A Comparison of Artificial Viscosity, Limiters, and Filters, for High Order Discontinuous Galerkin Solutions in Nonlinear Settings

Nonlinear systems of equations demonstrate complicated regularity features that are often obfuscated by overly diffuse numerical methods. Using a discontinuous Galerkin finite element method, we study a nonlinear system of advection-diffusion-reaction equations and aspects of its regularity. For numerical regularization, we present a family of solutions consisting of: (1) a sharp, computationally efficient slope limiter, known as the BDS limiter, (2) a standard spectral filter, and (3) a novel artificial diffusion algorithm with a solution-dependent entropy sensor. We analyze these three numerical regularization methods on a classical test in order to test the strengths and weaknesses of each, and then benchmark the methods against a large application model.