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High Order Finite Difference Discretization for Composite Grid Hierarchy and Its Applications

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems General theory of linear operators Elliptic equations and systems Partial differential equations, boundary value problems

Published online by Cambridge University Press:  23 November 2015

Qun Gu ,
Weiguo Gao  and
Carlos J. García-Cervera
Qun Gu
Affiliation:
School of Mathematical Sciences, Fudan University, Shanghai 200433, P.R. China
Weiguo Gao*
Affiliation:
MOE Key Laboratory of Computational Physical Sciences and School of Mathematical Sciences, Fudan University, Shanghai 200433, P.R. China
Carlos J. García-Cervera
Affiliation:
Department of Mathematics, University of California, Santa Barbara, CA 93106, USA
*
*Corresponding author. Email addresses: 081018018@f udan.edu.cn (Q. Gu), wggao@fudan.edu.cn (W. Gao), cgarcia@math.ucsb.edu (C.J.García-Cervera)
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Abstract

We introduce efficient approaches to construct high order finite difference discretizations for solving partial differential equations, based on a composite grid hierarchy. We introduce a modification of the traditional point clustering algorithm, obtained by adding restrictive parameters that control the minimal patch length and the size of the buffer zone. As a result, a reduction in the number of interfacial cells is observed. Based on a reasonable geometric grid setting, we discuss a general approach for the construction of stencils in a composite grid environment. The straightforward approach leads to an ill-posed problem. In our approach we regularize this problem, and transform it into solving a symmetric system of linear of equations. Finally, a stencil repository has been designed to further reduce computational overhead. The effectiveness of the discretizations is illustrated by numerical experiments on second order elliptic differential equations.

Keywords

High orderpoint clustering algorithmfinite difference methodcomposite gridstencilregularization

MSC classification

Secondary: 35J57: Boundary value problems for second-order elliptic systems 47A52: Ill-posed problems, regularization 65M50: Mesh generation and refinement 65N06: Finite difference methods
Type
Research Article
Information
Communications in Computational Physics , Volume 18 , Issue 5 , November 2015 , pp. 1211 - 1233
Copyright
Copyright © Global-Science Press 2015 

