New non-oscillatory central schemes on unstructured triangulations for hyperbolic systems of conservation laws

We discuss an extension of the Jiang–Tadmor and Kurganov–Tadmor fully-discrete non-oscillatory central schemes 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.