Analysis and Applied Mathematics Seminar
Maurice Fabien
University of Wisconsin Madison
A positivity-preserving discontinuous Galerkin scheme for hyperbolic PDEs with characteristics-informed augmentation
Abstract: We introduce a positivity-preserving discontinuous Galerkin (DG) scheme for hyperbolic PDEs on unstructured meshes in 2D and 3D. The standard DG spaces are augmented with either polynomial or non-polynomial basis functions. The primary purpose of these augmented basis functions is to ensure that the cell average from the unmodulated DG scheme remains positive. We explicitly obtain suitable basis functions by inspecting the method of characteristics on an auxiliary problem. A key result is proved which demonstrates that the augmented DG scheme will retain a positive cell average, provided that the inflow, source term, and variable coefficients are positive. Standard slope limiters can then be leveraged to produce a high-order conservative positivity-preserving DG scheme. Numerical experiments demonstrate the scheme is able to retain high-order accuracy as well as robustness for variable coefficients and nonlinear problems.
Monday April 21, 2025 at 4:00 PM in 636 SEO