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Volume 22, Issue 6
A Finite Difference Method for an Interface Problem with a Nonlinear Jump Condition

So-Hsiang Chou, Caleb Khaemba & Alice Wachira

DOI: 10.4208/ijnam2025-1035

Int. J. Numer. Anal. Mod., 22 (2025), pp. 801-823.

Published online: 2025-08

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  • 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.

  • Keywords

Finite difference method, nonlinear parabolic problems, ion sensors, transmission equation.

  • AMS Subject Headings

65N15, 65N30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-22-801, author = {Chou , So-HsiangKhaemba , Caleb and Wachira , Alice}, title = {A Finite Difference Method for an Interface Problem with a Nonlinear Jump Condition}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2025}, volume = {22}, number = {6}, pages = {801--823}, 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.

}, issn = {2617-8710}, doi = {https://doi.org/10.4208/ijnam2025-1035}, url = {http://global-sci.org/intro/article_detail/ijnam/24294.html} }
TY - JOUR T1 - A Finite Difference Method for an Interface Problem with a Nonlinear Jump Condition AU - Chou , So-Hsiang AU - Khaemba , Caleb AU - Wachira , Alice JO - International Journal of Numerical Analysis and Modeling VL - 6 SP - 801 EP - 823 PY - 2025 DA - 2025/08 SN - 22 DO - http://doi.org/10.4208/ijnam2025-1035 UR - https://global-sci.org/intro/article_detail/ijnam/24294.html KW - Finite difference method, nonlinear parabolic problems, ion sensors, transmission equation. AB -

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.

Chou , So-HsiangKhaemba , Caleb and Wachira , Alice. (2025). A Finite Difference Method for an Interface Problem with a Nonlinear Jump Condition. International Journal of Numerical Analysis and Modeling. 22 (6). 801-823. doi:10.4208/ijnam2025-1035
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