Document Zbl 1106.65093

Min, Chohong; Gibou, Frédéric; Ceniceros, Hector D.

A supra-convergent finite difference scheme for the variable coefficient Poisson equation on non-graded grids. (English) Zbl 1106.65093

J. Comput. Phys. 218, No. 1, 123-140 (2006).

The authors propose a finite difference algorithm for the Poisson equation that yields second order accuracy for the solutions and their gradients on non-graded grids. They employ quadtree (in 2D) and octree (in 3D) data structures as an efficient means to represent the Cartesian grid, allowing for constraint-free grid generation. The discretization at one cell’s node only uses nodes of two (2D) or three (3D) adjacent cells, producing schemes that are straightforward to implement. The linear systems obtained are nonsymmetric but are shown to be diagonally dominant. Numerical results in 2D and 3D demonstrate supra-convergence in the \( L^\infty \) norm.

Reviewer: Adrian Carabineanu (Bucureşti)

Cited in 2 Reviews

Cited in 45 Documents

MSC:

65N06	Finite difference methods for boundary value problems involving PDEs
65N12	Stability and convergence of numerical methods for boundary value problems involving PDEs
65N50	Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
35J05	Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation

Keywords:

quadtree and octree data structures; second-order accuracy; diagonally dominant matrix; finite difference algorithm; Poisson equation; grid generation; numerical results; supra-convergence

Software:

Gerris

Cite Review PDF

Full Text: DOI

References:

