An Immersed Interface Method for Parabolic Interface Problems with Nonlinear Jump Conditions
So-Hsiang Chou, Patrick Nyadjo Fonga
Abstract
We develop an immersed interface finite difference method for a one-dimensional nonlinear parabolic interface problem with jump condition \[ [u]α=λu+u-. \] The method combines a Crank--Nicolson immersed interface discretization with an \(s\)-parameter reduction of the nonlinear interface condition, thereby reducing the nonlinear coupling to a scalar quadratic equation. We also discuss two different viewpoints for combining Newton iteration with immersed interface discretization, namely the IIM--Newton and Newton--IIM formulations. Numerical experiments are presented to illustrate the behavior and accuracy of the method.
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