A numerical strategy to discretize and solve the Poisson equation on dynamically adapted multiresolution grids for time-dependent streamer discharge simulations. (English) Zbl 1352.65429
Summary: We develop a numerical strategy to solve multi-dimensional Poisson equations on dynamically adapted grids for evolutionary problems disclosing propagating fronts. The method is an extension of the multiresolution finite volume scheme used to solve hyperbolic and parabolic time-dependent PDEs. Such an approach guarantees a numerical solution of the Poisson equation within a user-defined accuracy tolerance. Most adaptive meshing approaches in the literature solve elliptic PDEs level-wise and hence at uniform resolution throughout the set of adapted grids. Here we introduce a numerical procedure to represent the elliptic operators on the adapted grid, strongly coupling inter-grid relations that guarantee the conservation and accuracy properties of multiresolution finite volume schemes. The discrete Poisson equation is solved at once over the entire computational domain as a completely separate process. The accuracy and numerical performance of the method are assessed in the context of streamer discharge simulations.
MSC:
| 65N08 | Finite volume methods for boundary value problems involving PDEs |
| 65N50 | Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs |
References:
| [1] | Chorin, A., Numerical solution of the Navier-Stokes equations, Math. Comput., 22, 745-762 (1968) · Zbl 0198.50103 |
| [2] | Témam, R., Sur l’approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires. II, Arch. Ration. Mech. Anal., 33, 377-385 (1969) · Zbl 0207.16904 |
| [3] | Guermond, J.; Minev, P.; Shen, J., An overview of projection methods for incompressible flows, Comput. Methods Appl. Mech. Eng., 195, 44-47, 6011-6045 (2006) · Zbl 1122.76072 |
| [4] | Babaeva, N.; Naidis, G., Two-dimensional modelling of positive streamer dynamics in non-uniform electric fields in air, J. Phys. D, Appl. Phys., 29, 2423-2431 (1996) |
| [5] | Kulikovsky, A., Positive streamer between parallel plate electrodes in atmospheric pressure air, J. Phys. D, Appl. Phys., 30, 441-450 (1997) |
| [6] | Berger, M.; Oliger, J., Adaptive mesh refinement for hyperbolic partial differential equations, J. Comput. Phys., 53, 484-512 (1984) · Zbl 0536.65071 |
| [7] | Berger, M.; Colella, P., Local adaptive mesh refinement for shock hydrodynamics, J. Comput. Phys., 82, 67-84 (1989) · Zbl 0665.76070 |
| [8] | Harten, A., Adaptive multiresolution schemes for shock computations, J. Comput. Phys., 115, 319-338 (1994) · Zbl 0925.65151 |
| [9] | Harten, A., Multiresolution algorithms for the numerical solution of hyperbolic conservation laws, Commun. Pure Appl. Math., 48, 1305-1342 (1995) · Zbl 0860.65078 |
| [10] | Cohen, A.; Kaber, S.; Müller, S.; Postel, M., Fully adaptive multiresolution finite volume schemes for conservation laws, Math. Comput., 72, 183-225 (2003) · Zbl 1010.65035 |
| [11] | Cohen, A.; Daubechies, I.; Feauveau, J.-C., Biorthogonal bases of compactly supported wavelets, Commun. Pure Appl. Math., 45, 5, 485-560 (1992) · Zbl 0776.42020 |
| [12] | Cohen, A., Wavelet methods in numerical analysis, (Handbook of Numerical Analysis, vol. 7 (2000), Elsevier: Elsevier Amsterdam) · Zbl 0976.65124 |
