A curved boundary treatment for discontinuous Galerkin schemes solving time dependent problems

For problems defined in a two-dimensional domain Ω with boundary conditions specified on a curve Γ, we consider discontinuous Galerkin (DG) schemes with high order polynomial basis functions on a geometry fitting triangular mesh. It is well known that directly imposing the given boundary conditions on a piecewise segment approximation boundary Γh will render any finite element method to be at most second order accurate. Unless the boundary conditions can be accurately transferred from Γ to Γh, in general curvilinear element method should be used to obtain high order accuracy. We discuss a simple boundary treatment which can be implemented as a modified DG scheme defined on triangles adjacent to Γh. Even though integration along the curve is still necessary, integrals over any curved element are avoided. If the domain Ω is convex, or if Ω is nonconvex and the true solutions can be smoothly extended to the exterior of Ω, the modified DG scheme is high order accurate. In these cases, numerical tests on first order and second order partial differential equations including hyperbolic systems and the scalar wave equation suggest that it is as accurate as the full curvilinear DG scheme.