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References

[1]Anderson, E., Bai, Z., Bischof, C., Blackford, S., Demmel, J., Dongarra, J., Du Croz, J., Greenbaum, A., Hammarling, S., McKenney, A., and Sorensen, D.. LAPACK Users’ Guide. Society for Industrial and Applied Mathematics, Philadelphia, PA, third edition, 1999. ISBN 0-89871-447-8 (paperback).Google Scholar
[2]Baker, N. A., Holst, M.J., and Wang, F.. Adaptive multilevel finite element solution of the Poisson-Boltzmann equation II. Refinement at solvent-accessible surfaces in biomolecular systems. J. Comput. Chem., 21(15): 1343–1352, 2000.3.0.CO;2-K>CrossRefGoogle Scholar
[3]Berger, M.J. and Colella, P.. Local adaptive mesh refinement for shock hydrodynamics. J. Comput. Phys., 82 (1): 64–84, 1989.CrossRefGoogle Scholar
[4]Berger, M.J. and Jameson, A.. Automatic adaptive grid refinement for the Euler equations. AIAA J., 23 (4): 561–568, 1985.Google Scholar
[5]Berger, M.J. and Oliger, J.. Adaptive mesh refinement for hyperbolic partial differential equations. J. Comput. Phys., 53 (3): 484–512, 1984.Google Scholar
[6]Berger, M.J. and Rigoutsos, I.. An algorithm for point clustering and grid generation. IEEE T. Syst. Man. Cy., 21 (5): 1278–1286, 1991.Google Scholar
[7]Bryan, G. L., Norman, M. L., O'Shea, B. W., Abel, T., Wise, J. H., Turk, M. J., Reynolds, D. R., Collins, D. C., Wang, P., Skillman, S. W., et al. Enzo: An adaptive mesh refinement code for astrophysics. arXiv preprint arXiv:1307.2265, 2013.Google Scholar
[8]Ceniceros, H. D. and Roma, A. M.. A nonstiff, adaptive, mesh refinement-based method for the Cahn-Hilliard equation, J. Comput. Phys., 225 (2): 1849–1862, 2007.Google Scholar
[9]Ceniceros, H. D., Nos, R. L., and Roma, A. M.. Three-dimensional, fully adaptive simulations of phase field fluid models, J. Comput. Phys., 229 (17): 6135–6155, 2010.Google Scholar
[10]Colella, P., Graves, D. T., Ligocki, T. J., Martin, D. F., Modiano, D., Serafini, D. B., and Van Straalen, B.. Chombo software package for amr applications-design document, 2000.Google Scholar
[11]Concus, P. and Golub, G. H.. Use of fast direct methods for the efficient numerical solution of nonseparable elliptic equations. SIAMJ. Numer. Anal., 10 (6): 1103–1120, 1973.CrossRefGoogle Scholar
[12]Debreu, L., Vouland, C., and Blayo, E.. AGRIF: Adaptive grid refinement in Fortran. Comput.Geosci., 34 (1): 8–13, 2008.Google Scholar
[13]Ding, H. and Shu, C.-W.. A stencil adaptive algorithm for finite difference solution of incompressible viscous flows. J. Comput. Phys., 214 (1): 397–420, 2006.CrossRefGoogle Scholar
[14]Fornberg, Bengt. Generation of finite difference formulas on arbitrarily spaced grids. Math.Comput., 51 (184): 699–706, 1988.Google Scholar
[15]Hayes, R. L., Fago, M., Ortiz, M., and Carter, E. A.. Prediction of dislocation nucleation during nanoindentation by the orbital free density functional theory local quasicontinuum method. Multiscale Model. Sim., 4 (2): 359–389, 2006.Google Scholar
[16]Holst, M.J., Baker, N. A., and Wang, F.. Adaptive multilevel finite element solution of the Poisson–Boltzmann equation I. Algorithms and examples. J. Comput. Chem., 21 (15): 1319–1342, 2000.Google Scholar
[17]Jenny, P., Lee, S. H., and Tchelepi, H. A.. Adaptive multiscale finite-volume method for multiphase flow and transport in porous media. Multiscale Model. Sim., 3 (1): 50–64, 2005.Google Scholar
[18]Lakin, W. D.. Differentiating matrices for arbitrarily spaced grid points. Int. J. Numer. Meth.Eng., 23 (2): 209–218, 1986.Google Scholar
[19]Lan, C.-W., Liu, C.-C., and Hsu, C.-M.. An adaptive finite volume method for incompressible heat flow problems in solidification. J. Comput. Phys., 178 (2): 464–497, 2002.Google Scholar
[20]Li, Y.-F., Liu, Z.-P., Liu, L.-L., and Gao, W.-G.. Mechanism and activity of photocatalytic oxygen evolution on titania anatase in aqueous surroundings. J. Am. Chem. Soc., 132 (37): 13008–13015, 2010.Google Scholar
[21]MacNeice, P., Olson, K. M., Mobarry, C., de Fainchtein, R., and Packer, C.. Paramesh: A parallel adaptive mesh refinement community toolkit. Comput. Phys. Commun., 126 (3): 330–354, 2000.Google Scholar
[22]Martin, D. F. and Cartwright, K. L.. Solving Poisson's equation using adaptive mesh refinement. Electronics Research Laboratory, College of Engineering, University of California, 1996.Google Scholar
[23]O'Shea, B. W., Bryan, G., Bordner, J., Norman, M. L., Abel, T., Harkness, R., and Kritsuk, A.. Introducing Enzo, an AMR cosmology application. arXiv preprint astro-ph/0403044, 2004.Google Scholar
[24]Saad, Y.. Iterative methods for sparse linear systems. SIAM, 2003.CrossRefGoogle Scholar
[25]Shenoy, V., Miller, R., Tadmor, E., Rodney, D., Phillips, R., and Ortiz, M.. An adaptive finite element approach to atomic-scale mechanics – the quasicontinuum method. J. Mech. Phys.Solid, 47: 611–642, 1999.Google Scholar
[26]Wang, Y. and Zhang, J.. Sixth order compact scheme combined with multigrid method and extrapolation technique for 2D Poisson equation. J. Comput. Phys., 228 (1): 137–146, 2009.Google Scholar
[27]Wise, S. M., Lowengrub, J. S., and Cristini, V.. An adaptive multigrid algorithm for simulating solid tumor growth using mixture models, Math. Comput. Modelling, 53: 1–20, 2011.Google Scholar
[28]Yang, X., James, A.J., Lowengrub, J. S., Zheng, X., and Cristini, V.. An adaptive coupled level-set/volume-of-fluid interface capturing method for unstructured triangular grids, J. Comput. Phys., 217: 364–394, 2006.Google Scholar
[29]Zhang, D.-E., Shen, L.-H., Zhou, A.-H., and Gong, X.-G.. Finite element method for solving Kohn-Sham equations based on self-adaptive tetrahedral mesh. Phys. Lett. A, 372 (30): 5071–5076, 2008.Google Scholar
[30]Zhang, J.. An explicit fourth-order compact finite difference scheme for three-dimensional convection-diffusion equation. Commun. Numer. Meth. Eng., 14 (3): 209–218, 1998.Google Scholar
[31]Zheng, X., Lowengrub, J., Anderson, A., and Cristini, V.. Adaptive unstructured volume remeshing II: Applications to two- and three-dimensional level set simulations of multiphase flow, J. Comp. Phys., 208: 626–650, 2005.Google Scholar
[32]Zhu, X.-L. and Oliger, J.. Data Structures and Implementations of Composite Adaptive Grid Methods. Citeseer, 1995.Google Scholar