New nonoscillatory central schemes on unstructured triangulations for hyperbolic systems of conservation laws

In this paper, we discuss an extension of the Jiang–Tadmor [SIAM J. Sci. Comput. 19 (1998) 1892–1917] and Kurganov–Tadmor [J. Comput. Phys. 160 (2000) 241–282] fully-discrete nonoscillatory central schemes for hyperbolic systems of conservation laws to unstructured triangular meshes. In doing so, we propose a new, “genuinely multidimensional,” nonoscillatory reconstruction — the minimum-angle plane reconstruction (MAPR). The MAPR is based on the selection of an interpolation stencil yielding a linear reconstruction (of the solution from its cell averages) with minimal angle with respect to the horizontal. Furthermore, by design, the MAPR is applicable to unstructured meshes consisting of elements with (almost) arbitrary geometry, and it does not bias the solution by using a coordinate direction-by-direction approach to the reconstruction. 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 nonconvex flux functions: the linear advection equation, the inviscid Burgers equation, a prototype scalar nonconvex equation, the Buckley–Leverett equation with gravity effects and the Euler equations. In particular, it is shown that, even though the MAPR is not designed with the goal of obtaining a scheme that satisfies a maximum principle or is total-variation diminishing (TVD) in mind, it provides a robust nonoscillatory reconstruction that captures composite waves accurately.