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New non-oscillatory central schemes on unstructured triangulations for hyperbolic systems of conservation laws. (English) Zbl 1151.65068
Summary: We discuss an extension of the fully-discrete non-oscillatory central schemes of G.-S. Jiang and E. Tadmor [SIAM J. Sci. Comput. 19, No. 6, 1892–1917 (1998; Zbl 0914.65095)] and of A. Kurganov and E. Tadmor [J. Comput. Phys. 160, No. 1, 241–282 (2000; Zbl 0987.65085)] for hyperbolic systems of conservation laws to unstructured triangular meshes. In doing so, we propose a new, “genuinely multidimensional,” non-oscillatory reconstruction – the minimum-angle plane reconstruction (MAPR). The MAPR is based on the selection of an interpolation stencil yielding a linear reconstruction with minimal angle with respect to the horizontal. This means that the MAPR does not bias the solution by using a coordinate direction-by-direction approach to the reconstruction, which is highly desirable when unstructured meshes consisting of elements with (almost) arbitrary geometry are used.
To show the “black-box solver” capabilities of the proposed schemes, numerical results are presented for a number of hyperbolic systems of conservation laws (in two spatial dimensions) with convex and non-convex flux functions. In particular, it is shown that, even though the MAPR is neither designed with the goal of obtaining a scheme that satisfies a maximum principle in mind nor is total-variation diminishing (TVD), it provides a robust non-oscillatory reconstruction that captures composite waves accurately.

MSC:
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
Software:
SINTEF
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