Higher Order Immersed Interface Methods for Hyperbolic Equations in Heterogeneous Media

Abstract

We extend the immersed interface method for solving linear hyperbolic equations in heterogeneous media to higher order accuracy. The coefficient functions of the hyperbolic equations considered are assumed to be piecewise constant. We extend the treatment on each interface of discontinuity in [8] from second order accurate to fourth order accurate. By combining the resulting high order interface treatment with high order finite difference schemes, e.g. ENO-ROE schemes, for solving the hyperbolic equations in smooth regions, the resulting immersed interface method is capable of providing sharp resolution of the wave propagation on heterogeneous media without using very fine grids.