A Jump Condition Capturing Scheme For Elliptic Interface Problems

The immersed interface method is general technique for solving differential equations with interface problems and now it is applied in various aspects of fluid dynamics. Interface problems usually lead to differential equations whose solutions and its derivatives have discontinuities or non-smoothness across some interface. Many differential equation numerical methods do not work efficiently for interface problems.In this paper the immersed interface method for solving elliptic interface prob-lems in 2D is extended to the situation when interface and jumps are implicitly provided by functions on the grid points in a neighborhood of the interface. The resulting scheme is particularly suitable for interface capturing methods such as the level set method. Formulations of the original immersed interface method are mod-ified for this implicit setting, including the interface jump relations, the algebraic equations for determining the coefficients of the finite difference scheme at irregular grid points and the correction terms. The local truncation error at grid points just adjacent to the interface is of first order. Since the interface is of co-dimension one, the global second order accuracy for the solution in maximum norm can be achieved, as demonstrated by numerical examples. It is easier to implement since standard Lagrange interpolation is used to compute the derivatives of the interface quantities.