A discontinuous Galerkin-front tracking scheme and its optimal–optimal error estimation

An error estimate of optimal convergence rates and optimal error propagation (optimal–optimal) was given for the numerical solutions produced by the Runge–Kutta discontinuous Galerkin (RKDG) method on the scalar nonlinear conservation laws in the case of smooth solutions in Sun and Rumsey (2013). This manuscript generalizes the problem to the case of a piecewise smooth solution containing one fully developed shock. A front tracking technique is incorporated in the RKDG scheme to produce a numerical solution with a truly high order error. The numerical smoothness approach of Sun and Rumsey (2013) is generalized to this particular case of a discontinuous solution.