High-order discontinuous Galerkin methods with Lagrange multiplier for hyperbolic systems of conservation laws

In this work, we present novel high-order discontinuous Galerkin methods with Lagrange multiplier (DGLM) for hyperbolic systems of 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 first establish stability results and provide conditions on the stabilization parameter, which plays an important role in resolving discontinuities as well. Accuracy tests are then performed, which shows optimal convergence in the L² norm. Extensive 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 2D circular dam break), and compressible Euler equations (Intersection of Mach 3 and Sod's shock tube).