Adaptive Moving Mesh Central-upwind Schemes
The development of accurate, efficient and robust numerical methods for the hyperbolic system of conservation/balance law is an important and challenging problem. One major numerical difficulty is related to the fact that the hyperbolic system admits non-smooth solutions. To achieve high resolution as well as to improve the efficiency of the numerical methods, we have developed new adaptive moving mesh (AMM) central-upwind schemes for the hyperbolic system on both adaptive one-dimensional (1-D) nonuniform grids and two-dimensional (2-D) structured quadrilateral meshes. The designed algorithm solves the PDE using the second-order semi-discrete central-upwind schemes and strong stability preserving Runge-Kutta ODE solver. After evolving the solutions to the new time level, the mesh points are then redistributed accordingly to the moving mesh PDE (MMPDE), together with a conservative projection strategy to project the solutions to the new mesh. We demonstrate the robustness and efficiency of AMM central-upwind schemes by a number of numerical examples of Euler equations of gas dynamics in both 1-D and 2-D cases. We also present the AMM central-upwind schemes for the Saint-Venant system of shallow water equations for both 1-D and 2-D cases. With bottom structures being considered, we implement a quadrature for the geometric source term, which is well-balanced in the sense that it is capable to exactly preserve the "lake at rest" steady state, and a draining time step technique to maintain the positivity of water depth during the PDE time-evolution step. Additionally, we introduce different projection strategy for semi-dry/dry area in the computational domain to ensure the positivity of water depth during mesh redistribution. Numerical experiments show that AMM central-upwind schemes can significantly improve the resolution of the solution.