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Stability of nonlinear convection-diffusion-reaction systems in discontinuous Galerkin methods. (English) Zbl 1361.65064

Summary: In this work we provide an extension of the classical von Neumann stability analysis for high-order accurate discontinuous Galerkin methods applied to generalized nonlinear convection-reaction-diffusion systems. We provide a partial linearization under which a sufficient condition emerges that guarantees stability in this context. The stability behavior of these systems is then closely analyzed relative to Runge-Kutta Chebyshev and strong stability preserving temporal discretizations over a nonlinear system of reactive compressible gases arising in the study of atmospheric chemistry.

MSC:

65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35K57 Reaction-diffusion equations
86A10 Meteorology and atmospheric physics
76N15 Gas dynamics (general theory)
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[1] Ammari, H.: Modeling and computations in electromagnetics: a volume dedicated to Jean-Claude Nedelec. Lecture Notes in Computational Science and Engineering. Springer, Dordrecht, (2007) · Zbl 1125.78002
[2] Arnold, D., Brezzi, F., Cockburn, B., Marini, D.: Discontinuous Galerkin methods for elliptic problems. In: Discontinuous Galerkin methods (Newport, RI, 1999), volume 11 of Lect. Notes Comput. Sci. Eng., pp. 89-101. Springer, Berlin, (2000) · Zbl 0948.65127
[3] Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39(5), 1749-1779 (2001) · Zbl 1008.65080 · doi:10.1137/S0036142901384162
[4] Atkinson, K.E.: An introduction to numerical analysis, 2nd edn. John Wiley & Sons Inc, New York (1989) · Zbl 0718.65001
[5] Atkinson, R., Baulch, D.L., Cox, R.A., Crowley, J.N., Hampson, R.F., Hynes, R.G., Jenkin, M.E., Rossi, M.J., Troe, J., Wallington, T.J.: Evaluated kinetic and photochemical data for atmospheric chemistry: volume IV—gas phase reactions of organic halogen species. Atmos. Chem. Phys. 8(15), 4141-4496 (2008) · doi:10.5194/acp-8-4141-2008
[6] Chapman, S., Cowling, T.G.: The mathematical theory of nonuniform gases. Cambridge Mathematical Library. Cambridge University Press, Cambridge, third edition. An account of the kinetic theory of viscosity, thermal conduction and diffusion in gases. In co-operation with D. Burnett, With a foreword by Carlo Cercignani (1990) · Zbl 0726.76084
[7] Dawson, C., Westerink, J., Feyen, J., Pothina, D.: Continuous, discontinuous and coupled discontinuous-continuous Galerkin finite element methods for the shallow water equations. Int. J. Numer. Methods Fluids 52(1), 63-68 (2006) · Zbl 1097.76048 · doi:10.1002/fld.1156
[8] Descombes, S., Massot, M.: Operator splitting for nonlinear reaction-diffusion systems with an entropic structure: singular perturbation and order reduction. Numer. Math. 97(4), 667-698 (2004) · Zbl 1060.65105 · doi:10.1007/s00211-003-0496-3
[9] Feireisl, E., Novotný, A., Petzeltová, H.: On the domain dependence of solutions to the compressible Navier-Stokes equations of a barotropic fluid. Math. Methods Appl. Sci. 25(12), 1045-1073 (2002) · Zbl 0996.35051 · doi:10.1002/mma.327
[10] Phys. Plasm. Two-fluid magnetic island dynamics in slab geometry. ii. islands interacting with resistive walls or resonant magnetic perturbations. 12(2), 022307 (2005) · Zbl 1168.65380
[11] Multicomponent flow modeling. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser Boston Inc., Boston, MA (1999) · Zbl 1073.65088
[12] Gottlieb, S., Ketcheson, D., Shu, C.-W.: High order strong stability preserving time discretizations. J. Sci. Comput. 38(3), 251-289 (2009) · Zbl 1203.65135 · doi:10.1007/s10915-008-9239-z
[13] Gottlieb, S., Shu, C.-W., Tadmor, E.: Strong stability-preserving high-order time discretization methods. SIAM Rev 43(1), 89-112 (2001). (electronic) · Zbl 0967.65098 · doi:10.1137/S003614450036757X
