Very High-Order Accurate Discontinuous Galerkin Method for Curved Boundaries with Polygonal Meshes

Preserving the optimal convergence order of discontinuous Galerkin (DG) discretisations in curved domains is a critical and well-known issue. The proposed approach relies on the reconstruction for off-site data (ROD) method developed originally within the finite volume framework. The main advantages are simplicity, since the PDE solver only considers polygonal domains, and versatility, since any type of boundary condition can be imposed. The developed DG–ROD method consists in splitting the boundary conditions treatment and the leading discrete equations from a classical DG formulation into two independent solvers coupled in a simple and efficient iterative procedure. A numerical benchmark is provided to assess the capability of the method with Dirichlet and Neumann boundary conditions prescribed on curved boundaries, demonstrating that the optimal convergence order is effectively achieved.