A Cartesian Grid Method for Solving Stokes Equations with Discontinuous Viscosity and Singular Sources

Abstract

In this paper, we present a new finite difference scheme for solving incompressible, steady stokes equations in discontinuous domains. While solving two phase Stokes equations, across some interfaces, there are discontinuities in the pressure, viscosity and velocity. Since, these jump conditions are coupled together, it is difficult to discretise and solve the system of equations accurately. We apply the augmented approach introduced by Li et al. (Int J Numer Meth. Fluids 44:33–53, 2004) to decouple these jump conditions. A new finite difference method is then developed and presented to solve the resulting augmented system of Stokes equations. Points of intersection of grid lines and the interface are used as a node in the finite difference stencil and jump conditions are then used to determine the values at these intersection points. Numerical solutions are compared with the corresponding analytical solutions and those of the augmented immersed interface method. The method is found to be second order accurate for almost all the variables in the infinity norm.