### Electronic Research Archive

2021, Issue 5: 3141-3170. doi: 10.3934/era.2021031
Special Issues

# A multigrid based finite difference method for solving parabolic interface problem

• 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.

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

### Related Papers:

• 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.

 [1] New geometric immersed interface multigrid solvers. SIAM J. Sci. Comput. (2004) 25: 1516-1533. [2] The immersed interface/multigrid methods for interface problems. SIAM J. Sci. Comput. (2002) 24: 463-479. [3] Convergence of an immersed finite element method for semilinear parabolic interface problems. Appl. Math. Sci. (Ruse) (2011) 5: 135-147. [4] A decomposed immersed interface method for variable coefficient elliptic equations with non-smooth and discontinuous solutions. J. Comput. Phys. (2004) 197: 364-386. [5] F. Bouchon and G. H. Peichl, An immersed interface technique for the numerical solution of the heat equation on a moving domain, Numerical Mathematics and Advanced Applications 2009, Springer Berlin Heidelberg, (2010), 181–189. doi: 10.1007/978-3-642-11795-4_18 [6] The immersed interface technique for parabolic problems with mixed boundary conditions. SIAM J. Numer. Anal. (2010) 48: 2247-2266. [7] Robust multigrid methods for nonsmooth coefficient elliptic linear systems. J. Comput. Appl. Math. (2000) 123: 323-352. [8] Finite element methods and their convergence for elliptic and parabolic interface problems. Numer. Math. (1998) 79: 175-202. [9] Finite-difference ghost-point multigrid methods on Cartesian grids for elliptic problems in arbitrary domains. J. Comput. Phys. (2013) 241: 464-501. [10] Second order finite-difference ghost-point multigrid methods for elliptic problems with discontinuous coefficients on an arbitrary interface. J. Comput. Phys. (2018) 361: 299-330. [11] On the numerical integration of $\frac{\partial ^2 u}{\partial x^2 } + \frac{\partial ^2 u}{\partial y^2 } = \frac{\partial u}{\partial t}$ by implicit methods. J. Soc. Indust. Appl. Math. (1955) 3: 42-65. [12] Numerical solution of two-dimensional heat-flow problems. AIChEJ. (1955) 1: 505-512. [13] Immersed finite element method for interface problems with algebraic multigrid solver. Commun. Comput. Phys. (2014) 15: 1045-1067. [14] An augmented matched interface and boundary (MIB) method for solving elliptic interface problem. J. Comput. Appl. Math. (2019) 361: 426-443. [15] H. Feng and S. Zhao, FFT-based high order central difference schemes for the three-dimensional Poisson's equation with various types of boundary conditions, J. Comput. Phys., 410 (2020), 109391, 24 pp. doi: 10.1016/j.jcp.2020.109391 [16] H. Feng and S. Zhao, A fourth order finite difference method for solving elliptic interface problems with the FFT acceleration, J. Comput. Phys., 419 (2020), 109677, 25 pp. doi: 10.1016/j.jcp.2020.109677 [17] A fourth order accurate discretization for the Laplace and heat equations on arbitrary domains, with applications to the Stefan problem. J. Comput. Phys. (2005) 202: 577-601. [18] The immersed interface method for a nonlinear chemical diffusion equation with local sites of reactions. Numer. Algorithms (2004) 36: 285-307. [19] The immersed interface method for two-dimensional heat-diffusion equations with singular own sources. Appl. Numer. Math. (2007) 57: 486-497. [20] On multiscale ADI methods for parabolic PDEs with a discontinuous coefficient. Multiscale Model. Simul. (2018) 16: 1623-1647. [21] Accurate solution and gradient computation for elliptic interface problems with variable coefficients. SIAM J. Numer. Anal. (2017) 55: 570-597. [22] ADI methods for heat quations with discontinuous along an arbitrary interface. Proc. Sympos. Appl. Math. (1993) 48: 311-315. [23] Alternating direction ghost-fluid methods for solving the heat equation with interfaces. Comput. Math. Appl. (2020) 80: 714-732. [24] A matched Peaceman–Achford ADI method for solving parabolic interface problems. Appl. Math. Comput. (2017) 299: 28-44. [25] Partially penalized immersed finite element methods for parabolic interface problems. Numer. Methods Partial Differential Equations (2015) 31: 1925-1947. [26] IIM-based ADI finite difference scheme for nonlinear convection–diffusion equations with interfaces. Appl. Math. Model. (2013) 37: 1196-1207. [27] A dimension by dimension splitting immersed interface method for heat conduction equation with interfaces. J. Comput. Appl. Math. (2014) 261: 221-231. [28] S. F. McCormick, Multigrid Methods, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1987. doi: 10.1137/1.9781611971057 [29] The numerical solution of parabolic and elliptic equations. J. Soc. Indust. Appl. Math. (1955) 3: 28-41. [30] Efficient symmetric discretization for the Poisson, heat and Stefan-type problems with Robin boundary conditions. J. Comput. Phys. (2010) 229: 875-889. [31] W. H. Press and S. A. Teukolsky, Numerical Recipes in FORTRAN, the Art of Scientific Computing, 2$^{nd}$ edition, Cambridge University Press, New York, 1992. [32] Optimal error estimates for linear parabolic problems with discontinuous coefficients. SIAM J. Numer. Anal. (2005) 43: 733-749. [33] Finite element methods for semilinear elliptic and parabolic interface problems. Appl. Numer. Math. (2009) 59: 1870-1883. [34] Symmetric interior penalty Galerkin approaches for two-dimensional parabolic interface problems with low regularity solutions. J. Comput. Appl. Math. (2018) 330: 356-379. [35] A boundary condition-capturing multigrid approach to irregular boundary problems. SIAM J. Sci. Comput. (2004) 25: 1982-2003. [36] A spatially second order alternating direction implicit (ADI) method for three dimensional parabolic interface problems. Comput. Math. Appl. (2018) 75: 2173-2192. [37] The explicit-jump immersed interface method: Finite difference methods for PDEs with piecewise smooth solutions. SIAM J. Numer. Anal. (2000) 37: 827-862. [38] MIB method for elliptic equations with multi-material interfaces. J. Comput. Phys. (2011) 230: 4588-4615. [39] Discontinuous Galerkin immersed finite element methods for parabolic interface problems. J. Comput. Appl. Math. (2016) 299: 127-139. [40] A matched alternating direction implicit (ADI) method for solving the heat equation with interfaces. J. Sci. Comput. (2015) 63: 118-137. [41] High-order FDTD methods via derivative matching for Maxwell's equations with material interfaces. J. Comput. Phys. (2004) 200: 60-103. [42] High order matched interface and boundary method for elliptic equations with discontinuous coefficients and singular source. J. Comput. Phys. (2006) 213: 1-30.
###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

1.604 0.8

Article outline

## Figures and Tables

Figures(8)  /  Tables(13)

• On This Site