A high-order nonlinear limiter for discontinuous Galerkin method on parallel adaptive Cartesian grids

The discontinuous Galerkin (DG) method has been widely adopted due to its excellent properties. However, the problem of designing a class of high-order limiter that takes into account accuracy, compactness, efficiency, and robustness has long been an open question in simulating compressible flow with strong discontinuities. In this paper, a high-order multi-resolution weighted essentially non-oscillatory (MR-WENO) limiter is designed for the DG method on a parallel adaptive Cartesian grid, based directly on the weak solution to a polynomial obtained by the DG method. It can gradually be reduced to first-order accuracy in the vicinity of discontinuities while maintaining the excellent features of the DG method. Thus, it essentially has non-oscillatory characteristics in non-smooth regions with respect to the adaptive Cartesian grids. An improved shock detection technique is adopted as an indicator to identify troubled cells, which forms a high-order limiting procedure. A high-order MR-WENO limiter is used for both two- and three-dimensional cases to reconstruct different degrees of freedom on adaptive Cartesian grids. If the mesh is refined or coarsened, the details of the implementation algorithm are presented to determine how the hanging nodes are modulated and how the numerical solutions are redefined on such adaptive Cartesian grids. The parallelization of this method can be achieved by linking to the octree-based adaptive mesh refinement library called p4est. Finally, the low dissipation, shock capture ability, and load balancing of the high-order DG method with an MR-WENO limiter may enhance the resolutions of nearby strong discontinuities in adaptive Cartesian grids.