Robust high-order space-time conservative schemes for solving conservation laws on hybrid meshes

In this paper, the second-order space-time conservation element and solution element (CE/SE) method proposed by Chang (1995) [3] is implemented on hybrid meshes for solving conservation laws. In addition, the present scheme has been extended to high-order versions including third and fourth order. Most methodologies of proposed schemes are consistent with that of the original CE/SE method, including: (i) a unified treatment of space and time (thereby ensuring good conservation in both space and time); (ii) a highly compact node stencil (the solution node is calculated using only the neighboring mesh nodes) regardless of the order of accuracy at the cost of storing all derivatives. A staggered time marching strategy is adopted and the solutions are updated alternatively between cell centers and vertexes. To construct explicit high-order schemes, second- and third-order derivatives are calculated by a modified finite-difference/weighted-average procedure which is different from that used to calculate the first-order derivatives. The present schemes can be implemented on a wide variety of meshes, including triangular, quadrilateral and hybrid (consisting of both triangular and quadrilateral elements). Beyond that, it can be easily extended to arbitrary-order schemes and arbitrary shape of polygonal elements by using the present methodologies. A series of common benchmark examples are used to confirm the accuracy and robustness of the proposed schemes.