Accurate Higher-order Embedded Boundary Methods without Small Cell Issues

We present a cut-cell finite volume approach for discretizing fluid dynamics conservation laws in complex geometries. We use an embedded boundary method, where the geometry is defined by the contour of an implicit function on a Cartesian grid, which greatly simplifies meshing for constructive solid geometry or data-driven interfaces. However, it does introduce the problem of arbitrarily "small cells": if different flux reconstruction functions are used on a given small volume, careful cancellations must be maintained to leading order. Naive approaches, such as one-sided differences, redistribution, or grid-line interpolations, can deteriorate accuracy and stability. We introduce a higher-order least squares reconstruction that maintains high accuracy and creates stencils with good stability and conditioning properties. We demonstrate this with classical fluids test problems on simple domains, where we can demonstrate stability and convergence. We will also show results for very complex geometries that still maintain accuracy and stability. With careful refinement of the geometric representation, we can also show very high convergence rates, with at least 4th order in space and time for a conservative, arbitrarily small-cell mesh.