A second-order-accurate symmetric discretization of the Poisson equation on irregular domains. (English) Zbl 0996.65108

Summary: We consider the variable coefficient Poisson equation with Dirichlet boundary conditions on an irregular domain and show that one can obtain second-order accuracy with a rather simple discretization. Moreover, since our discretization matrix is symmetric, it can be inverted rather quickly as opposed to the more complicated nonsymmetric discretization matrices found in other second-order-accurate discretizations of this problem. Multidimensional computational results are presented to demonstrate the second-order accuracy of this numerical method. In addition, we use our approach to formulate a second-order-accurate symmetric implicit time discretization of the heat equation on irregular domains. Then we briefly consider Stefan problems.


65N06 Finite difference methods for boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35K05 Heat equation
35R35 Free boundary problems for PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
80M20 Finite difference methods applied to problems in thermodynamics and heat transfer
80A22 Stefan problems, phase changes, etc.
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[1] Adalsteinsson, D.; Sethian, J., The fast construction of extension velocities in level set methods, J. comput. phys., 148, 2, (1999) · Zbl 0919.65074
[2] Almgren, R., Variational algorithms and pattern formation in dendritic solidification, J. comput. phys., 106, 337, (1993) · Zbl 0787.65095
[3] Atkins, P., physical chemistry, (1994), Freeman New York
[4] Beyer, R.; LeVeque, R., Analysis of a one-dimensional model for the immersed boundary method, SIAM J. numer. anal., 29, 332, (1992) · Zbl 0762.65052
[5] Chen, S.; Merriman, B.; Osher, S.; Smereka, P., A simple level set method for solving Stefan problems, J. comput. phys., 135, 8, (1997) · Zbl 0889.65133
[6] Fedkiw, R., A symmetric spatial discretization for implicit time discretization of Stefan type problems, (June 1998)
[7] Fedkiw, R.; Aslam, T.; Merriman, B.; Osher, S., A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method), J. comput. phys., 152, 457, (1999) · Zbl 0957.76052
[8] Fedkiw, R.; Aslam, T.; Xu, S., The ghost fluid method for deflagration and detonation discontinuities, J. comput. phys., 154, 393, (1999) · Zbl 0955.76071
[9] Fedkiw, R.; Marquina, A.; Merriman, B., An isobaric fix for the overheating problem in multimaterial compressible flows, J. comput. phys., 148, 545, (1999) · Zbl 0933.76075
[10] Golub, G.; Van Loan, C., matrix computations, (1989), Johns Hopkins Univ. Press Baltimore
[11] Johansen, H., Cartesian grid embedded boundary finite difference methods for elliptic and parabolic differential equations on irregular domains, (1997), Univ. of California Berkeley
[12] Johansen, H.; Colella, P., A cartesian grid embedded boundary method for Poisson’s equation on irregular domains, J. comput. phys., 147, 60, (1998) · Zbl 0923.65079
[13] Jones, W.; Menzies, K., Analysis of the cell-centered finite volume method for the diffusion equation, J. comput. phys., 165, 45, (2000) · Zbl 0972.65087
[14] Juric, D.; Tryggvason, G., A front-tracking method for dendritic solidification, J. comput. phys., 123, 127, (1996) · Zbl 0843.65093
[15] Kang, M.; Fedkiw, R.; Liu, X.-D., A boundary condition capturing method for multiphase incompressible flow, J. sci. comput., 15, 323, (2000) · Zbl 1049.76046
[16] Karma, A.; Rappel, W.-J., Quantitative phase field modeling of dendritic growth in two and three dimensions, Phys. rev., 57, 4323, (1998) · Zbl 1086.82558
[17] Kim, Y.-T.; Goldenfeld, N.; Dantzig, J., Computation of dendritic microstructures using a level set method, Phys. rev. E, 62, 2471, (2000)
[18] LeVeque, R.; Li, Z., The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM J. numer. anal., 31, 1019, (1994) · Zbl 0811.65083
[19] Li, Z., A fast iterative algorithm for elliptic interface problems, SIAM J. numer. anal., 35, 230, (1998) · Zbl 0915.65121
[20] Liu, X.-D.; Fedkiw, R.; Kang, M., A boundary condition capturing method for Poisson’s equation on irregular domains, J. comput. phys., 154, 151, (2000) · Zbl 0958.65105
[21] Nguyen, D.; Fedkiw, R.; Kang, M., A boundary condition capturing method for incompressible flame discontinuities, J. comput. phys., 172, 71, (2001) · Zbl 1065.76575
[22] Osher, S.; Sethian, J., Fronts propagating with curvature dependent speed: algorithms based on hamilton – jacobi formulations, J. comput. phys., 79, 12, (1988) · Zbl 0659.65132
[23] Peskin, C., Numerical analysis of blood flow in the heart, J. comput. phys., 25, 220, (1977) · Zbl 0403.76100
[24] Plapp, M.; Karma, A., Multiscale finite-difference-diffusion-monte – carlo method for simulating dendritic solidification, J. comput. phys., 165, 592, (2000) · Zbl 0979.80009
[25] Schmidt, A., Computation of three dimensional dendrites with finite elements, J. comput. phys., 125, 293, (1996) · Zbl 0844.65096
[26] Sethian, J., Fast marching methods, SIAM rev., 41, 199, (1999) · Zbl 0926.65106
[27] Sethian, J.; Strain, J., Crystal growth and dendritic solidification, J. comput. phys., 98, 231, (1992) · Zbl 0752.65088
[28] Shu, C.W.; Osher, S., Efficient implementation of essentially non-oscillatory shock capturing schemes, J. comput. phys., 77, 439, (1988) · Zbl 0653.65072
[29] Sussman, M.; Smereka, P.; Osher, S., A level set approach for computing solutions to incompressible two-phase flow, J. comput. phys., 114, 146, (1994) · Zbl 0808.76077
[30] Udaykumar, H.; Mittal, R.; Shyy, W., Computation of solid – liquid phase fronts in the sharp interface limit on fixed grids, J. comput. phys., 153, 535, (1999) · Zbl 0953.76071
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