| [13] | Müller, S., Adaptive Multiscale Schemes for Conservation Laws, vol. 27 (2003), Springer-Verlag · Zbl 1016.76004 |
| [14] | Müller, S., Multiresolution schemes for conservation laws, (DeVore, R.; Kunoth, A.; etal., Multiscale, Nonlinear and Adaptive Approximation (2009), Springer: Springer Berlin, Heidelberg), 379-408 · Zbl 1190.65131 |
| [15] | Brix, K.; Melian, S.; Müller, S.; Bachmann, M., Adaptive multiresolution methods: practical issues on data structures, implementation and parallelization, ESAIM Proc., 34, 151-183 (2011) · Zbl 1302.65219 |
| [16] | Domingues, M.; Gomes, S.; Roussel, O.; Schneider, K., Adaptive multiresolution methods, ESAIM Proc., 34, 1-96 (2011) · Zbl 1302.65185 |
| [17] | Roussel, O.; Schneider, K.; Tsigulin, A.; Bockhorn, H., A conservative fully adaptive multiresolution algorithm for parabolic PDEs, J. Comput. Phys., 188, 2, 493-523 (2003) · Zbl 1022.65093 |
| [18] | Bürger, R.; Ruiz-Baier, R.; Schneider, K.; Sepúlveda, M., Fully adaptive multiresolution schemes for strongly degenerate parabolic equations in one space dimension, ESAIM: Math. Model. Numer. Anal., 42, 535-563 (2008) · Zbl 1147.65066 |
| [19] | Duarte, M.; Massot, M.; Descombes, S.; Tenaud, C.; Dumont, T.; Louvet, V.; Laurent, F., New resolution strategy for multi-scale reaction waves using time operator splitting, space adaptive multiresolution and dedicated high order implicit/explicit time integrators, SIAM J. Sci. Comput., 34, 1, A76-A104 (2012) · Zbl 1243.65107 |
| [20] | Dumont, T.; Duarte, M.; Descombes, S.; Dronne, M.-A.; Massot, M.; Louvet, V., Simulation of human ischemic stroke in realistic 3D geometry, Commun. Nonlinear Sci. Numer. Simul., 18, 6, 1539-1557 (2013) · Zbl 1322.92019 |
| [21] | Duarte, M.; Descombes, S.; Tenaud, C.; Candel, S.; Massot, M., Time-space adaptive numerical methods for the simulation of combustion fronts, Combust. Flame, 160, 1083-1101 (2013) |
| [22] | Vasilyev, O.; Kevlahan, N.-R., An adaptive multilevel wavelet collocation method for elliptic problems, J. Comput. Phys., 206, 2, 412-431 (2005) · Zbl 1070.65122 |
| [23] | Vasilyev, O.; Bowman, C., Second-generation wavelet collocation method for the solution of partial differential equations, J. Comput. Phys., 165, 2, 660-693 (2000) · Zbl 0984.65105 |
| [24] | Vasilyev, O., Solving multi-dimensional evolution problems with localized structures using second generation wavelets, Int. J. Comput. Fluid Dyn., 17, 151-168 (2003) · Zbl 1034.76045 |
| [25] | Schneider, K.; Vasilyev, O., Wavelet methods in computational fluid dynamics, Annu. Rev. Fluid Mech., 42, 473-503 (2010) · Zbl 1345.76085 |
| [26] | Jaffard, S., Wavelet methods for fast resolution of elliptic problems, SIAM J. Numer. Anal., 29, 4, 965-986 (1992) · Zbl 0761.65083 |
| [27] | Dahlke, S.; Dahmen, W.; DeVore, R., Nonlinear approximation and adaptive techniques for solving elliptic operator equations, (Dahmen, W.; Kurdila, A.; Oswald, P., Multiscale Wavelet Methods for Partial Differential Equations. Multiscale Wavelet Methods for Partial Differential Equations, Wavelet Analysis and Its Applications, vol. 6 (1997), Academic Press), 237-283 · Zbl 1528.65002 |
| [28] | Cohen, A.; Masson, R., Wavelet methods for second-order elliptic problems, preconditioning, and adaptivity, SIAM J. Sci. Comput., 21, 3, 1006-1026 (1999) · Zbl 0981.65132 |
