Using High‐Order Methods on Adaptively Refined Block‐Structured Meshes: Derivatives, Interpolations, and Filters

Abstract

Block‐structured adaptive mesh refinement (AMR) is used for efficient resolution of partial differential equations (PDEs) solved on large computational domains by clustering mesh points only where required by large gradients. Previous work has indicated that fourth‐order convergence can be achieved on such meshes by using a suitable combination of high‐order derivative, interpolation, and filter stencils and can deliver significant computational savings over conventional second‐order methods at engineering error tolerances. In this paper, we explore the interactions between the errors introduced by the discretized operators for derivatives, interpolations, and filters. We develop general expressions for high‐order derivative, interpolation, and filter stencils applicable in multiple dimensions, using a Fourier approach, to facilitate high‐order block‐structured AMR implementations. These stencils are derived under the assumption that the fields that they are applied to are smooth. For a given derivative stencil (and thus a given order of accuracy), the necessary interpolation order is found to be dependent on the highest spatial derivative in the PDE being solved. This is demonstrated empirically, using one‐ and two‐dimensional model equations, by observing the increase in delivered accuracy as the order of accuracy of the interpolation stencil is increased. We also examine the empirically observed order of convergence, as the effective resolution of the mesh is increased by successively adding levels of refinement, with different orders of derivative, interpolation, and filter stencils. The procedure devised here is modular and its various pieces, i.e., the derivative, interpolation, and filter stencils, can be extended independently to higher orders and dimensions (in the case of interpolation stencils). The application of these methods often requires nontrivial logic, especially near domain boundaries; this logic, along with the stencils used in this study, are available as a freely downloadable software library.