A Jiang–Tadmor Scheme on Unstructured Triangulations

Nonoscillatory central schemes are a class of Godunov-type (i.e., shock-capturing, finite volume) numerical methods for solving hyperbolic systems of conservation laws (e.g., the Euler equations of gas dynamics). Throughout the last decade, central (Godunov-type) schemes have gained popularity due to their simplicity and efficiency. In particular, the latter do not require the solution of a Riemann problem or a characteristic decomposition to compute the intercell flux. One example of a 2D central Godunov-type scheme is that of Jiang and Tadmor [SIAM J. Sci. Comput. 19 (1998) 1892–1917]. Unfortunately, the latter scheme is constructed on a uniform Cartesian (tensor product) grid and uses a directionby-direction reconstruction. Therefore, the JT scheme is not applicable to problems with complicated domain geometry, and may not be able to achieve the full second order of accuracy (in time and space) when the solution is not aligned with the coordinate directions. In this paper, an extension of the JT scheme to unstructured triangular grids will be discussed. To this end, a new, “genuinely multidimensional,” nonoscillatory reconstruction — the minimum-angle plane reconstruction (MAPR) — is discussed. 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, numerical results are presented for hyperbolic systems of conservation laws with convex and nonconvex flux functions. In particular, it will be shown that the MAPR is able to capture composite waves accurately.