Towards a High-Order Embedded Boundary Finite Volume Method for the Incompressible Navier-Stokes Equations with Complex Geometries

Abstract

A fourth-order finite volume embedded boundary (EB) method is presented for the unsteady Stokes equations. The algorithm represents complex geometries on a Cartesian grid using EB, employing a technique to mitigate the "small cut-cell" problem without mesh modifications, cell merging, or state redistribution. Spatial discretizations are based on a weighted least-squares technique that has been extended to fourth-order operators and boundary conditions, including an approximate projection to enforce the divergence-free constraint. Solutions are advanced in time using a fourth-order additive implicit-explicit Runge-Kutta method, with the viscous and source terms treated implicitly and explicitly, respectively. Formal accuracy of the method is demonstrated with several grid convergence studies, and results are shown for an application with a complex bioinspired material. The developed method achieves fourth-order accuracy and is stable despite the pervasive small cells arising from complex geometries.

A High Order Cartesian Grid, Finite Volume Method for Elliptic Interface Problems

“…In order to compute ∇ • β∇u , we apply (18) to each surface integral in (15) and divide by the cell volume.…”

“…See Figure 4. As is done in [12] and [15], we employ a weighted least-squares approach to force stencil weights to decay with distance faster than the growth of the highest polynomial term. To each piece of data in d we assign a weight that is inversely related to its distance from the centroid of volume V p,i .…”