An Immersed Interface Method for Parabolic Interface Problems with Nonlinear Jump Conditions

We develop an immersed interface finite difference method for a one-dimensional nonlinear parabolic interface problem with jump condition $$[u{]}_{\alpha }=\lambda u^{+}{u}^{-}\mathrm{.}$$ 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.