Immersed interface methods for moving interface problems*


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Abstract

A second order difference method is developed for the nonlinear moving interface problem of the formwhere 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.

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