| [29] | Cohen, A.; Masson, R., Wavelet adaptive method for second order elliptic problems: boundary conditions and domain decomposition, Numer. Math., 86, 2, 193-238 (2000) · Zbl 0961.65106 |
| [30] | Barinka, A.; Barsch, T.; Charton, P.; Cohen, A.; Dahlke, S.; Dahmen, W.; Urban, K., Adaptive wavelet schemes for elliptic problems - implementation and numerical experiments, SIAM J. Sci. Comput., 23, 3, 910-939 (2001) · Zbl 1016.65090 |
| [31] | Cohen, A.; Dahmen, W.; DeVore, R., Adaptive wavelet methods for elliptic operator equations: convergence rates, Math. Comput., 70, 27-75 (2001) · Zbl 0980.65130 |
| [32] | 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, 1-46 (1998) · Zbl 0933.76055 |
| [33] | Teyssier, R., Cosmological hydrodynamics with adaptive mesh refinement - a new high resolution code called RAMSES, Astron. Astrophys., 385, 1, 337-364 (2002) |
| [34] | Montijn, C.; Hundsdorfer, W.; Ebert, U., An adaptive grid refinement strategy for the simulation of negative streamers, J. Comput. Phys., 219, 2, 801-835 (2006) · Zbl 1330.76087 |
| [35] | Martin, D.; Colella, P.; Graves, D., A cell-centered adaptive projection method for the incompressible Navier-Stokes equations in three dimensions, J. Comput. Phys., 227, 3, 1863-1886 (2008) · Zbl 1137.76040 |
| [36] | Safta, C.; Ray, J.; Najm, H., A high-order low-Mach number AMR construction for chemically reacting flows, J. Comput. Phys., 229, 24, 9299-9322 (2010) · Zbl 1222.80024 |
| [37] | Ebert, U.; Brau, F.; Derks, G.; Hundsdorfer, W.; Kao, C.-Y.; Li, C.; Luque, A.; Meulenbroek, B.; Nijdam, S.; Ratushnaya, V.; Schäfer, L.; Tanveer, S., Multiple scales in streamer discharges, with an emphasis on moving boundary approximations, Nonlinearity, 24, 1, C1-C26 (2011) · Zbl 1214.82100 |
| [38] | Pancheshnyi, S.; Ségur, P.; Capeillère, J.; Bourdon, A., Numerical simulation of filamentary discharges with parallel adaptive mesh refinement, J. Comput. Phys., 227, 13, 6574-6590 (2008) · Zbl 1144.76040 |
| [39] | Unfer, T.; Boeuf, J.-P.; Rogier, F.; Thivet, F., Multi-scale gas discharge simulations using asynchronous adaptive mesh refinement, Comput. Phys. Commun., 181, 2, 247-258 (2010) · Zbl 1333.76063 |
| [40] | Duarte, M.; Bonaventura, Z.; Massot, M.; Bourdon, A.; Descombes, S.; Dumont, T., A new numerical strategy with space-time adaptivity and error control for multi-scale streamer discharge simulations, J. Comput. Phys., 231, 1002-1019 (2012) · Zbl 1406.76063 |
| [41] | Duarte, M., Méthodes numériques adaptatives pour la simulation de la dynamique de fronts de réaction multi-échelles en temps et en espace (2011), Ecole Centrale: Ecole Centrale Paris, France, Ph.D. thesis |
| [42] | Bihari, B.; Harten, A., Multiresolution schemes for the numerical solution of 2-D conservation laws I, SIAM J. Sci. Comput., 18, 2, 315-354 (1997) · Zbl 0878.35007 |
| [43] | Morrow, R.; Lowke, J., Streamer propagation in air, J. Phys. D, Appl. Phys., 30, 614-627 (1997) |
| [44] | Benilov, M.; Naidis, G., Modelling of low-current discharges in atmospheric-pressure air taking account of non-equilibrium effects, J. Phys. D, Appl. Phys., 36, 15, 1834-1841 (2003) |
