Exploration of consistent numerical integration for a 2D Lagrangian discontinuous Galerkin (DG) hydro dynamic method

摘要

Cell-centered discontinuous Galerkin (DG) methods have been developed for solving the 2D gas dynamic equations in Lagrangian hydrodynamics. Different from finite volume (FV) methods, a volume integral and a surface integral must be calculated with DG methods. The challenge is that the surface integration accuracy needs to be consistent with the volume integration; otherwise, a spurious velocity component is produced that is perpendicular to the flow direction on ID problems using a 2D code with a square mesh. Gauss-Lobatto quadrature is commonly used with Lagrangian FV and DG methods for the surface integration because the integration points coincide with the vertices of the element. The volume integrals in DG methods are commonly approximated using Gauss-Legendre quadrature, which is exact for a polynomial up to a degree of In - 1 with n quadrature points. Since the Gauss-Lobatto quadrature is only exact for a polynomial up to a degree of In - 3, this necessitates the usage of more quadrature points on the face (i.e., higher-order elements) for ensuring that the surface integration has commensurate accuracy to the volume integration. To obtain a consistent numerical integration with linear elements, we propose a consistent mixed Gauss quadrature rule for the surface and volume integrals in the DG evolution equations. The term 'consistent' means that the surface integration method is sufficiently accurate to eliminate the spurious velocity component perpendicular to the flow direction for ID problems on 2D meshes, and the term 'mixed' means that the Gauss-Lobatto and Gauss-Legendre quadrature rules are both used for approximating the integrals. The consistent mixed quadrature approach will integrate the surface fluxes involving the velocity using the Gauss-Labatto quadrature rule, and then integrate all other integrals using the Gauss-Legendre quadrature rule including the surface fluxes involving the mechanical stress. The commonly used integration approach with DG methods is a mixed quadrature approach because the surface integration rule is different from the volume integration rule; however, it is not consistent in terms of accuracy. In this work, we will show that the consistent mixed Gauss quadrature rule will remove the spurious velocity component and that it delivers the designed second-order accuracy on linear elements. The accuracy and robustness of the Lagrangian DG hydrodynamic method with this consistent mixed Gauss quadrature rule are demonstrated by calculating a suite of challenging test problems with linear meshes. We also present results using a combination of various quadrature rules.