High-Order Maximum Principle Preserving (MPP) Techniques for Solving Conservation Laws with Applications on Multiphase Flow

Abstract

We develop numerical methods to solve the linear scalar conservation law fulfilling the maximum principle. To do this we use continuous and discontinuous Galerkin finite elements and achieve the preservation on the maximum principle via the Flux Corrected Transport (FCT) method. We use high-order polynomial spaces with Bernstein basis functions and obtain the optimal convergence rates with spaces of up to third order for smooth solutions that are monotone. This methodology produces good quality results for spaces up to (around) third order. However, when higher-order spaces are used non-physical oscillations are introduced, which is true nevertheless the methods are maximum principle preserving. These oscillations can be highly reduced by defining tighter bounds. Using discontinuous Galerkin finite elements we present a new FCT-like methodology based on single cell flux corrections. This method combines a mass conservative low-order Maximum Principle Preserving (MPP) solution with a non-mass conservative high-order MPP solution. The process is designed to recover mass conservation locally (with respect to degrees of freedom). Using this scheme we obtain the optimal convergence rates with spaces of up to third order for smooth solutions that are monotone. The method is designed to overcome problems when high-order spaces are used and, under this context, we obtained better results than with the standard FCT method. We present two methods to transport a smoothed Heaviside level set function using a one-stage reinitialization based on artificial compression. The first method allows arbitrarily large compression which might lead to non-physical behavior. To overcome this difficulty the second method self-balances the artificial dissipation and compression. Finally, we use the level set solver with a Navier-Stokes solver to simulate incompressible two-phase flow.