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) + lambda uu(x) = (beta u(x))(x) - f(x, t), x is an element of [0,alpha) boolean OR (alpha, 1], d alpha/dt = w(t, alpha; u, u(x)), where alpha(t) is the moving interface. The coefficient beta(x, t) and the source term f(x, t) can be discontinuous across alpha(t) and moreover, f(x, t) may have a delta or/and delta-prime function singularity there. As a result, although the equation is parabolic, the solution u and its derivatives may be discontinuous across alpha(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 alpha(t) simultaneously. Several numerical examples, including models of ice-melting and glaciation, are presented. Second order accuracy on uniform grids is confirmed both for the solution and the position of the interface.