Moving mesh method for simulating high-dimensional time dependent PDEs with fast propagating shock waves

摘要

When solutions of the high dimensional PDE involve fast propagating shock waves, existing variational moving mesh methods are rather unstable and time consuming. To circumvent the problem, we improve the existing variational method by adding one term which adapts the mesh to the dynamic variations of the PDEs during time iterations. This is an extension of the moving mesh strategy in (Gao, 2018) for one dimensional (1-D) problems. We prove the stability and convergence of the present moving mesh equation for arbitrary initial mesh. Consequently, by enlarging the time step sizes of mesh generation, the computational time is shortened without causing instability. The moving mesh equation and the PDE are simulated step by step applying the alternate finite difference method. Both theoretical analysis and numerical results imply that the method is highly effective and robust when simulating high dimensional time dependent PDEs with fast moving shock waves.