×

Compact integration factor methods for complex domains and adaptive mesh refinement. (English) Zbl 1194.65111

Summary: The implicit integration factor (IIF) method, a class of efficient semi-implicit temporal schemes, was introduced recently for stiff reaction-diffusion equations. To reduce the cost of IIF, a compact implicit integration factor (cIIF) method was later developed for efficient storage and calculation of exponential matrices associated with the diffusion operators in two and three spatial dimensions for Cartesian coordinates with regular meshes. Unlike IIF, cIIF cannot be directly extended to other curvilinear coordinates, such as polar and spherical coordinates, due to the compact representation for the diffusion terms in cIIF.
In this paper, we present a method to generalize cIIF for other curvilinear coordinates through examples of polar and spherical coordinates. The new cIIF method in polar and spherical coordinates has similar computational efficiency and stability properties as the cIIF in Cartesian coordinate. In addition, we present a method for integrating cIIF with adaptive mesh refinement (AMR) to take advantage of the excellent stability condition for cIIF. Because the second order cIIF is unconditionally stable, it allows large time steps for AMR, unlike a typical explicit temporal scheme whose time step is severely restricted by the smallest mesh size in the entire spatial domain. Finally, we apply those methods to simulating a cell signaling system described by a system of stiff reaction-diffusion equations in both two and three spatial dimensions using AMR, curvilinear and Cartesian coordinates. Excellent performance of the new methods is observed.

MSC:

65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35K57 Reaction-diffusion equations
65Y20 Complexity and performance of numerical algorithms
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
65L05 Numerical methods for initial value problems involving ordinary differential equations

Software:

