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

• Applicant: Professor Dr. Dmitri Kuzmin
• Subject Area: Mathematics
• Term: since 2017
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 387630025

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.

• DFG Programme: Research Grants