Sparse invariant domain preserving discontinuous Galerkin methods with subcell convex limiting

Abstract

In this paper, we develop high-order nodal discontinuous Galerkin (DG) methods for hyperbolic conservation laws that satisfy invariant domain preserving properties using a subcell flux corrections and convex limiting. These methods are based on a subcell flux corrected transport (FCT) methodology that involves blending a high-order target scheme with a robust, low-order invariant domain preserving method that is obtained using a graph viscosity technique. The new low-order discretizations are based on sparse stencils which do not increase with the polynomial degree of the high-order DG method. As a result, the accuracy of the low-order method does not degrade when used with high-order target methods. The method is applied to both scalar conservation laws, for which the discrete maximum principle is naturally enforced, and to systems of conservation laws such as the Euler equations, for which positivity of density and a minimum principle for specific entropy are enforced. Numerical results are presented on a number of benchmark test cases.

arXiv comment27 pages, 13 figures

References in corpus (11)

• MFEM: a modular finite element methods library
• On discretely entropy conservative and entropy stable discontinuous Galerkin methods
• Viscous Shock Capturing in a Time-Explicit Discontinuous Galerkin Method
• Invariant domain preserving discretization-independent schemes and convex limiting for hyperbolic systems
• An optimization-based approach for high-order accurate discretization of conservation laws with discontinuous solutions
• Algebraic entropy fixes and convex limiting for continuous finite element discretizations of scalar hyperbolic conservation laws
• Approximate tensor-product preconditioners for very high order discontinuous Galerkin methods
• On thin plate spline interpolation
• Subcell flux limiting for high-order Bernstein finite element discretizations of scalar hyperbolic conservation laws
• Analysis and entropy stability of the line-based discontinuous Galerkin method
• High-order matrix-free incompressible flow solvers with GPU acceleration and low-order refined preconditioners

Cited by in corpus (12)

• Subcell limiting strategies for discontinuous Galerkin spectral element methods
• A Conservative Discontinuous Galerkin Discretization for the Chemically Reacting Navier-Stokes Equations
• An Entropy Stable Nodal Discontinuous Galerkin Method for the resistive MHD Equations. Part II: Subcell Finite Volume Shock Capturing
• Limiter-based entropy stabilization of semi-discrete and fully discrete schemes for nonlinear hyperbolic problems
• A positivity preserving strategy for entropy stable discontinuous Galerkin discretizations of the compressible Euler and Navier-Stokes equations
• Positivity-preserving and entropy-bounded discontinuous Galerkin method for the chemically reacting, compressible Euler equations. Part I: The one-dimensional case
• A new perspective on flux and slope limiting in discontinuous Galerkin methods for hyperbolic conservation laws
• On error-based step size control for discontinuous Galerkin methods for compressible fluid dynamics
• On the robustness of high-order upwind summation-by-parts methods for nonlinear conservation laws
• Conservative and accurate solution transfer between high-order and low-order refined finite element spaces
• Local subcell monolithic DG/FV convex property preserving scheme on unstructured grids and entropy consideration
• Analysis of algebraic flux correction for semi-discrete advection problems