A quasi-Lagrangian moving mesh discontinuous Galerkin method for hyperbolic conservation laws
For computational scientists solving hyperbolic conservation laws, this method offers a way to dynamically adapt meshes without interpolation, improving accuracy for problems with shocks and smooth regions.
The paper presents a moving mesh discontinuous Galerkin method for hyperbolic conservation laws that combines DG with a mesh movement strategy based on moving mesh PDEs, achieving theoretically predicted convergence orders for smooth solutions and capturing shocks without interpolation. Numerical results demonstrate advantages over uniform meshes and insensitivity to mesh smoothness.
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.