A multigrid based finite difference method for solving parabolic interface problem

Hongsong Feng; Shan Zhao; Hongsong Feng; Shan Zhao

doi:10.3934/era.2021031

Electronic Research Archive

2021, Volume 29, Issue 5: 3141-3170. doi: 10.3934/era.2021031

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A multigrid based finite difference method for solving parabolic interface problem

Hongsong Feng ,
Shan Zhao ^,

Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487, USA

Received: 01 November 2020 Revised: 01 February 2021 Published: 15 April 2021
65M06, 65M55, 65M12

In this paper, a new Cartesian grid finite difference method is introduced to solve two-dimensional parabolic interface problems with second order accuracy achieved in both temporal and spatial discretization. Corrected central difference and the Matched Interface and Boundary (MIB) method are adopted to restore second order spatial accuracy across the interface, while the standard Crank-Nicolson scheme is employed for the implicit time stepping. In the proposed augmented MIB (AMIB) method, an augmented system is formulated with auxiliary variables introduced so that the central difference discretization of the Laplacian could be disassociated with the interface corrections. A simple geometric multigrid method is constructed to efficiently invert the discrete Laplacian in the Schur complement solution of the augmented system. This leads a significant improvement in computational efficiency in comparing with the original MIB method. Being free of a stability constraint, the implicit AMIB method could be asymptotically faster than explicit schemes. Extensive numerical results are carried out to validate the accuracy, efficiency, and stability of the proposed AMIB method.
- Parabolic interface problem,
- matched interface and boundary (MIB),
- multigrid method,
- Schur complement,
- gradient recovery
Citation: Hongsong Feng, Shan Zhao. A multigrid based finite difference method for solving parabolic interface problem[J]. Electronic Research Archive, 2021, 29(5): 3141-3170. doi: 10.3934/era.2021031

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Abstract

In this paper, a new Cartesian grid finite difference method is introduced to solve two-dimensional parabolic interface problems with second order accuracy achieved in both temporal and spatial discretization. Corrected central difference and the Matched Interface and Boundary (MIB) method are adopted to restore second order spatial accuracy across the interface, while the standard Crank-Nicolson scheme is employed for the implicit time stepping. In the proposed augmented MIB (AMIB) method, an augmented system is formulated with auxiliary variables introduced so that the central difference discretization of the Laplacian could be disassociated with the interface corrections. A simple geometric multigrid method is constructed to efficiently invert the discrete Laplacian in the Schur complement solution of the augmented system. This leads a significant improvement in computational efficiency in comparing with the original MIB method. Being free of a stability constraint, the implicit AMIB method could be asymptotically faster than explicit schemes. Extensive numerical results are carried out to validate the accuracy, efficiency, and stability of the proposed AMIB method.

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