Foundations for high-order, conservative cut-cell methods: Stable discretizations on degenerate meshes

Abstract

Cut-cell methods for unsteady flow problems can greatly simplify the grid generation process and allow for high-fidelity simulations on complex geometries. However, cut-cell methods have been limited to low orders of accuracy. This is driven, largely, by the variety of procedures typically introduced to evaluate derivatives in a stable manner near the highly irregular embedded geometry. In the present work, a completely new approach, termed TEMO (truncation error matching and optimization), is taken to solve this problem. The approach is based on two simple and intuitive design principles. These principles directly allow for the construction of stable 8th order approximations to elliptic and parabolic problems. In addition, when combined with the non-linear optimization process of Ref. [6], these principles allow for stable and conservative 4th order approximations to hyperbolic problems without the addition of numerical dissipation. To the best of the authors' knowledge, these are the highest orders ever achieved for a cut-cell discretization by a significant margin. This is done for both explicit and compact finite differences and is accomplished without any geometric transformations or artificial stabilization procedures.