Open AccessA multigrid based finite difference method for solving parabolic interface problem
Author(s) -
Hongsong Feng,
Shan Zhao
Publication year - 2021
Publication title -
electronic research archive
Language(s) - English
Resource type - Journals
ISSN - 2688-1594
DOI - 10.3934/era.2021031
Subject(s) - multigrid method , discretization , finite difference , laplace operator , cartesian coordinate system , interface (matter) , mathematics , stability (learning theory) , finite difference method , boundary (topology) , grid , schur complement , algorithm , computer science , mathematical analysis , partial differential equation , eigenvalues and eigenvectors , geometry , parallel computing , physics , bubble , quantum mechanics , maximum bubble pressure method , machine learning
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.