[14] Heath, R.E., Gamba, I.M., Morrison, P.J., Michler, C.: A discontinuous Galerkin method for the Vlasov-Poisson system. J. Comput. Phys. 231(4), 1140-1174 (2012) · Zbl 1244.82081 · doi:10.1016/j.jcp.2011.09.020
[15] Hirschfelder, J., Curtiss, C., Bird, R.: The Molecular Theory of Gases and Liquids. Structure of Matter Series. Wiley-Interscience, New York (1954) · Zbl 0057.23402
[16] Janenko, N. N.: The method of fractional steps for the solution of problems in continuum mechanics. In: Partial differential equations (Proc. Sympos.) (Russian), pp. 239-249. Izdat. “Nauka”, Moscow (1970) · Zbl 1425.65117
[17] Kubatko, E., Dawson, C., Westerink, J.: Time step restrictions for Runge-Kutta discontinuous Galerkin methods on triangular grids. J. Comput. Phys. 227(23), 9697-9710 (2008) · Zbl 1154.65071 · doi:10.1016/j.jcp.2008.07.026
[18] Kubatko, E., Westerink, J., Dawson, C.: An unstructured grid morphodynamic model with a discontinuous Galerkin method for bed evolution. Ocean Modell. 15(1-2) 71-89, (2006) 3rd International Workshop on Unstructured Mesh Numerical Modelling of Coastal, Shelf and Ocean Flows, Toulouse, France. SEP 20-22, (2004) · Zbl 1364.35278
[19] Kubatko, E., Westerink, J., Dawson, C.: Semi discrete discontinuous Galerkin methods and stage-exceeding-order, strong-stability-preserving Runge-Kutta time discretizations. J. Comput. Phys. 222(2), 832-848 (2007) · Zbl 1113.65093 · doi:10.1016/j.jcp.2006.08.005
[20] Kubatko, E., Yeager, B., Ketcheson, D.: Optimal strong-stability-preserving Runge-Kutta time discretizations for discontinuous Galerkin methods. J. Sci. Comput. 60(2), 313-344 (2014) · Zbl 1304.65219 · doi:10.1007/s10915-013-9796-7
[21] LeVeque, R.: Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems (Classics in Applied Mathematics Classics in Applied Mathemat). Society for Industrial and Applied Mathematics, Philadelphia, PA, USA (2007) · Zbl 1127.65080
[22] Liu, H., Yu, H.: The entropy satisfying discontinuous Galerkin Method for Fokker-Planck equations. J. Sci. Comput. 62(3), 803-830 (2015) · Zbl 1320.65143 · doi:10.1007/s10915-014-9878-1
[23] Liu, X., Nie, Q.: Compact integration factor methods for complex domains and adaptive mesh refinement. J. Comput. Phys. 229(16), 5692-5706 (2010) · Zbl 1194.65111 · doi:10.1016/j.jcp.2010.04.003
[24] Loverich, J., Hakim, A., Shumlak, U.: A discontinuous Galerkin method for ideal two-fluid plasma equations. Commun. Comput. Phys. 9(2), 240-268 (2011) · Zbl 1167.76384 · doi:10.4208/cicp.250509.210610a
[25] Lu, B., Zhou, Y.C.: Poisson-Nernst-Planck equations for simulating biomolecular diffusion-reaction processes II: size effects on ionic distributions and diffusion-reaction rates. Biophys. J. 100(10), 2475-2485 (2011) · doi:10.1016/j.bpj.2011.03.059
[26] Mellet, A., Vasseur, A.: On the barotropic compressible Navier-Stokes equations. Comm. Partial Differ. Equ. 32(1-3), 431-452 (2007) · Zbl 1149.35070 · doi:10.1080/03605300600857079
[27] Michoski, C., Evans, J., Schmitz, P.: Discontinuous galerkin-adaptive methods for multiscale chemical reactors: Quiescent reactors. Comput. Methods Appl. Mech. Eng. 279, 163-197 (2014) · Zbl 1425.65117 · doi:10.1016/j.cma.2014.06.020
[28] Michoski, C., Vasseur, A.: Existence and uniqueness of strong solutions for a compressible multiphase Navier-Stokes miscible fluid-flow problem in dimension n = 1. Math. Models Methods Appl. Sci. 19(03), 443-476 (2009) · Zbl 1166.76047 · doi:10.1142/S0218202509003498
[29] Naldi, G., Pareschi, L., Toscani, G. (eds.): Mathematical modeling of collective behavior in socio-economic and life sciences. Birkhäuser Boston, Boston (2010) · Zbl 1200.91010
[30] Ropp, D.L., Shadid, J.N.: Stability of operator splitting methods for systems with indefinite operators: reaction-diffusion systems. J. Comput. Phys. 203(2), 449-466 (2005) · Zbl 1073.65088 · doi:10.1016/j.jcp.2004.09.004
