| To simulate fluid-solid interaction or two-fluid flows on Cartesian grids by the immersed interface method, we incorporate into a numerical scheme the jump conditions of the first and second order Cartesian derivatives of the velocity and pressure [SIAM Journal on Scientific Computing Vol. 27, No. 6, pp. 1948-1980]. These Cartesian jump conditions can be systematically derived from the principal jump conditions for the velocity and pressure , i.e. the jump conditions of the velocity and pressure, their normal derivatives and their Laplacians. However, the previous derivation requires the global parameterization of a fluid-solid or two-fluid interface. Here, we propose a new derivation that is based on the triangulation of an interface to avoid the global interface parameterization. This new derivation makes the immersed interface method more robust for applications. We test our new derivation by solving a Poisson equation with a discontinuous solution across triangulated interfaces shaped as a sphere, a cube, a cylinder and a cone. Furthermore, we parallelize our serial code so that we can solve the pressure Poisson equation with a large number of interfaces in multiple subdomains using a parallel multigrid Poisson solver that utilizes the Message Passing Interface (MPI). |
