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A vertex-centered, dual discontinuous Galerkin method. (English) Zbl 1091.65098
Summary: This note introduces a new version of the discontinuous Galerkin method for discretizing first-order hyperbolic partial differential equations. The method uses piecewise polynomials that are continuous on a macroelement surrounding the nodes in the unstructured mesh but discontinuous between the macroelements. At lowest order, the method reduces to a vertex-centered finite-volume method with control volumes based on a dual mesh, and the method can be implemented using an edge-based data structure. The method provides therefore a strategy to extend existing vertex-centered finite-volume codes to higher order using the discontinuous Galerkin method. Preliminary tests on a model linear hyperbolic equation in two-dimensional indicate a favorable qualitative behavior for nonsmooth solutions and optimal convergence rates for smooth solutions.
MSC:
65M60Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (IVP of PDE)
35L45First order hyperbolic systems, initial value problems
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