Positivity-Preserving Discontinuous Galerkin Methods with Lax–Wendroff Time Discretizations

Abstract

This work introduces a single-stage, single-step method for the compressible Euler equations that is provably positivity-preserving and can be applied on both Cartesian and unstructured meshes. This method is the first case of a single-stage, single-step method that is simultaneously high-order, positivity-preserving, and operates on unstructured meshes. Time-stepping is accomplished via the Lax---Wendroff approach, which is also sometimes called the Cauchy---Kovalevskaya procedure, where temporal derivatives in a Taylor series in time are exchanged for spatial derivatives. The Lax---Wendroff discontinuous Galerkin (LxW-DG) method developed in this work is formulated so that it looks like a forward Euler update but with a high-order time-extrapolated flux. In particular, the numerical flux used in this work is a convex combination of a low-order positivity-preserving contribution and a high-order component that can be damped to enforce positivity of the cell averages for the density and pressure for each time step. In addition to this flux limiter, a moment limiter is applied that forces positivity of the solution at finitely many quadrature points within each cell. The combination of the flux limiter and the moment limiter guarantees positivity of the cell averages from one time-step to the next. Finally, a simple shock capturing limiter that uses the same basic technology as the moment limiter is introduced in order to obtain non-oscillatory results. The resulting scheme can be extended to arbitrary order without increasing the size of the effective stencil. We present numerical results in one and two space dimensions that demonstrate the robustness of the proposed scheme.