Title:
Moving mesh split form discontinuous Galerkin methods to solve conservation laws
Abstract:
The construction of high order nodal discontinuous Galerkin (DG) methods to solve conservation laws and related equations includes the approximation of non-linear flux terms in the volume integrals. These terms can lead to aliasing and stability issues in the DG approximation. Aliasing issues arises when the flux terms are composed of products of polynomials, and are then interpolated by a polynomial basis of a lower order than the product of the polynomials. The split form DG framework provides a tool to construct the numerical approximation in a way that aliasing issues caused by the discretization of the DG volume integrals are avoided. In particular the split form DG framework can be used to construct entropy stable DG methods.
On the other hand the r-adaptive method or moving mesh method involves the re-distribution of the mesh nodes in regions of rapid variation of the solution. In comparison with h-adaptive discretizations, where the mesh is refined and coarsened by changing the number of elements in the tessellation, the r-adaptive method has some advantages, e.g. no hanging nodes appear and the number of elements does not change. In this talk, the focus is on the construction of moving mesh entropy stable split form DG methods. Thereby, a proper methodology to compute the grid point distribution to move the mesh will be not discussed. Numerical experiments for the three dimensional compressible Euler equations will be presented to show the capability of the moving mesh split form DG methods.