n n n qi x = qi + x − xi σi
n
n 1
dt
u*dt
dx
1 (^ )^1 (^1 ) 1
n n n qi (^) − x = qi (^) − + x − xi − σi−
( )
( (^ ))
1/ 2
1 1/ 2
n i i n n i i
−
− −
n
n 1
exactly evaluate each of the integrals:
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
σ
σ σ
=
=
=
∫
∫
∫
( )
1
n n
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
(^) Docsity.com
Lax-Wendroff Reconstruction (down wind slope)
1
n
n
1
n