Stabilization and Limiting Techniques for Galerkin Approximations of Hyperbolic Conservation Laws With High Order Finite Elements

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This project is aimed at further development of novel monolithic convex limiting (MCL) techniques for finite element discretizations of nonlinear hyperbolic problems. The principal investigator's MCL methodology blends a standard high-order Galerkin discretization and a low-order algebraic version of the Lax-Friedrichs method in a manner which guarantees the validity of relevant maximum principles and entropy inequalities. The resulting nonlinear scheme is provably positivity-preserving and entropy stable. No free parameters are involved and the sparse form of the MCL-constrained discretization has the compact stencil of the piecewise-linear approximation on a submesh with the same nodes. The main focus of the first funding period was on the analysis and design of algebraic flux correction tools for continuous Galerkin methods using high-order Bernstein finite elements. The proposed sequel project will extend the algorithmic framework and theoretical foundations of MCL to general Runge-Kutta time discretizations, stationary problems, and discontinuous Galerkin (DG) methods. A task of particular importance will be the development of entropy correction tools that are suitable not only for scalar nonlinear semi-discrete problems but also for fully discrete approximations to hyperbolic systems. The proposed research endeavors will also include the development of hp-adaptive DG schemes equipped with a new kind of flux and slope limiters for the piecewise-linear subcell approximation in non-smooth macrocells. All new features will be implemented in the open-source C++ finite element library MFEM (https://mfem.org). Detailed theoretical studies and a comparison to other high-resolution DG schemes will be performed to assess the accuracy, robustness, and efficiency of the proposed algorithms.