Applications of Mathematics, Vol. 62, No. 4, pp. 405-432, 2017
Error analysis of splitting methods for semilinear evolution equations
Masahito Ohta, Takiko Sasaki
Received January 30, 2017. First published July 12, 2017.
Abstract: We consider a Strang-type splitting method for an abstract semilinear evolution equation \partial_t u = Au+F(u). Roughly speaking, the splitting method is a time-discretization approximation based on the decomposition of the operators $A$ and $F.$ Particularly, the Strang method is a popular splitting method and is known to be convergent at a second order rate for some particular ODEs and PDEs. Moreover, such estimates usually address the case of splitting the operator into two parts. In this paper, we consider the splitting method which is split into three parts and prove that our proposed method is convergent at a second order rate.
References: [1] C. Besse, B. Bidégaray, S. Descombes: Order estimates in the time of splitting methods for the nonlinear Schrödinger equation. SIAM J. Numer. Anal. 40 (2002), 26-40. DOI 10.1137/S0036142900381497 | MR 1921908 | Zbl 1026.65073 [2] J. P. Borgna, M. De Leo, D. Rial, C. Sánchez de la Vega: General splitting methods for abstract semilinear evolution equations. Commun. Math. Sci. 13 (2015), 83-101. DOI 10.4310/CMS.2015.v13.n1.a4 | MR 3238139 | Zbl 1311.65106 [3] T. Cazenave, A. Haraux: An Introduction to Semilinear Evolution Equations. Oxford Lecture Series in Mathematics and Its Applications 13, Clarendon Press, Oxford (1998). MR 1691574 | Zbl 0926.35049 [4] S. Descombes, M. Thalhammer: An exact local error representation of exponential operator splitting methods for evolutionary problems and applications to linear Schrödinger equations in the semi-classical regime. BIT 50 (2010), 729-749. DOI 10.1007/s10543-010-0282-4 | MR 2739463 | Zbl 1205.65250 [5] E. Hairer, C. Lubich, G. Wanner: Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations. Springer Series in Computational Mathematics 31, Springer, Berlin (2006). DOI 10.1007/3-540-30666-8 | MR 2221614 | Zbl 1094.65125 [6] T. Jahnke, C. Lubich: Error bounds for exponential operator splittings. BIT 40 (2000), 735-744. DOI 10.1023/A:1022396519656 | MR 1799313 | Zbl 0972.65061 [7] C. Lubich: On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations. Math. Comput. 77 (2008), 2141-2153. DOI 10.1090/S0025-5718-08-02101-7 | MR 2429878 | Zbl 1198.65186
Affiliations: Masahito Ohta, Department of Mathematics, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo, Japan; Takiko Sasaki, Department of Mathematics, Meiji University, 1-1-1 Higashimita Tama-ku, Kawasaki, Kanagawa, Japan, e-mail: takiko@meiji.ac.jp
