Date of Award

Spring 5-2022

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics & Statistics

Program/Concentration

Computational and Applied Mathematics

Committee Director

Yan Peng

Committee Member

Richard D. Noren

Committee Member

Ruhai Zhou

Committee Member

Zhilin Li

Abstract

Interface problems have many applications in physics. In this dissertation, we develop a direct method for solving three-dimensional elliptic interface problems and study their application in solving parabolic interface problems. As many of the physical applications of interface problems can be approximated with partial differential equations (PDE) with piecewise constant coefficients, our derivation of the model is focused on interface problems with piecewise constant coefficients but have a finite jump across the interface. The critical characteristic of the method is that our computational framework is based on a finite difference scheme on a uniform Cartesian grid system and does not require an augmented variable as in the augmented approach. So the implementation of the method is easier to understand for non-experts in the area. The discretization of the PDE uses the standard seven-point central difference scheme for grid points away from the interface and a twenty-seven-point compact scheme that considers the jump discontinuities in the solution, flux, and jump ratio for grid points near or on the interface. As a result, the developed model can obtain second-order accuracy globally for both the solution and the solution's gradient. Moreover, our numerical experiment indicates that eigenvalues of the coefficient matrix of the resulting linear system for the finite difference scheme are located in the left half-plane, implicating our method's stability. We have also developed a model for solving two and three-dimensional parabolic interface problems using the Crank-Nicolson scheme and some modifications into the direct immersed interface method (IIM). The developed model can accurately capture the discontinuities in the solution across the interface and achieve second-order accuracy for both the solution and the solution's gradient in both space and time.

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DOI

10.25777/1kwa-v554

ISBN

9798819393352

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