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 Peter April 27, 2003 16:12

Godunov scheme

Hi

I was woundering what is the Godunov scheme ?

Also compared to upwind, QUICK, central etc.

Regards

Peter

 Andy April 28, 2003 14:55

Re: Godunov scheme

The Godunov scheme is the exact, but very expensive, solution to the Riemann problem at an interface. It solves the fully non-linear implicit Jacobian, which is where the expense is entailed. An approximate Riemann solver freezes the Jocobian at the interface, and essentially linearises it, so that the problem may be solved faster, an example is the Roe approximate Riemann solver or the Osher scheme. Nobody bothers to solve the exact Riemann problem at every interface, since it is too expensive to evaluate these non-linear flux functions.

You now have for instance chosen your scheme; lets take Roe. The terms you have mentioned, are purely the methods of doing the convective differencing, they solve the convective part of the Navier-stokes equations. You should be careful that you do not refer to these as a scheme, they are not, they are just an algebraic expression for solving the convective terms of the equations, however, you will see all the time things like an upwind scheme or a central difference scheme, just beware. QUICK is from the MUSCL family, and is formulated for structured grids, it is meant to be second order accurate, and upwind. Upwind means that you weight the value at the interface based on the wind direction of the flow, in a proper TVD edge based solver this can be seen easily, but again as with the term scheme, the term upwind is used less strictly than it should be. Central refers to central difference, second order by construction; you can read up on differencing.

I have tried to put the above in some short context, for instance if you want to find out about QUICK, then read about MUSCL, Van Leer was the inventor. I make the cautious warnings because when you write a solver, you will soon become aware that you are not writing an upwind scheme, or a quick scheme, or linear reconstruction, but are using these formulations to calculate the fluxes at an interface, in a higher order manner, and that these in tern will be used in the scheme, Roe, Osher, and so on, to calculate the convective part of the equations as well as being used to work out the correct amount of artificial viscosity that should be added to maintain a stable scheme, this is done via the Jacobian, which you have linearised and is essential to the scheme. In the context of the Roe scheme for example this Jacobian will manifests itself in your code as the Roe averages, at an interface.

Hope this helps, rather brief but should provide a starting point.

Cheers

Andy

 J.J. Smyth April 29, 2003 05:28

Re: Godunov scheme

Hi,

You say

"The Godunov scheme is the exact, but very expensive, solution to the Riemann problem at an interface. It solves the fully non-linear implicit Jacobian, which is where the expense is entailed. An approximate Riemann solver freezes the Jocobian at the interface, and essentially linearises it, so that the problem may be solved faster, an example is the Roe approximate Riemann solver or the Osher scheme"

 Tom April 29, 2003 05:43

Re: Godunov scheme

Try Randall LeVeque's book

Finite Volume Methods for Hyperbolic Problems (Cambridge Texts in Applied Mathematics)

Actually the Riemann problem is only exactly solvable in one space-dimension (i.e. the classic dam burst problem in shallow water theory). In higher dimensions you have to resort to approximate solvers.

Tom.

 Andy April 29, 2003 06:24

Re: Godunov scheme

An excellent book that will put it in context as well as the approximate solvers like Roe, and completly different approches like ENO and WENO, and HLLC, can be found in Computational Gas Dynamics by Laney (might have spelt that wrongly). He also has a very nice web page where you can down load fortran files for different schemes and try them out.

Tom is right about the one-d problem, however, even when you solve the approximate riemann problem you are still solving an approximate one-d problem, be clear that you understand why we bother to use approximate riemann solvers, it is only due to the non-linear implicit nature of the Jacobian. An active area of research is multi-dimensional upwind where you resolve for the absoulute direction of the flow at an interface, instead of taking it normal. This introduces the multi-dimensional nature into the upwind.

Hope this helps, again the book is excellent.

Andy

 versi May 6, 2003 05:37

Re: Godunov scheme

also E.Toro's book.

 Killercam November 27, 2009 11:51

"The Godunov scheme is the exact, but very expensive, solution to the Riemann problem at an interface. It solves the fully non-linear implicit Jacobian, which is where the expense is entailed..."

This is not true. A Godunov scheme is an upwind numerical method that is positively conservative. IT does not solve the Riemann problem. The Riemann problem stems from the use of a Godunov scheme, it is the solution of the Riemann problem that provides you with the intercell Godunov fluxes...

The books you suggest however, are correct.

Cheers.

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