High-order bound-preserving discontinuous Galerkin methods for multicomponent chemically reacting flows

Abstract

In this paper, we design high-order bound-preserving discontinuous Galerkin (DG) methods for multicomponent chemically reacting flows. In this problem, the density and pressure are positive and the mass fractions are between 0 and 1. There are three main difficulties. First of all, it is not easy to construct high-order positivity-preserving schemes for convection-diffusion equations. In this paper, we design a special penalty term to the diffusion term and construct the positivity-preserving flux for the system. The proposed idea is locally conservative, high-order accurate and easy to implement. Secondly, the positivity-preserving technique cannot preserve the upper bound 1 of the mass fractions. To bridge this gap, we apply the positivity-preserving technique to each mass fraction and develop consistent numerical fluxes in the system and conservative time integrations to preserve the summation of the mass fractions to be 1. Therefore, each mass fraction would be between 0 and 1. Finally, most previous bound-preserving DG methods are based on Euler forward time discretization. However, due to the rapid reaction rates, the target system may contain stiff sources, leading to restricted time step sizes. To fix this and preserve conservative property, we apply the conservative modified exponential Runge-Kutta method. The method is third-order accurate and keeps the summation of the mass fractions to be 1. Numerical experiments will be given to demonstrate the good performance of the proposed schemes.