Inverse Lax-Wendroff Boundary Treatment of Discontinuous Galerkin Method for 1D Conservation Laws

In this paper, we propose a new class of discontinuous Galerkin (DG) methods for solving 1D conservation laws on unfitted meshes. The standard DG method is used in the interior cells. For the small cut elements around the boundaries, we directly design approximation polynomials based on inverse Lax-Wendroff (ILW) principles for the inflow boundary conditions and introduce the post-processing to preserve the local conservation properties of the DG method. The theoretical analysis shows that our proposed methods have the same stability and numerical accuracy as the standard DG method in the inner region. An additional nonlinear limiter is designed to prevent spurious oscillations if a shock is near the boundary. Numerical results indicate that our methods achieve optimal numerical accuracy for smooth problems and do not introduce additional oscillations in discontinuous problems.