[1]	M.J. Aftosmis, M.J. Berger, J.E. Melton, Adaptive cartesian mesh generation, CRC Handbook of Mesh Generation (Contributed Chapter), 1998.; M.J. Aftosmis, M.J. Berger, J.E. Melton, Adaptive cartesian mesh generation, CRC Handbook of Mesh Generation (Contributed Chapter), 1998.
[2]	A. Almgren, A fast adaptive vortex method using local corrections, Ph.D. Thesis, University of California, Berkeley, 1991.; A. Almgren, A fast adaptive vortex method using local corrections, Ph.D. Thesis, University of California, Berkeley, 1991.
[3]	Almgren, A.; Bell, J.; Colella, P.; Howell, L.; Welcome, M., A conservative adaptive projection method for the variable density incompressible Navier-Stokes equations, J. Comput. Phys., 142, 1-46 (1998) · Zbl 0933.76055
[4]	Almgren, A.; Buttke, R.; Colella, P., A fast adaptive vortex method in three dimensions, J. Comput. Phys., 113, 177-200 (1994) · Zbl 0809.76071
[5]	Berger, M.; Colella, P., Local adaptive mesh refinement for shock hydrodynamics, J. Comput. Phys., 82, 64-84 (1989) · Zbl 0665.76070
[6]	Berger, M.; Oliger, J., Adaptive mesh refinement for hyperbolic partial differential equations, J. Comput. Phys., 53, 484-512 (1984) · Zbl 0536.65071
[7]	Ceniceros, H. D.; Roma, A. M., Study of long-time dynamics of a viscous vortex sheet with a fully adaptive non-stiff method, Phys. Fluids, 16, 12, 4285-4318 (2004) · Zbl 1187.76085
[8]	Chorin, A., A numerical method for solving incompressible viscous flow problems, J. Comput. Phys., 2, 12-26 (1967) · Zbl 0149.44802
[9]	P. Ciarlet, Introduction to Numerical Linear and Optimization, Cambridge Texts in Applied Mathematics, 40 West 20th street, New York, NY 10011, 1998.; P. Ciarlet, Introduction to Numerical Linear and Optimization, Cambridge Texts in Applied Mathematics, 40 West 20th street, New York, NY 10011, 1998.
[10]	Gibou, F.; Fedkiw, R., A fourth order accurate discretization for the Laplace and heat equations on arbitrary domains, with applications to the Stefan problem, J. Comput. Phys., 202, 577-601 (2005) · Zbl 1061.65079
[11]	Gibou, F.; Fedkiw, R.; Cheng, L.-T.; Kang, M., A second-order-accurate symmetric discretization of the Poisson equation on irregular domains, J. Comput. Phys., 176, 205-227 (2002) · Zbl 0996.65108
[12]	Golub, G.; Loan, C., Matrix Computations (1989), The John Hopkins University Press: The John Hopkins University Press Baltimore, MD · Zbl 0733.65016
[13]	Ham, F.; Lien, F.; Strong, A., A fully conservative second-order finite difference scheme for incompressible flow on nonuniform grids, J. Comput. Phys., 117, 117-133 (2002) · Zbl 1066.76044
[14]	Johansen, H.; Colella, P., A Cartesian grid embedded boundary method for Poisson’s equation on irregular domains, J. Comput. Phys., 147, 60-85 (1998) · Zbl 0923.65079
[15]	Kincaid, D.; Cheney, W., Numerical Analysis: Mathematics of Scientific Computing (2002), Brooks/Cole Publishing Co.: Brooks/Cole Publishing Co. Pacific Grove, CA, USA
[16]	Kreiss, H. O.; Manteuffel, H.-O.; Schwartz, T. A.; Wendroff, B.; White, A. B., Supra-convergent schemes on irregular grids, Math. Comp., 47, 537-554 (1986) · Zbl 0619.65055
[17]	LeVeque, R.; Li, Z., The immersed interface method for elliptic equations with discontinuous coefficients and singular sources, SIAM J. Numer. Anal., 31, 1019-1044 (1994) · Zbl 0811.65083
[18]	Li, Z., A fast iterative algorithm for elliptic interface problems, SIAM J. Numer. Anal., 35, 230-254 (1998) · Zbl 0915.65121
[19]	Lipnikov, K.; Morel, J.; Shashkov, M., Mimetic finite difference methods for diffusion equations on non-orthogonal non-conformal meshes, J. Comput. Phys., 199, 589-597 (2004) · Zbl 1057.65071
[20]	F. Losasso, R. Fedkiw, S. Osher, Spatially adaptive techniques for level set methods and incompressible flow, Comput. Fluids (in press). <Available online: www.sciencedirect.com; F. Losasso, R. Fedkiw, S. Osher, Spatially adaptive techniques for level set methods and incompressible flow, Comput. Fluids (in press). <Available online: www.sciencedirect.com · Zbl 1177.76295
[21]	F. Losasso, F. Gibou, R. Fedkiw, Simulating water and smoke with an octree data structure, ACM Trans. Graph. (SIGGRAPH Proc.), 2004, pp. 457-462.; F. Losasso, F. Gibou, R. Fedkiw, Simulating water and smoke with an octree data structure, ACM Trans. Graph. (SIGGRAPH Proc.), 2004, pp. 457-462.
[22]	Manteuffel, T.; White, A., The numerical solution of second-order boundary value problems on nonuniform meshes, Math. Comput., 47, 176, 511-535 (1986) · Zbl 0635.65092
[23]	Mayo, A., The fast solution of Poisson’s and the biharmonic equations on irregular regions, SIAM J. Numer. Anal., 21, 285-299 (1984) · Zbl 1131.65303
[24]	McCorquodale, P.; Colella, P.; Grote, D.; Vay, J.-L., A node-centered local refinement algorithm for Poisson’s equation in complex geometries, J. Comput. Phys., 201, 34-60 (2004) · Zbl 1059.65094
[25]	McKenney, A.; Greengard, L., A fast Poisson solver for complex geometries, J. Comput. Phys., 118, 348-355 (1995) · Zbl 0823.65115
[26]	Popinet, S., Gerris: a tree-based adaptive solver for the incompressible Euler equations in complex geometries, J. Comput. Phys., 190, 572-600 (2003) · Zbl 1076.76002
[27]	Saad, Y., Iterative Methods for Sparse Linear Systems (1996), PWS Publishing: PWS Publishing New York, NY · Zbl 1002.65042
[28]	Samet, H., The Design and Analysis of Spatial Data Structures (1989), Addison-Wesley: Addison-Wesley New York
[29]	Samet, H., Applications of Spatial Data Structures: Computer Graphics, Image Processing and GIS (1990), Addison-Wesley: Addison-Wesley New York
[30]	Sussman, M.; Algrem, A. S.; Bell, J. B.; Colella, P.; Howell, L. H.; Welcome, M. L., An adaptive level set approach for incompressible two-phase flow, J. Comput. Phys, 148, 81-124 (1999) · Zbl 0930.76068
[31]	Young, D.; Melvin, R.; Bieterman, M.; Johnson, F.; Samant, S.; Bussoletti, J., A locally refined rectangular grid finite element method: application to computational fluid dynamics and computational physics, J. Comput. Phys., 92, 1-66 (1991) · Zbl 0709.76078

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.