## Iterative operator-splitting methods with higher-order time integration methods and applications for parabolic partial differential equations.(English)Zbl 1144.65062

Computational simulations of bioremediation, heat transfer, transport and chemical reaction problems require accurate numerical methods. One of the widely spread methods is to transform time and space dependent problems described by one or a system of partial differential equations to a system of ordinary differential equations (ODEs) using a certain space discretization method like the finite element method and then solve the system of ODEs.
The presented method is a combination of two efficient methods: an iterative and an operator-splitting method by using the advantages of both of them. Splitting methods have the benefit in decoupling multiphysics problems into simpler physical problems and solve this reduced problems in each time step. The iterative method is used in each time step for solving reduced problems iteratively. An analysis of the convergence and the rate of convergence of this iterative operator-splitting method is presented.
To obtain high-order methods Runge-Kutta and backward differentiation formula methods are used. For both commutative and noncommutative operator of the continuous equations the A-stability of the method is proved under certain conditions. Various numerical experiments for systems of ODEs or parabolic equations also with nonlinearities are presented. Comparisons of various methods of higher order together with various splitting methods are shown.

### 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 35K55 Nonlinear parabolic equations 65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
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### References:

 [1] Butcher, J.C., Numerical methods for ordinary differential equations, (2003), Wiley Chichester · Zbl 1032.65512 [2] Chau, K.W.; Wu, C.L.; Li, Y.S., Comparison of several flood forecasting models in yangtze river, J. hydrologic eng. ASCE, 10, 6, 485-491, (2005) [3] Cheng, C.T.; Chau, K.W., Fuzzy iteration methodology for reservoir flood control operation, J. amer. water resources assoc., 37, 5, 1381-1388, (2001) [4] Csomós, P.; Faragó, I.; Havasi, Á, Weighted sequential splittings and their analysis, Comput. math. appl., 50, 1017-1031, (2005) · Zbl 1086.65053 [5] Daoud, D.; Geiser, J., Overlapping Schwarz wave form relaxation for the solution of coupled and decoupled system of convection diffusion reaction equation, Appl. math. comput., 190, 1, 946-964, (2007) · Zbl 1126.65087 [6] Engel, K.-J.; Nagel, R., One-parameter semigroups for linear evolution equations, (2000), Springer New York · Zbl 0952.47036 [7] R.E. Ewing, Up-scaling of biological processes and multiphase flow in porous media, in: IIMA Volumes in Mathematics and its Applications, vol. 295, Springer, Berlin, 2002, pp. 195-215. · Zbl 1102.76332 [8] I. Faragó, Splitting methods for abstract Cauchy problems, in: Z. Li, L. Vulkov, J. Was’niewski (eds.), Numerical Analysis and Its Application, Lecture Notes in Computer Science, vol. 3401, Springer, Berlin, 2005, pp. 35-45. [9] I. Faragó, A modified iterated operator splitting method, Appl. Math. Model., (to appear) Published online: July 6, 2007, $$\langle$$doi:10.1016/j.apm.2007.04.018⟩. [10] I. Faragó, J. Geiser, Iterative Operator-Splitting Methods for Linear Problems. Preprint no. 1043 of the Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany, June 2005. [11] Faragó, I.; Havasi, Á, On the convergence and local splitting error of different splitting schemes, Progr. comput. fluid dyn., 5, 495-504, (2005) · Zbl 1189.76427 [12] P. Frolkovič, J. Geiser, Numerical Simulation of Radionuclides Transport in Double Porosity Media with Sorption, in: Proceedings of Algorithmy 2000, Conference of Scientific Computing, 2000, pp. 28-36. [13] J. Geiser, Numerical Simulation of a Model for Transport and Reaction of Radionuclides, in: Proceedings of the Large Scale Scientific Computations of Engineering and Environmental Problems, Sozopol, Bulgaria, 2001. · Zbl 1136.65336 [14] J. Geiser, Gekoppelte Diskretisierungsverfahren für Systeme von Konvektions-Dispersions-Diffusions-Reaktionsgleichungen, Ph.D. Thesis, University of Heidelberg, Germany, 2004. [15] J. Geiser, $$R^3 T$$ : radioactive-retardation-reaction-transport-program for the simulation of radioactive waste disposals, Technical Report, Institute for Scientific Computation, TX A&M University, College Station, April 2004. [16] J. Geiser, Weighted iterative operator-splitting methods : stability-theory. Lecture Notes in Computer Science, Springer, Berlin, vol. 4310, 2007 pp. 40-47, Proceedings of the 6th International Conference, NMA, Borovets, Bulgaria. · Zbl 1137.65405 [17] J. Geiser, Linear and Quasi-Linear Iterative Splitting Methods: Theory and Applications. International Mathematical Forum, Hikari Ltd., vol.2, (49), 2007, pp. 2391-2416. · Zbl 1144.65060 [18] J. Geiser, Decomposition methods for partial differential equations: theory and applications in multiphysics problems, Habilitation Thesis, Humboldt University of Berlin, Germany, reviewed, July 2007. [19] J. Geiser, R.E. Ewing, J. Liu, Operator splitting methods for transport equations with nonlinear reactions, in: Proceedings of the Third MIT Conference on Computational Fluid and Solid Mechanic, Cambridge, MA, June 14-17, 2005. [20] W. Hundsdorfer, L. Portero, A note on iterated splitting schemes, CWI Report MAS-E0404, Amsterdam, Netherlands, 2005. [21] Hundsdorfer, W.H.; Verwer, J., Numerical solution of time-dependent advection – diffusion – reaction equations, (2003), Springer Berlin · Zbl 1030.65100 [22] Kanney, J.; Miller, C.; Kelley, C., Convergence of iterative split-operator approaches for approximating nonlinear reactive transport problems, Adv. water resources, 26, 247-261, (2003) [23] R.J. LeVeque, Finite volume methods for hyperbolic problems, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002. · Zbl 1010.65040 [24] Marchuk, G.I., Some applicatons of splitting-up methods to the solution of problems in mathematical physics, Apl. mat., 1, 103-132, (1968) · Zbl 0159.44702 [25] Strang, G., On the construction and comparision of difference schemes, SIAM J. numer. anal., 5, 506-517, (1968) · Zbl 0184.38503 [26] J.G. Verwer, B. Sportisse, A note on operator splitting in a stiff linear case, MAS-R9830, ISSN 1386-3703, 1998. [27] Wu, C.L.; Chau, K.W.; Huang, J.S., Modelling coupled water and heat transport in a soil – much – plant – atmosphere continuum (SMPAC) system, Appl. math. modelling, 31, 2, 152-169, (2007) · Zbl 1197.76021 [28] Yoshida, H., Construction of higher order symplectic integrators, Phys. lett. A, 150, 5-7, (1990) [29] Zlatev, Z., Computer treatment of large air pollution models, (1995), Kluwer Academic Publishers Dordrecht · Zbl 0852.65058
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