Open Access

ARTICLE

A new sixth-order immersed interface method for solving Poisson equations with straight interfaces

R. Itza Balam^1,2, M. Uh Zapata^1,2, J. Montalvo-Urquizo³

1 Consejo Nacional de Humanidades, Ciencias y Tecnologías, CONAHCYT, México
2 Centro de Investigación en Matemáticas A. C., CIMAT Unidad Mérida, México
3 School of Science and Engineering, Tecnológico de Monterrey, México

* Corresponding Authors: R. Itza Balam (), M. Uh Zapata (), J. Montalvo-Urquizo ()

Revista Internacional de Métodos Numéricos para Cálculo y Diseño en Ingeniería 2023, 39(4), 1-12. https://doi.org/10.23967/j.rimni.2023.12.002

Received 05 September 2023; Accepted 18 December 2023; Issue published 28 December 2023

Abstract

This paper introduces a sixth-order Immersed Interface Method (IIM) for addressing 2D Poisson problems characterized by a discontinuous forcing function with straight interfaces. In the presence of this discontinuity, the problem exhibits a non-smooth solution at the interface that divides the domain into two regions. Here, the IIM is employed to compute the solution on a fixed Cartesian grid. This method integrates necessary jump conditions resulting from the interface into the numerical schemes. In order to achieve a sixth-order method, the proposed approach combines implicit finite differences with the IIM. The proposed scheme is efficient because the matrix arising from discretization remains the same as in the smooth problem, and changes are made to the resulting linear system by introducing new terms on the right side. These supplementary terms account for the discontinuities in the solution and its derivatives, with calculations restricted near the interface. The paper demonstrates the accuracy of the proposed method through various numerical examples.

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Keywords

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