Well-balanced discontinuous Galerkin scheme for 2 × 2 hyperbolic balance law

Abstract

The numerical solutions to nonlinear hyperbolic balance laws at (or near) steady state may develop spurious oscillations due to the imbalance between flux and source terms. In the present article, we study a high order well-balanced discontinuous Galerkin (DG) scheme for balance law with subsonic flow, which preserves equilibrium solutions of the flow exactly, and also provides non-oscillatory solutions for flow near equilibrium. The key technique is to reformulate the DG scheme in terms of global equilibrium variables which remain constant in space and time, and are obtained by rewriting the balance law in conservative form. We show that the proposed scheme is well-balanced and validate the scheme for various flows given by 2×2 hyperbolic balance law. We also extend the scheme to flows on networks, particularly to include the coupling conditions at nodes of the network.