Superconvergence of discontinuous Galerkin solutions for a nonlinear scalar hyperbolic problem

Abstract

In this paper we study the superconvergence of the discontinuous Galerkin solutions for nonlinear hyperbolic partial differential equations. On the first inflow element we prove that the p-degree discontinuous finite element solution converges at Radau points with an O( h p+2 ) rate. We further show that the solution flux converges on average at O( h 2 p+2 ) on element outflow boundary when no reaction terms are present. For reaction–convection problems we establish an O( h min(2 p+2, p+4) ) superconvergence rate of the flux on element outflow boundary. Globally, we prove that the flux converges at O( h 2 p+1 ) on average at the outflow of smooth-solution regions for nonlinear conservation laws. Numerical computations indicate that our results extend to nonrectangular meshes and nonuniform polynomial degrees. We further include a numerical example which shows that discontinuous solutions are superconvergent to the unique entropy solution away from shock discontinuities.