Immersed Interface Methods For Moving Interface Problems

. A second order difference method is developed for the nonlinear moving interface problem of the form u t + uux = (fiux) x \Gamma f(x; t); x 2 [ 0; ff ) [ ( ff; 1 ]; dff dt = w (t; ff; u; ux ) ; where ff(t) is the moving interface. The coefficients fi(x; t) and the source term f(x; t) can be discontinuous across ff(t) and moreover, f(x; t) may have a delta function singularity there. As a result, although the equation is parabolic, the solution u and its derivatives may be discontinuous across ff(t). Two typical interface conditions are considered. One condition occurs in Stefan-like problems in which the solution is known on the interface. A new stable interpolation strategy is proposed. The other type occurs in a one-dimensional model of Peskin's immersed boundary method in which only jump conditions are given across the interface. The Crank-Nicolson difference scheme with modifications near the interface is used to solve for the solution u(x; t) and the interface ff(t) simultan..