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Monolithic convex limiting in discontinuous Galerkin discretizations of hyperbolic conservation laws. (English) Zbl 1524.65539

Summary: In this work we present a framework for enforcing discrete maximum principles in discontinuous Galerkin (DG) discretizations. The developed schemes are applicable to scalar conservation laws as well as hyperbolic systems. Our methodology for limiting volume terms is similar to recently proposed methods for continuous Galerkin approximations, while DG flux terms require novel stabilization techniques. Piecewise Bernstein polynomials are employed as shape functions for the DG spaces, thus facilitating the use of very high order spatial approximations. We discuss the design of a new, provably invariant domain preserving DG scheme that is then extended by state-of-the-art subcell flux limiters to obtain a high-order bound preserving approximation. The limiting procedures can be formulated in the semi-discrete setting. Thus convergence to steady state solutions is not inhibited by the algorithm. We present numerical results for a variety of benchmark problems. Conservation laws considered in this study are linear and nonlinear scalar problems, as well as the Euler equations of gas dynamics and the shallow water system.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
76M10 Finite element methods applied to problems in fluid mechanics
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
76N15 Gas dynamics (general theory)
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35Q35 PDEs in connection with fluid mechanics

Software:

MFEM; GitHub; FESTUNG; SHASTA
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Full Text: DOI arXiv

References:

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