Preserving Nonnegativity in Discontinuous Galerkin Approximations to Scalar Transport via Truncation and Mass Aware Rescaling (TMAR)

Abstract

n accurate nonnegativity preserving limiter is presented for use with discontinuous Galerkin (DG) discretizations of scalar advection equations. The nonnegativity of the tracer field is preserved through the application of a mass conservative limiter that truncates negatives within each element and linearly rescales the resulting DG polynomials to preserve element-mean mass. As a preliminary step, the DG fluxes through each side of the element are limited in a manner similar to flux-corrected transport to ensure that the element-mean mass remains nonnegative during each individual stage of the time integration. In this paper, it is proven that such a truncation and mass aware rescaling (TMAR) does not change the order of accuracy of the underlying unlimited DG approximation. Numerical tests with two-dimensional deforming flows confirm that the method remains accurate and efficient while preserving nonnegativity. In comparison to some popular previous approaches, TMAR limiting is particularly well suited to approximations that use high-degree polynomial expansions (quartics or higher) to capture features that are only moderately well resolved.