A spacetime discontinuous Galerkin method for scalar conservation laws

Abstract

We present a spacetime discontinuous Galerkin (SDG) finite element method for scalar hyperbolic conservation laws. The method is consistent with the integral forms of the conservation law and its associated entropy condition written in terms of the physical Godunov flux on each spacetime element. The discrete basis functions are piecewise continuous in spacetime, admitting discontinuities across element boundaries. The resulting Bubnov–Galerkin method is high-order stable for both convex and non-convex flux functions; it does not require stabilization beyond the basic Galerkin projection. The SDG method is applicable to both layered and unstructured spacetime grids. An element-by-element solution scheme delivers O( N) computational complexity (where N is the number of elements) when applied to spacetime meshes that conform to a special causality constraint. We employ two different limiters to control the local overshoot and undershoot that the basic SDG projection generates near shocks.