A multigrid based finite difference method for solving parabolic interface problem

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.

Alternating Direction Implicit (ADI) Methods for Solving Two-Dimensional Parabolic Interface Problems with Variable Coefficients

“…Consequently, carefully designed numerical procedures are required to account for jump conditions (4) in the numerical formulations so that numerical accuracy near the interface can be recovered. Such Cartesian grid interface algorithms have been successfully developed in many finite difference methods [4,6,8,[10][11][12][13][14][15][16][17] and finite element methods [18][19][20].…”

“…Comparing with general interface algorithms for parabolic PDEs [6,8,11,[16][17][18][19][20], the interface treatment in the ADI framework is more subtle. This is because the jump conditions (4) are prescribed in the normal direction so that the usual interface treatments naturally couple spatial derivatives in all Cartesian directions.…”