CMPGRD; LAPACK; Overture
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Kassam, A.-K.; Trefethen, L. N., Fourth-order time stepping for stiff PDEs, SIAM Journal on Scientific Computing, 26, 1214-1233 (2005) · Zbl 1077.65105
[2] Beylkin, G.; Keiser, J. M.; Vozovoi, L., A new class of time discretization schemes for the solution of nonlinear PDEs, Journal of Computational Physics, 147, 362-387 (1998) · Zbl 0924.65089
[3] Hou, T. Y.; Lowengrub, J.; Shelley, M. J., Removing the stiffness from interfacial flows with surface tension, Journal of Computational Physics, 114, 312 (1994) · Zbl 0810.76095
[4] Leo, P. H.; Lowengrub, J. S.; Nie, Q., Microstructural evolution in orthotropic elastic media, Journal of Computational Physics, 157, 44-88 (2000) · Zbl 0960.74076
[5] Jou, H. J.; Leo, P. H.; Lowengrub, J. S., Microstructual evolution in inhomogeneous elastic media, Journal of Computational Physics, 131, 109 (1997) · Zbl 0880.73050
[6] Du, Q.; Zhu, W., Stability analysis and applications of the exponential time differencing schemes, Journal of Computational Mathematics, 22, 200 (2004) · Zbl 1052.65081
[7] Du, Q.; Zhu, W., Modified exponential time differencing schemes: analysis and applications, BIT, Numerical Mathematics, 45, 307-328 (2005) · Zbl 1080.65074
[8] Nie, Q.; Zhang, Y.-T.; Zhao, R., Efficient semi-implicit schemes for stiff systems, Journal of Computational Physics, 214, 521-537 (2006) · Zbl 1089.65094
[9] Nie, Q.; Wan, F. Y.M.; Zhang, Y.-T.; Liu, X. F., Integration factor methods for high spatial dimensions, Journal of Computational Physics, 227, 5238-5255 (2008) · Zbl 1142.65072
[10] Henshaw, W. D.; Schwendeman, D. W., An adaptive numerical method for high-speed reactive flows on overlapping grids, Journal of Computational Physics, 191, 420-447 (2003) · Zbl 1134.76427
[11] Dawson, C.; Kirby, R., High resolution schemes for conservation laws with locally varying time steps, SIAM Journal on Scientific Computing, 22, 6, 2256-2284 (2001) · Zbl 0980.35015
[12] Kirby, R., On the convergence of high resolution methods with multiple time scales for hyperbolic conservation laws, Mathematics of Computation, 72, 243, 1239-1250 (2003) · Zbl 1018.65112
[13] Tang, H.; Warnecke, G., A Class of High Resolution Schemes for Hyperbolic Conservation Laws and Convection Diffusion Equations with Varying Time and Space Grids (2003), Univ., Fak. für Mathematik
[14] Constantinescu, E. M.; Sandu, A., Multirate timestepping methods for hyperbolic conservation laws, Journal of Scientific Computing, 33, 3, 239-278 (2007) · Zbl 1127.76033
[15] Berger, M. J.; Oliger, J., Adaptive mesh refinement for hyperbolic partial differential equations, Journal of Computational Physics, 53, 484-512 (1984) · Zbl 0536.65071
[16] Chesshire, G.; Henshaw, W. D., Composite overlapping meshes for the solution of partial differential equations, Journal of Computational Physics, 90, 1, 1-64 (1990) · Zbl 0709.65090
[17] Steger, J. L.; Benek, J. A., On the use of composite grid schemes in computational aerodynamics, Computer Methods in Applied Mechanics and Engineering, 64, 1-3, 301-320 (1987) · Zbl 0607.76061
[18] Henshaw, W. D.; Chesshire, G., Multigrid on composite meshes, SIAM Journal on Scientific and Statistical Computing, 8, 914 (1987) · Zbl 0655.65117
[19] Hinatsu, M.; Ferziger, J. H., Numerical computation of unsteady incompressible flow in complex geometry using a composite multigrid technique, International Journal for Numerical Methods in Fluids, 13, 8 (1991) · Zbl 0741.76044
[20] R.L. Meakin, Moving Body Overset Grid Methods for Complete Aircraft Tiltrotor Simulations, AIAA Paper, 1993, pp. 93-3350.; R.L. Meakin, Moving Body Overset Grid Methods for Complete Aircraft Tiltrotor Simulations, AIAA Paper, 1993, pp. 93-3350.
[21] Tu, J. Y.; Fuchs, L., Calculation of flows using three-dimensional overlapping grids and multigrid methods, International Journal for Numerical Methods in Engineering, 38, 2 (1995), ISSN 0029-5981 · Zbl 0823.76059
[22] Petersson, N. A., Hole-cutting for three-dimensional overlapping grids, SIAM Journal on Scientific Computing, 21, 2, 646-665 (1999) · Zbl 0947.65128
[23] Berger, M. J.; Colella, P., Local adaptive mesh refinement for shock hydrodynamics, Journal of Computational Physics, 82, 1, 64-84 (1989) · Zbl 0665.76070
[24] Torabi, S.; Wise, S.; Lowengrub, J.; Ratz, A.; Voigt, A., A new method for simulating strongly anisotropic Cahn-Hilliard equations, Materials Science and Technology-Association for Iron and Steel Technology, 3, 1432 (2007)
[25] Lee, H. G.; Lowengrub, J.; Goodman, J., Modeling pinchoff and reconnection in a Hele-Shaw cell. II. Analysis and simulation in the nonlinear regime, Physics of Fluids, 14, 514 (2002) · Zbl 1184.76317