| [45] | Kossyi, I.; Kostinsky, A.; Matveyev, A.; Silakov, V., Kinetic scheme of the non-equilibrium discharge in nitrogen-oxygen mixtures, Plasma Sources Sci. Technol., 1, 3, 207-220 (1992) |
| [46] | Bourdon, A.; Pasko, V.; Liu, N.; Celestin, S.; Ségur, P.; Marode, E., Efficient models for photoionization produced by non-thermal gas discharges in air based on radiative transfer and the Helmholtz equations, Plasma Sources Sci. Technol., 16, 3, 656-678 (2007) |
| [47] | Amestoy, P.; Duff, I.; L’Excellent, J.-Y., Multifrontal parallel distributed symmetric and unsymmetric solvers, Comput. Methods Appl. Mech. Eng., 184, 2-4, 501-520 (2000) · Zbl 0956.65017 |
| [48] | Amestoy, P.; Duff, I.; Koster, J.; L’Excellent, J.-Y., A fully asynchronous multifrontal solver using distributed dynamic scheduling, SIAM J. Matrix Anal. Appl., 23, 1, 15-41 (2001) · Zbl 0992.65018 |
| [49] | Henson, V.; Yang, U., BoomerAMG: a parallel algebraic multigrid solver and preconditioner, Appl. Numer. Math., 41, 155-177 (2002) · Zbl 0995.65128 |
| [50] | Notay, Y., An aggregation-based algebraic multigrid method, Electron. Trans. Numer. Anal., 37, 123-146 (2010) · Zbl 1206.65133 |
| [51] | Napov, A.; Notay, Y., An algebraic multigrid method with guaranteed convergence rate, SIAM J. Sci. Comput., 34, A1079-A1109 (2012) · Zbl 1248.65037 |
| [52] | Notay, Y., Aggregation-based algebraic multigrid for convection-diffusion equations, SIAM J. Sci. Comput., 34, A2288-A2316 (2012) · Zbl 1250.76139 |
| [53] | Saad, Y.; Schultz, M., GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 7, 3, 856-869 (1986) · Zbl 0599.65018 |
| [54] | Briels, T.; Kos, J.; van Veldhuizen, E.; Ebert, U., Circuit dependence of the diameter of pulsed positive streamers in air, J. Phys. D, Appl. Phys., 39, 24, 5201-5210 (2006) |
| [55] | Cummer, S.; Jaugey, N.; Li, J.; Lyons, W.; Nelson, T.; Gerken, E., Submillisecond imaging of sprite development and structure, Geophys. Res. Lett., 33, 4, L04104 (2006) |
| [56] | Nijdam, S.; Geurts, C.; van Veldhuizen, E.; Ebert, U., Reconnection and merging of positive streamers in air, J. Phys. D, Appl. Phys., 42, 4, 045201 (2009) |
| [57] | Luque, A.; Ebert, U.; Hundsdorfer, W., Interaction of streamer discharges in air and other oxygen-nitrogen mixtures, Phys. Rev. Lett., 101, 7, 075005 (2008) |
| [58] | Bonaventura, Z.; Duarte, M.; Bourdon, A.; Massot, M., Derivation of a merging condition for two interacting streamers in air, Plasma Sources Sci. Technol., 21, 5, 052001 (2012) |
| [59] | Larsen, E.; Thommes, G.; Klar, A.; Seaid, M.; Gotz, T., Simplified P-N approximations to the equations of radiative heat transfer and applications, J. Comput. Phys., 183, 2, 652-675 (2002) · Zbl 1016.65105 |
| [60] | Liu, N.; Celestin, S.; Bourdon, A.; Pasko, V.; Ségur, P.; Marode, E., Application of photoionization models based on radiative transfer and the Helmholtz equations to studies of streamers in weak electric fields, Appl. Phys. Lett., 91, 21, 211501 (2007) |
| [61] | Zheleznyak, M.; Mnatsakanyan, A.; Sizykh, S., Photo ionization of nitrogen and oxygen mixtures by radiation from a gas discharge, High Temp., 20, 3, 357-362 (1982) |
| [62] | Liu, N.; Pasko, V., Effects of photoionization on propagation and branching of positive and negative streamers in sprites, J. Geophys. Res., 109, A04301 (2004) |
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