A Finite Difference Method for an Interface Problem with a Nonlinear Jump Condition

Abstract

We propose a finite difference approach to numerically solve an interface heat equation in one dimension with discontinuous conductivity and nonlinear interface condition. The discontinuous physical solution is sought among the multiple solutions of the nonlinear equation. Our method finds the approximate jump of the exact solution by two auxiliary linear problems with finite jumps. The approximate physical solution is then obtained by a weighted sum. The convergence and stability of the method are analyzed by the method of nonnegative matrices. Numerical examples are given to confirm the theory. In particular, numerical simulations are demonstrated in regards to the study of polymetric ion-selective electrodes and ion sensors.