A high-order vertex-centered quasi-Lagrangian discontinuous Galerkin method for compressible Euler equations in two-dimensions

Abstract This article introduces a new vertex-centered quasi-Lagrangian discontinuous Galerkin method for two-dimensional compressible flows that is third-order accurate both in space and time. The computational domain is divided into structured quadrilateral cells with straight edges. Nodal control volumes with curved edges are constructed surrounding the grid vertices. The Euler equations in arbitrary Lagrangian-Eulerian (ALE) form are discretized on these nodal control volumes using a discontinuous Galerkin method. The time marching is implemented by the third-order strong-stability-preserving Runge-Kutta method. A compact HWENO reconstruction algorithm is used as limiter to eliminate spurious oscillations near discontinuities. The polynomial expression of fluid velocity defined in a nodal control volume is also obtained from the reconstruction procedure, and is used to calculate the moving velocity of corresponding grid vertex. In this way, the grid vertices are moved in a rigorously Lagrangian manner, although there are still mass fluxes between neighbouring nodal control volumes. Therefore the scheme is called as a quasi-Lagrangian one. The scheme is conservative for mass, momentum and total energy. Some numerical tests are carried out to demonstrate the accuracy and robustness of the scheme. It has a favorable qualitative behavior for discontinuous problems and optimal convergence rates for smooth problems.