Conservative solution transfer between anisotropic meshes for time-accurate hybridized discontinuous Galerkin methods

We present a hybridized discontinuous Galerkin (HDG) solver for general time-dependent balance laws. In particular, we focus on a coupling of the solution process for unsteady problems with our anisotropic mesh refinement framework. The goal is to properly resolve all relevant unsteady features with the smallest possible number of mesh elements, and hence to reduce the computational cost of numerical simulations while maintaining its accuracy. A crucial step is then to transfer the numerical solution between two meshes, as the anisotropic mesh adaptation is producing highly skewed, non-nested sequences of triangular grids. For this purpose, we adopt the Galerkin projection for the HDG solution transfer as it preserves the conservation of physically relevant quantities and does not compromise the accuracy of high-order method. We present numerical experiments verifying these properties of the anisotropically adaptive HDG method. Previously developed anisotropic mesh adaptation framework is coupled with a hybridized discontinuous Galerkin (HDG) solver for general time-dependent balance laws. Special emphasis is placed on the solution transfer between anisotropically adapted meshes such that the conservation of physically relevant quantities is preserved and the accuracy of high-order method is not compromised. This is achieved by so called Galerkin projection on each element of the mesh. These properties are verified by means of test cases having both smooth and discontinuous solutions. image