Dongmi Luo

Dongmi Luo, Weizhang Huang, Jianxian Qiu

A moving mesh discontinuous Galerkin method is presented for the numerical solution of hyperbolic conservation laws. The method is a combination of the discontinuous Galerkin method and the mesh movement strategy which is based on the moving mesh partial differential equation approach and moves the mesh continuously in time and orderly in space. It discretizes hyperbolic conservation laws on moving meshes in the quasi-Lagrangian fashion with which the mesh movement is treated continuously and no interpolation is needed for physical variables from the old mesh to the new one. Two convection terms are induced by the mesh movement and their discretization is incorporated naturally in the DG formulation. Numerical results for a selection of one- and two-dimensional scalar and system conservation laws are presented. It is shown that the moving mesh DG method achieves the theoretically predicted order of convergence for problems with smooth solutions and is able to capture shocks and concentrate mesh points in non-smooth regions. Its advantage over uniform meshes and its insensitiveness to mesh smoothness are also demonstrated.

Dongmi Luo, Weizhang Huang, Jianxian Qiu

A hybrid LDG-HWENO scheme is proposed for the numerical solution of KdV-type partial differential equations. It evolves the cell averages of the physical solution and its moments (a feature of Hermite WENO) while discretizes high order spatial derivatives using the local DG method. The new scheme has the advantages of both LDG and HWENO methods, including the ability to deal with high order spatial derivatives and the use of a small number of global unknown variables. The latter is independent of the order of the scheme and the spatial order of the underlying differential equations. One and two dimensional numerical examples are presented to show that the scheme can attain the same formal high order accuracy as the LDG method.