A Fourth-Order Embedded Boundary Finite Volume Method for the Unsteady Stokes Equations with Complex Geometries

TL;DR

This paper introduces a fourth-order finite volume embedded boundary method for unsteady Stokes equations that accurately handles complex geometries on Cartesian grids without mesh modifications, ensuring stability and high accuracy.

Contribution

The paper presents a novel fourth-order embedded boundary finite volume method that mitigates small cell issues and enforces divergence-free conditions without mesh modifications.

Findings

Achieves fourth-order accuracy in complex geometries

Remains stable despite small cut cells

Demonstrated effectiveness with bio-inspired material application

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 bio-inspired material. The developed method achieves fourth-order accuracy and is stable despite the pervasive small cells arising from complex geometries.