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Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems. (English) Zbl 1110.65077
Summary: In this article we present a non-oscillatory finite volume scheme of arbitrary accuracy in space and time for solving linear hyperbolic systems on unstructured grids in two and three space dimensions using the arbitrary high order schemes using derivatives (ADER) approach. The key point is a new reconstruction operator that makes use of techniques developed originally in the discontinuous Galerkin finite element framework.
First, we use a hierarchical orthogonal basis to perform reconstruction. Second, reconstruction is not done in physical coordinates, but in a reference coordinate system which eliminates scaling effects and thus avoids ill-conditioned reconstruction matrices.
In order to achieve non-oscillatory properties, we propose a new weighted essential nonoscillatory (WENO) reconstruction technique that does not reconstruct point-values but entire polynomials which can easily be evaluated and differentiated at any point. We show that due to the special reconstruction the WENO oscillation indicator can be computed in a mesh-independent manner by a simple quadratic functional. Our WENO scheme does not suffer from the problem of negative weights as previously described in the literature, since the linear weights are not used to increase accuracy. Accuracy is obtained by merely putting a large linear weight on the central stencil.
The resulting one-step ADER finite volume scheme obtained in this way performs only one nonlinear WENO reconstruction per element and time step and thus can be implemented very efficiently even for unstructured grids in three space dimensions. We show convergence results obtained with the proposed method up to sixth order in space and time on unstructured triangular and tetrahedral grids in two and three space dimensions, respectively.

MSC:
65M06 Finite difference 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
76M12 Finite volume methods applied to problems in fluid mechanics
35L45 Initial value problems for first-order hyperbolic systems
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