Optimal Error Estimate of a Discontinuous Galerkin Method for One-Dimensional Linear Hyperbolic Equation with Degenerate Points Moving Along Space-Time Curves

In this paper we consider a discontinuous Galerkin method with purely upwind numerical flux to solve one-dimensional linear variable-coefficient hyperbolic equation, where the flow speed has different positive and negative signs when passing through the degenerate points. Our purpose is to give a rigorous proof of the optimal L2-norm error estimate when the degenerate points move along smooth curves depending on both space and time. The main difficulty is how to deal with the signs’ change of the flow speed regarding time. To this end, we propose a novel analysis framework with the help of a time-dependent projection based on the hybrid application of Gauss–Radau projections. First of all, we give a mesh-dependent subdivision of the computational domain and elaborately determine the space-time distribution of troubled locations, in which the type of Gauss–Radau projections suddenly switches with respect to the time variable. Then, we propose a union of bilinear forms (UBFs) with jump conditions to reflect the jumps along the time direction on troubled locations, for which a sharp boundedness concerning the accumulation of all involved jumps is proved. Finally, the optimal convergence order is derived by using the approximation properties of the time-dependent projection and the sharp boundedness for the UBFs. Numerical experiments are also given to validate the optimal order of accuracy.