I have just started studying the shallow water equations. I have solved and implemented the Riemann Problem for the 1-D case. I now want to implement a finite volume scheme for the same 1-D SWEs but the result I get seem totally different from the exact solution. I evaluated numerical fluxes F at boundary interfaces using F(U_) where U_ is the variable value at boundary cells and evaluated inter-cell fluxes using F(Ustar) where Ustar is a solution of the Riemann Problem with initial data from the average values of U in the bounding cells.
What could be the cause of the disparity?
I could paste the exact functions here.
Some sort of upwind scheme is necessary. It should provide you a sufficient dissipation, or artificial viscosity, without too much dispersion effect. You might want to read some CFD textbooks.
Dear Chinedu, First of all I strongly recommend you to read the book "Reimann Solvers and Numerical Methods for Fluid Dynamic" by E.F.Toro
now these comments may be helpfull:
1. For implementation of boundary condition, one can add one (or as many required by method) extra grid point to each boundary( in 1D 2 point one at x=-dx and one at x=L+dx) . then update these points by U @ timestep n. Now to solve the eq over domain loop over interior points and now you can evaluate flux at boundary face just like other places.
2. In comparison a 2d results to 1d exact make sure that you add a propper source term in your 1D eq.
3. For the start, try final times in which your solution won't reach the boundary. In this case non-reflecting boundary condition will surly help
4. Start your experiments with FVM by 1rst order Godunov method, then extend it to higher order, e.g. MUSCL-Hancok
I have been written a 1D FORTRAN code by MUSCL-Hanckok method if you think it is usefull I can email it to you( so leave your mail for me)
5. And Finally you should note that Full 2D code will need much restricted CFL condition
CHEERS
Thanks a million, Moshen. My email is [email protected]. Please also send 2D codes.
I appreciate.
For internal fluxes use F(U finite volume) NOT F(URIEMMANN). How to do this see N.C.Markatos Ironmaking and Steelmaking,Vol 16,no4,p266-273,1989 or Asia-Pacific J.of Chem.Engineering,7(2),182-205,2012.
When you go 2D use the depth-averaged Navier-Stoke equations
Dear Chinedu,
Moshen replay was an exhaustive one. I just also recommend you to take a look to the great Le Veque TextBook:
"Finite Volume Methods for Hyperbolic Problems", Randall J. Le Veque. (Chapter 13)
It is in my opinion the clearest as well rigorous book I have ever read on Finite Volume Methods,
Regards,
Andrea
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