[31] Ropp, D.L., Shadid, J.N.: Stability of operator splitting methods for systems with indefinite operators: advection-diffusion-reaction systems. J. Comput. Phys. 228(9), 3508-3516 (2009) · Zbl 1168.65380 · doi:10.1016/j.jcp.2009.02.001
[32] Ruuth, S.: Global optimization of explicit strong-stability-preserving Runge-Kutta methods. Math. Comp 75(253), 183-207 (2006). (electronic) · Zbl 1080.65088 · doi:10.1090/S0025-5718-05-01772-2
[33] Shu, C.-W., Osher, S.: Efficient implementation of essentially nonoscillatory shock-capturing schemes. J. Comput. Phys. 77(2), 439-471 (1988) · Zbl 0653.65072 · doi:10.1016/0021-9991(88)90177-5
[34] Smoller, J.: Shock waves and reaction-diffusion equations, volume 258 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, New York, second edition, (1994) · Zbl 0807.35002
[35] Solonnikov, V. A., Tani, A.: Evolution free boundary problem for equations of motion of viscous compressible barotropic liquid. In The Navier-Stokes equations II—theory and numerical methods (Oberwolfach, 1991), volume 1530 of Lecture Notes in Math., pages 30-55. Springer, Berlin, (1992) · Zbl 0786.35106
[36] Sportisse, B.: An analysis of operator splitting techniques in the stiff case. J. Comput. Phys. 161(1), 140-168 (2000) · Zbl 0953.65062 · doi:10.1006/jcph.2000.6495
[37] Srinivasan, B., Hakim, A., Shumlak, U.: Numerical methods for two-fluid dispersive fast MHD phenomena. Commun. Comput. Phys. 10(1), 183-215 (2011) · Zbl 1364.76120 · doi:10.4208/cicp.230909.020910a
[38] Strikwerda, J.C.: Finite difference schemes and partial differential equations, 2nd edn. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2004) · Zbl 1071.65118 · doi:10.1137/1.9780898717938
[39] Sutherland, J., Kennedy, C.: Improved boundary conditions for viscous, reacting, compressible flows. J. Comput. Phys. 191, 502-524 (2003) · Zbl 1134.76736 · doi:10.1016/S0021-9991(03)00328-0
[40] Tabei, M., Mast, T., Waag, R.: A k-space method for coupled first-order acoustic propagation equations. J. Acoust. Soc. Am. 111(1, 1), 53-63 (2002) · doi:10.1121/1.1421344
[41] Torrilhon, M., Jeltsch, R.: Essentially optimal explicit Runge-Kutta methods with application to hyperbolic-parabolic equations. Numer. Math. 106(2), 303-334 (2007) · Zbl 1113.65074 · doi:10.1007/s00211-006-0059-5
[42] Trefethen, L. N.: Finite difference and spectral methods for ordinary and partial differential equations (1996)
[43] van Der Houwen, P.J., Sommeijer, B.P.: On the internal stability of explicit, m-stage Runge-Kutta methods for large m-values. ZAMM—J. Appl. Math. Mech./Zeitschrift für Angewandte Mathematik und Mechanik 60(10), 479-485 (1980) · Zbl 0455.65052
[44] Verwer, J .G., Sommeijer, B .P.: An implicit-explicit Runge-Kutta-Chebyshev scheme for diffusion-reaction equations. SIAM J. Sci. Comput 25(5), 1824-1835 (2004). (electronic) · Zbl 1061.65090 · doi:10.1137/S1064827503429168
[45] Wang, F.-B.: A PDE system modeling the competition and inhibition of harmful algae with seasonal variations. Nonlinear Anal. Real World Appl. 25, 258-275 (2015) · Zbl 1327.35380 · doi:10.1016/j.nonrwa.2015.02.010
[46] Wanner, G., Hairer, E., Norsett, S.: Order stars and stability theorems. BIT Numer. Math. 18(4), 475-489 (1978) · Zbl 0444.65039 · doi:10.1007/BF01932026
[47] Wiebe, N., Berry, D., Høyer, P., Sanders, B.C.: Higher order decompositions of ordered operator exponentials. J. Phys. A Math. Theor. 43(6), 065203 (2010) · Zbl 1191.47055 · doi:10.1088/1751-8113/43/6/065203
[48] Wu, F., Carr, R.W.: Kinetics of CH2ClO radical reactions with O2 and NO, and the unimolecular elimination of HCl. J. Phys. Chem. A 105(9), 1423-1432 (2001) · doi:10.1021/jp001953m
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