Immersed interface methods for moving interface problems

A second order difference method is developed for the nonlinear moving interface problem of the form\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$u_t + \lambda uu_x = \left( {\beta u_x } \right)_x - f\left( {x,t} \right),x \in \left[ {\left. {0,\alpha } \right) \cup \left( {\left. {\alpha ,1} \right]} \right.,} \right.$$ \end{document}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\frac{{d\alpha }}{{dt}} = w\left( {t,\alpha ;u,u_x } \right),$$ \end{document}, where α (t) is the moving interface. The coefficient β(x,t) and the source term f(x,t) can be discontinuous across α (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 α (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 α (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.