Non-oscillatory central schemes on unstructured grids for two-dimensional hyperbolic conservation laws

Abstract

We propose a new central finite volume scheme on unstructured triangular grids to approximate the solution of general two-dimensional hyperbolic systems of conservation laws. The proposed method is an unstructured two-dimensional extension of the original Nessyahu and Tadmor scheme, and a generalization of the barycentric central methods of Arminjon et al. Starting with a conformal finite element triangulation, the proposed method evolves a piecewise linear numerical solution on two staggered grids, thus avoiding the resolution of the Riemann problems arising at the cell interfaces. The control cells of the original grid are the triangles of a finite element mesh, while the dual cells are the staggered quadrilaterals constructed on adjacent triangles. The resulting central scheme is second-order accurate both in space and time and is oscillations-free thanks to numerical gradients limiting. The extension of the staggered Lax–Friedrichs scheme on unstructured grids is easily obtained from the Nessyahu and Tadmor extension by simply evolving a piecewise constant solution instead of a linear one. We validate the developed schemes and solve classical two-dimensional problems arising in gas dynamics. The quality of the obtained numerical results confirms the efficiency and robustness of the proposed schemes.