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An analysis of operator splitting techniques in the stiff case. (English) Zbl 0953.65062
Operator splitting methods are widely used in many applications, such as air pollution modeling, combustion, and reactive flows. The author regards the case where the evolution equations to be simulated are stiff. He considers systems with two operators: a stiff one and a non-stiff one. For example, a linear evolution system under a singular perturbation has the form ${{d z}\over{d t}} = \Biggl({A\over \varepsilon} + B \Biggr) z,\quad z(0) = z_0,$ where $$\varepsilon$$ is a small positive parameter. The author shows some splitting schemes for solving of the evolution systems (“first-order”, “second-order” schemes). The “first-order” scheme has the form \begin{aligned} {{d z^*}\over{d t}} = B^* z^*,\quad z^*(0) = z_0, &\quad \text{on }[0,\Delta t],\\ {{d z^{**}}\over{d t}} = B^* z^{**},\quad z^{**}(0) = z^*(\Delta t), &\quad \text{on }[0,\Delta t],\end{aligned} where the final value is given by $$z^{**}(\Delta t)$$. The main results deals with the choice of the sequential order for the operators: the stiff operator must always be last in the splitting scheme.
Reviewer: A.Dishliev (Sofia)

##### MSC:
 65M20 Method of lines for initial value and initial-boundary value problems involving PDEs 92D40 Ecology 92E20 Classical flows, reactions, etc. in chemistry 34A30 Linear ordinary differential equations and systems 34E15 Singular perturbations for ordinary differential equations 35K15 Initial value problems for second-order parabolic equations 65L05 Numerical methods for initial value problems involving ordinary differential equations 80A25 Combustion 80M20 Finite difference methods applied to problems in thermodynamics and heat transfer
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##### References:
 [1] Browning, G.; Kreiss, H.O., Splitting methods for problems with different timescales, Monthly weather rev., 122, 2614, (1994) [2] Burman, E.; Sainsaulieu, L., Numerical analysis of two operator splitting methods for an hyperbolic system of conservation laws with stiff relaxation terms, (April 1994) [3] Caflish, R.E.; Jin, Shi; Russo, G., Uniformly accurate schemes for hyperbolic systems with relaxation, SIAM J. numer. anal., 34, 246, (1997) · Zbl 0868.35070 [4] Dekker, K.; Verwer, J.G., Stability of runge – kutta methods for stiff non-linear differential equations, (1984) · Zbl 0571.65057 [5] Graf, J.; Moussiopoulos, N., Intercomparison of two models for the dispersion of chemically reacting pollutants, Beitr. phys. atmosph., 64, 13, (Feb. 1991) [6] Hesstvedt, E.; Hov, O.; Isaksen, I.S.A., Qssas in air polution modelling: comparison of two numerical schemes for oxydant prediction, Int. J. chem. kinet., 10, 971, (1978) [7] Hindmarsh, A.C., Scientific computing, 55-74, (1983) [8] Hundsdorfer, W.H., Numerical solution of advection-diffusion-reaction equations, (1996) [9] Jin, Shi, Runge – kutta methods for hyperbolic conservation laws with stiff relaxation terms, J. comp. phys., 122, 51, (1995) · Zbl 0840.65098 [10] Kim, J.; Cho, S.Y., Computation accuracy and efficiency of the time-splitting method in solving atmospheric transport-chemistry equations, Atmos. environ., 31, 2215, (1997) [11] Knoth, O.; Wolke, R., Air pollution modelling and its applications X, (1994) [12] Kreiss, H.O., Numerical solution of problems with different timescales II, 86, (1997) [13] L. Lanser, and, J. G. Verwer, Analysis of operator splitting for advection-diffusion-reaction problems from air pollution modeling, in, Proceedings 2nd Meeting on Numerical Methods for Differential Equations, Coimbra, Portugal, February 1998. · Zbl 0949.65090 [14] Larrouturou, B.; Sportisse, B., Some mathematical and numerical aspects of reduction in chemical kinetics, Computational science for the 21st century, conference in honor of Professor R. glowinski, (1997) · Zbl 0911.65059 [15] LeVeque, R.J.; Yee, H.C., A study of numerical methods for hyperbolic conservation laws with stiff source terms, J. comp. phys., 86, 187, (1990) · Zbl 0682.76053 [16] Lubich, Ch.; Nipp, K.; Stoffer, D., Rk solutions of stiff differential equations near stationary points, SIAM J. numer. anal., 32, 1296, (1995) · Zbl 0840.65077 [17] McRae, G.J.; Goodin, W.R.; Seinfeld, J.H., Numerical solution of the atmospheric diffusion equation for chemically reacting flows, J. comp. phys., 45, (1982) · Zbl 0502.76098 [18] Nipp, K.; Stoffer, D., Invariant manifolds and global error estimates of numerical integration schemes applied to stiff systems of singular perturbation types. aprt i: rk methods, Numer. math., 70, 245, (1995) · Zbl 0824.65058 [19] Otey, G.R.; Dywer, H.A., Numerical study of the interaction of fast chemistry and diffusion, Aiaa j., 17, 606, (1978) [20] Prothero, A.; Robinson, A., On the stability and accuracy of one-step methods for solving stiff systems of odes, Math. comput., 28, 145, (1974) · Zbl 0309.65034 [21] P. Rouchon, and, B. Sportisse, Slow and fast kinetic scheme with slow diffusion, in, Proceedings of the Workshop Numerical Aspects of Reduction in chemical kinetics, ENPC-CERMICS, Sept. 1997. [22] Sanz-Serna, J.M., The state of the art in numerical analysis, 121-143, (1997) [23] Sportisse, B., Contribution à la modélisation des écoulements réactifs: réduction des modèles de cinétique chimique et simulation de la pollution atmosphérique, (April 1999) [24] B. Sportisse, Reducing chemical kinetics: A mathematical point of view, in, Proceedings of the Workshop Numerical Aspects of Reduction in chemical kinetics, ENPC-CERMICS, Sept. 1997. · Zbl 0911.65059 [25] B. Sportisse, G. Bencteux, and, P. Plion, Method of lines versus operator splitting methods for reaction-diffusion systems in air pollution modelling, in, Proceedings of the Conference APMS ’98, ENPC-INRIA, Oct. 26-29 1998. [26] Sportisse, B.; Djouad, R., Reduction of chemical kinetics in air pollution modelling, (July 1999) [27] Sportisse, B.; Jaubertie, A.; Plion, P., Preprocessed reduction of a simplified photochemical mechanism describing the tropospheric chemistry of ozone, (1998) [28] Sportisse, B.; Rouchon, P., Reduction of slow-fast chemistry with slow processes, (July 1999) [29] Strang, G., On the construction and comparison of difference schemes, SIAM J. numer. anal., 5, 506, (1968) · Zbl 0184.38503 [30] Sun, Pu, A pseudo non-time splitting method in air quality modeling, J. comp. phys., 127, 152, (1996) · Zbl 0859.65133 [31] Tikhonov, A.N.; Vasileva, A.B.; Sveshnikov, A.G., Differential equations, (1985) · Zbl 0553.34001 [32] Verwer, J.G.; de Vries, H.B., Global extrapolation and first-order splitting method, SIAM J. sci. stat. comput., 6, (1985) · Zbl 0592.65060 [33] Verwer, J.G.; Sportisse, B., A note on operator splitting analysis in the stiff linear case, (1998) [34] Yanenko, N.N., The method of fractional steps, (1971) · Zbl 0209.47103
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