×

zbMATH — the first resource for mathematics

Self-adaptive time integration of flux-conservative equations with sources. (English) Zbl 1093.65083
Summary: We present a novel, flux-conserving, asynchronous method for the explicit time integration of multi-scale, flux-conservative partial differential equations with source terms. Unlike the conventional explicit and implicit integration schemes, it is based on a discrete-event simulation paradigm, which describes time advance in terms of increments to physical quantities and causality rules rather than time stepping. This method exerts self-adaptive control over local update rates of solution by predicting and correcting changes to simulation variables in accordance with local physical scales. The discrete-event simulation paradigm is independent of the underlying spatial mesh and thus can be incorporated into block-structured and unstructured mesh refinement techniques.
The effectiveness and robustness of the new method is demonstrated on a number of one-dimensional, uniform mesh models based on diffusion-convection-reaction equations. The event-driven integration reduces numerical approximation errors due to large local time derivatives, prevents explosive numerical instabilities in locally super-Courant calculations and automatically reduces the CPU overhead associated with stiff terms and inactive regions in computation space.

MSC:
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35L60 First-order nonlinear hyperbolic equations
35L65 Hyperbolic conservation laws
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] ()
[2] Ascher, U.M.; Ruuth, S.J.; Wetton, B.T.R., Implicit-explicit methods for time dependent partial differential equations, SIAM J. numer. anal., 32, 3, 797, (1995) · Zbl 0841.65081
[3] Rider, W.J.; Knoll, D.N.; Olson, G.L., A multigrid Newton-Krylov method for multimaterial equilibrium radiation diffusion, J. comput. phys., 152, 164, (1999) · Zbl 0944.85002
[4] Rider, W.J.; Knoll, D.N., Time step size selection for radiation diffusion calculations, J. comput. phys., 152, 790, (1999) · Zbl 0940.65101
[5] Lowrie, R.B., A comparison of implicit time integration methods for nonlinear relaxation and diffusion, J. comput. phys., 196, 566, (2004) · Zbl 1053.65080
[6] Ober, C.C.; Shadid, J.N., Studies on the accuracy of time-integration methods for the radiation diffusion equations, J. comput. phys., 195, 743, (2004) · Zbl 1053.65082
[7] Verwer, J.G.; Sommeijer, B.P., An implicit-explicit Runge-Kutta-Chebychev scheme for diffusion-reaction equations, SIAM J. sci. comput., 25, 5, 1824, (2004) · Zbl 1061.65090
[8] Hundsdorfer, W.; Jaffre, J., Implicit-explicit time stepping with spatial discontinuous finite elements, Appl. numer. math., 45, 231, (2003) · Zbl 1029.65115
[9] Chacón, L.; Knoll, D.A.; Finn, J.M., An implicit, nonlinear reduced resistive MHD solver, J. comput. phys., 178, 15, (2002) · Zbl 1139.76328
[10] Baldwin, C.; Brown, P.N.; Falgout, R.; Graziani, F.; Jones, J., Iterative linear solvers in a 2D radiation-hydrodynamics code: methods and performance, J. comput. phys., 154, 1, (1999) · Zbl 0964.76036
[11] Berger, M.J.; Oliger, J., Adaptive mesh refinement for hyperbolic partial differential equations, J. comput. phys., 53, 484, (1984) · Zbl 0536.65071
[12] Berger, M.J.; Colella, P., Local adaptive mesh refinement for shock hydrodynamics, J. comput. phys., 82, 64, (1989) · Zbl 0665.76070
[13] Bryan, G.L., Fluids in the universe: adaptive mesh refinement in cosmology, Comput. sci. eng., 1, 2, 46, (1999)
[14] Bell, J.B.; Day, M.S.; Rendleman, C.A.; Woosley, S.E.; Zingale, M.A., Adaptive low Mach number simulations of nuclear flame microphysics, J. comput. phys., 195, 677, (2004) · Zbl 1115.85302
[15] Cherry, E.M.; Greenside, H.S.; Henriquez, C.S., Efficient simulation of three-dimensional anisotropic cardiac tissue using an adaptive mesh refinement method, Chaos, 13, 3, 853, (2003) · Zbl 1080.92513
[16] Dawson, C.; Kirby, R., High resolution schemes for conservation laws with locally varying time steps, SIAM J. sci. comput., 22, 6, 2256, (2001) · Zbl 0980.35015
[17] Kirby, R., On the convergence of high resolution methods with multiple time scales for hyperbolic conservation laws, Math. comput., 72, 243, 1239, (2002) · Zbl 1018.65112
[18] Quan, W.; Evans, S.J.; Hastings, H.M., Efficient integration of a realistic two-dimensional cardiac tissue model by domain decomposition, IEEE trans. biomed. eng., 45, 3, 372, (1998)
[19] Amitai, D.; Averbuch, A.; Israeli, M.; Itzikowitz, S., Implicit-explicit parallel asynchronous solver of parabolic pdes, SIAM J. sci. comput., 19, 4, 1366, (1998) · Zbl 0914.65092
[20] Otani, N.F., Computer modeling in cardiac electrophysiology, J. comput. phys., 161, 21, (2000) · Zbl 0943.92014
[21] Abedi, R.; Chung, S.-H.; Erickson, J.; Fan, Y.; Garland, M.; Guoy, D.; Haber, R.; Sullivan, J.M.; Thite, S.; Zhou, Y., Spacetime meshing with adaptive refinement and coarsening, (), 300-308 · Zbl 1422.65242
[22] Lew, A.; Mardsen, J.E.; Ortiz, M.; West, M., Asynchronous variational integrators, Arch. rational mech. anal., 167, 85, (2003) · Zbl 1055.74041
[23] Karimabadi, H.; Driscoll, J.; Omelchenko, Y.A.; Omidi, N., A new asynchronous methodology for modeling of physical systems: breaking the curse of Courant condition, J. comput. phys., 205, 2, 755, (2005) · Zbl 1087.76538
[24] ()
[25] Fujimoto, R.M., Parallel and distributed simulation systems, (2000), Wiley Interscience New York
[26] J. Nutaro, Parallel discrete event simulation with application to continuous systems, Ph.D. Thesis, Department of Electrical and Computer Engineering, University of Arizona, 2003.
[27] Nutaro, J.; Zeigler, B.P.; Jammalamadaka, R.; Akerkar, S., Discrete event simulation of gas dynamics within the DEVS framework, Lect. note comp. sci., 2660, 319, (2003)
[28] Hundsdorfer, W.; Montijn, C., A note on flux limiting for diffusion discretizations, IMA J. numer. anal., 24, 4, 635, (2004) · Zbl 1062.65085
[29] Omelchenko, Y.A.; Karimabadi, H., Event-driven hybrid particle-in-cell simulation: a new paradigm for multi-scale plasma modeling, J. comp. phys., 216, 1, 153-178, (2006) · Zbl 1126.82043
[30] Karimabadi H., J. Driscoll, J. Dave, Y. Omelchenko, K. Perumalla, R. Fujimoto, N. Omidi, Parallel discrete event simulations of grid-based models: asynchronous electromagnetic hybrid code, in: Workshop on State-of-the-art in Scientific Computing, Springer LNCS Proceedings, 580, 2005.
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.