High order discontinuous Galerkin methods for hyperbolic conservation laws

초록

In this talk, we present a novel high-order discontinuous Galerkin method with Lagrange multiplier (DGLM) for hyperbolic conservation laws. Lagrange multipliers are introduced on the inter-element boundaries via the concept of weak divergence. Static condensation on element unknowns considerably reduces globally coupled degrees of freedom, resulting in the stiffness equations in the Lagrange multipliers only. We establish stability results and provide conditions on the stabilization parameter, which plays a role in capturing shocks and discontinuities as well. The error estimates are derived in energy norm. Accuracy tests are performed, which shows optimal convergence in L2 norms. Numerical results indicate that the DGLM has potentials in delivering high order accurate information for various problems in hyperbolic conservation laws. Numerical examples include inviscid Burgers’ equations, shallow water equations (subcritical flow and supercritical upstream, subcritical downstream flow), and compressible Euler equations (Sod's Shock Tube and Intersection of Mach 3).