Behavior of Finite Volume Schemes for Hyperbolic Conservation Laws on Adaptive Redistributed Spatial Grids

Abstract

In this work we consider finite volume schemes combined with dynamic spatial mesh redistribution. We study whether appropriate mesh redistribution is a satisfactory mechanism for increasing the resolution of numerical solutions for problems of scalar and systems of conservation laws (CL) in one space dimension, while being at the same time a stabilization mechanism for selecting the appropriate entropy solution. In order to increase the resolution around shock areas and keep the computational cost low, our redistribution policy is to reconstruct spatially the numerical solution on a new mesh, where the solution’s curvature is almost uniformly distributed, while the node’s cardinality is kept constant. We examine the stabilization properties of that redistribution process by adding it as a substep on the time evolution step of some classical schemes with known (unstable) characteristics. Testing the resulting method for several such schemes and on a large number of CL problems that have solutions with special characteristics (shocks, rarefaction areas, steady states) and comparing the results with those produced by schemes with extra stabilization mechanisms (like slope/flux limiters, entropy corrections), we conclude that indeed the proposed redistribution adds such stabilization properties while at the same time increasing the resolution.