[26] Cherry, E. M.; Greenside, H. S.; Henriquez, C. S., A space-time adaptive method for simulating complex cardiac dynamics, Physical Review Letters, 84, 6, 1343-1346 (2000)
[27] Cherry, E. M.; Greenside, H. S.; Henriquez, C. S., Efficient simulation of three-dimensional anisotropic cardiac tissue using an adaptive mesh refinement method, Chaos: An Interdisciplinary Journal of Nonlinear Science, 13, 853 (2003) · Zbl 1080.92513
[28] Trangenstein, J. A.; Kim, C., Operator splitting and adaptive mesh refinement for the Luo-Rudy I model, Journal of Computational Physics, 196, 2, 645-679 (2004) · Zbl 1056.92014
[29] K. Brislawn, D.L. Brown, G. Chesshire, J. Saltzman, Adpative-refined Overlapping Grids for the Numerical Solution of Hyperbolic Systems of Conservation Laws, Report LA-UR-95-257, Los Alamos National Laboratory, 1995.; K. Brislawn, D.L. Brown, G. Chesshire, J. Saltzman, Adpative-refined Overlapping Grids for the Numerical Solution of Hyperbolic Systems of Conservation Laws, Report LA-UR-95-257, Los Alamos National Laboratory, 1995.
[30] E.P. Boden, E.F. Toro, A combined Chimera-AMR technique for computing hyperbolic pdes, in: Djilali (Ed.), Proceedings of the Fifth Annual Conference of the CFD Society of Canada, 1997, pp. 5.13-5.18.; E.P. Boden, E.F. Toro, A combined Chimera-AMR technique for computing hyperbolic pdes, in: Djilali (Ed.), Proceedings of the Fifth Annual Conference of the CFD Society of Canada, 1997, pp. 5.13-5.18.
[31] Meankin, R. L., Composite overset structured grids, (Thompson, J. F.; Soni, B. K.; Weatherill, N. P., Handbook of Grid Generation (1999), CRC Press), 1-20, (Chapter 11)
[32] Brown, D. L.; Henshaw, W. D.; Quinlan, D. J., Overture: an object-oriented framework for solving partial differential equations, Lecture Notes in Computer Science, 177-184 (1997)
[33] D.L. Brown, G.S. Chesshire, W.D. Henshaw, D.J. Quinlan, Overture: an object oriented software system for solving partial differential equations in serial and parallel environments, in: Proceedings of the Eighth SIAM Conference on Parallel Processing for Scientific Computing, 1997.; D.L. Brown, G.S. Chesshire, W.D. Henshaw, D.J. Quinlan, Overture: an object oriented software system for solving partial differential equations in serial and parallel environments, in: Proceedings of the Eighth SIAM Conference on Parallel Processing for Scientific Computing, 1997.
[34] Horn, R. A.; Johnson, C. R., Matrix Analysis (1990), Cambridge University Press · Zbl 0704.15002
[35] Sleijpen, G. L.G.; Van der Vorst, H. A., A Jacobi-Davidson iteration method for linear eigenvalue problems, SIAM Review, 42, 2, 267-293 (2000) · Zbl 0949.65028
[36] Anderson, E.; Bai, Z.; Bischof, C., LAPACK Users’ Guide (1999), Society for Industrial Mathematics · Zbl 0755.65028
[37] L. Bardwell, X.F. Liu, R.D. Moore, Q. Nie, Spatially-localized Scaffold Proteins May Simultaneously Boost and Suppress Signaling, Preprint, 2009.; L. Bardwell, X.F. Liu, R.D. Moore, Q. Nie, Spatially-localized Scaffold Proteins May Simultaneously Boost and Suppress Signaling, Preprint, 2009.
[38] Whitmarsh, A. J.; Davis, R. J., Structural organization of MAP-kinase signaling modules by scaffold proteins in yeast and mammals, Trends in Biochemical Sciences, 23, 481-485 (1998)
[39] Morrison, D. K.; Davis, R. J., Regulation of MAP kinase signaling modules by scaffold proteins in mammals, Annual Review of Cell and Developmental Biology, 19, 91-118 (2003)
[40] Wong, W.; Scott, J. D., AKAP signalling complexes: focal points in space and time, Nature Reviews Molecular Cell Biology, 5, 959-970 (2004)
[41] Park, S. H.; Zarrinpar, A.; Lim, W. A., Rewriting MAP kinase pathways using alternative scaffold assembly mechanisms, Science, 299, 1061-1064 (2003)
[42] Harris, K.; Lamson, R. E.; Nelson, B.; Hughes, T. R.; Marton, M. J.; Roberts, C. J.; Boone, C.; Pryciak, P. M., Role of scaffolds in MAP kinase pathway specificity revealed by custom design of pathway-dedicated signaling proteins, Current Biology, 11, 1815-1824 (2001)
[43] Dickens, M.; Rogers, J. S.; Cavanagh, J.; Raitano, A.; Xia, Z.; Halpern, J. R.; Greenberg, M. E.; Sawyers, C. L.; Davis, R. J., A cytoplasmic inhibitor of the JNK signal transduction pathway, Science, 277, 693-696 (1997)
[44] Cohen, L.; Henzel, W. J.; Baeuerle, P. A., IKAP is a scaffold protein of the IkB kinase complex, Nature, 395, 292-296 (1998)
[45] Kortum, R. L.; Lewis, R. E., The molecular scaffold KSR1 regulates the proliferative and oncogenic potential of cells, Molecular and Cellular Biology, 24, 4407-4416 (2004)
[46] Brigham, E. O., The Fast Fourier Transform and its Applications (1988), Prentice Hall, Englewood Cliffs: Prentice Hall, Englewood Cliffs NJ
[47] Van Loan, C., Computational Frameworks for the Fast Fourier Transform (1992), Society for Industrial Mathematics · Zbl 0757.65154
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.