A Second Order Virtual Node Algorithm for Stokes Flow Problems with Interfacial Forces and Irregular Domains

Abstract

We present a numerical method for the solution of the Stokes equations that handles both interfacial discontinuities and geometrically irregular flow domains. The method is efficient, easy to implement, second order accurate and yields discretely divergence free velocities. We discretize the equations using an embedded approach on a uniform MACgrid employing virtual nodes at interfaces and boundaries. Interfaces and boundaries are represented with a hybrid Lagrangian/level set method. We rewrite the Stokes equations as three Poisson equations and use the techniques developed in Bedrossian et al. (2010) [1] to impose jump and boundary conditions. We also use a final Poisson equation to enforce a discrete divergence-free condition. All four linear systems involved are symmetric positive definite with three of the four having the standard 5-point Laplace stencil everywhere. Numerical results indicate second order accuracy in L1 for both velocities and pressure.