Abstract
The matched interface and boundary method (MIB) and ghost fluid method (GFM) are two well-known methods for solving elliptic interface problems. Moreover, they can be coupled with efficient time advancing methods, such as the alternating direction implicit (ADI) methods, for solving time-dependent partial differential equations (PDEs) with interfaces. However, to our best knowledge, all existing interface ADI methods for solving parabolic interface problems concern only constant coefficient PDEs, and no efficient and accurate ADI method has been developed for variable coefficient PDEs. In this work, we propose to incorporate the MIB and GFM in the framework of the ADI methods for generalized methods to solve two-dimensional parabolic interface problems with variable coefficients. Various numerical tests are conducted to investigate the accuracy, efficiency, and stability of the proposed methods. Both the semi-implicit MIB-ADI and fully-implicit GFM-ADI methods can recover the accuracy reduction near interfaces while maintaining the ADI efficiency. In summary, the GFM-ADI is found to be more stable as a fully-implicit time integration method, while the MIB-ADI is found to be more accurate with higher spatial and temporal convergence rates.
Highlights
This work aims to develop alternating direction implicit (ADI) finite difference methods for solving a two-dimensional (2D) parabolic equation ∂u ∂= ∇ · ( β∇u) + f = β u+ β u + f, (1) ∂t ∂x ∂y∂y over a rectangular domain Ω ⊂ R2 with a Dirichlet boundary condition imposed on the boundary ∂Ω and an initial condition defined at an initial time, i.e., t = 0
The matched ADI (mADI) method [10,12,13] will be generalized for solving variable coefficient parabolic interface problems, in which the ADI scheme serves as a time stepper
Based on results presented in Examples 1 and 2, it is reasonable to believe that ADID1 is superior to ADIPR to be used for temporal discretization in terms of convergence and stability—both ADID1-involved methods converge in almost all tested cases, and the two ADIPR-involved methods converge only when the time step ∆t is small, and the results they obtain are not seen to be more accurate than those obtained by matched interface and boundary (MIB)-ADID1
Summary
This work aims to develop alternating direction implicit (ADI) finite difference methods for solving a two-dimensional (2D) parabolic equation. In solving a simplified parabolic interface problem with the material coefficient β being a constant throughout the domain, the IIMADI scheme introduces some correction terms in the ADI formulation so that the accuracy near interfaces can be recovered back to second order. Later, this IIM-ADI scheme was applied to other parabolic PDEs [28,29]. The mADI method [10,12,13] will be generalized for solving variable coefficient parabolic interface problems, in which the ADI scheme serves as a time stepper.
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