Comparison of different numerical schemes for 1D conservation laws

Abstract

The importance of numerical flux solvers (NFS) in constructing conservative methods is well known, but it is not always clear which solvers more suitable for given conservation laws. To elucidate this, we compare some NFS for some conservation laws. The finite volume method is adopted as the base scheme and the selected models include linear, nonlinear and also a system of conservation laws. The NFS are the HLL, the Lax-Friedrich’s, Modified Lax- Friedrich’s and the Rusanov’s schemes. We first perform experimental order of convergence to ascertain that the base scheme is correctly implemented. The methods are then compared for the models. The results show that the HLL is a very superior method compared to the other methods. In particular, only the HLL is able to compute physically correct solution of the Buckley-Leverett model. On the other hand, the Lax-Friedrich’s solvers demonstrated to be the most inferior of all the methods. Our conclusion is that, for practical cases, the Lax- Friedrich solver should be avoided, especially when the analytical flux is nonlinear.

Subject Classification: (2010):

• 76M12

Keywords:

• Finite volume schemes
• HLL
• Rusanov
• Lax-Friedrichs
• Burger’s equation
• Linear advection equations
• Shallow water models
• Buckley-Leverett equation
• Numerical fluxes