Lawrence Berkeley National Laboratory

In this paper, we present components of a fourth-order finite volume method that will eventually allow for solving the incompressible Navier-Stokes equations. The algorithm represents complex geometries on a Cartesian embedded boundary grid, and takes special steps to achieve stable and accurate discretizations. Spatial discretizations are based on a weighted least-squares technique that mitigates the “small cut cell” problem, without mesh modifications, cell merging, or redistribution. Solutions are advanced in time using a projection method coupled with a higher-order additive implicit-explicit (ImEx) Runge-Kutta method, where the diffusion and advection terms are treated implicitly and explicitly, respectively. We demonstrate convergence tests for a scalar advection-diffusion equation and an approximate projection solver and confirm fourth-order accuracy for complex geometries. These methods are key components of solvers for the incompressible Navier-Stokes equations, and indicate feasibility of achieving higher-order accuracy on an arbitrary cut cell mesh.