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Maximum-principle-satisfying and positivity-preserving high order discontinuous Galerkin schemes for conservation laws on triangular meshes. (English) Zbl 1247.65131
An extension of higher order finite volume and discontinuous Galerkin (DG) schemes satisfying a strict maximum principle to triangular meshes is presented. Such an extension is highly nontrivial. In order to obtain a maximum principle satisfying the finite volume schemes or the discontinuous Galerkin method, a special quadrature rule is constructed. The same method can preserve the maximum principle for DG or finite volume schemes solving two-dimensional incompressible Euler equations in the vorticity stream-function formulation. Also the positivity preserving high order DG or finite volume schemes solving compressible Euler equations on triangular meshes are obtained. Numerical tests for the third order Runge-Kutta DG method on unstructured meshes are reported.

MSC:
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
76N15 Gas dynamics, general
76D05 Navier-Stokes equations for incompressible viscous fluids
76M10 Finite element methods applied to problems in fluid mechanics
76M12 Finite volume methods applied to problems in fluid mechanics
35B50 Maximum principles in context of PDEs
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
Software:
EasyMesh
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References:
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