×

zbMATH — the first resource for mathematics

Dimension splitting for evolution equations. (English) Zbl 1149.65084
The authors study splitting methods for the time integration of abstract evolution equations \(u_t = (A+B)u\). There are some works which yield optimal convergence order, however they require one of the following restrictions: the operator \(B\) is relatively bounded with respect to \(A\), or the evolution equation is either given on an unbounded domain or on a domain equipped with periodic boundary conditions.
The aim of the paper is to propose an analytic framework which enables the derivation of optimal convergence orders for various splitting methods (including the Lie and Peaceman-Rachford splitting) without these restrictions. Some numerical experiments are presented to illustrate the theoretical results.

MSC:
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35K90 Abstract parabolic equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Descombes S. and Ribot M. (2003). Convergence of the Peaceman–Rachford approximation for reaction–diffusion systems. Numer. Math. 95(3): 503–525 · Zbl 1034.65077
[2] Descombes S. and Schatzman M. (2002). Strang’s formula for holomorphic semi-groups. J. Math. Pures Appl. 81(1): 93–114 · Zbl 1030.35095
[3] Faou, E.: Analysis of splitting methods for reaction–diffusion problems in the light of stochastic calculus. Preprint (2007)
[4] Faragó I. and Havasi Á. (2007). Consistency analysis of operator splitting methods for C 0-semigroups. Semigroup Forum 74(1): 125–139 · Zbl 1125.47033
[5] Gyöngy I. and Krylov N. (2005). An accelerated splitting-up method for parabolic equations. SIAM J. Math. Anal. 37(4): 1070–1097 · Zbl 1101.35038
[6] Hundsdorfer W. and Verwer J. (2003). Numerical Solution of Time-Dependent Advection–Diffusion–Reaction Equations. Springer, Berlin · Zbl 1030.65100
[7] Ichinose, T., Tamura, H.: The norm convergence of the Trotter–Kato product formula with error bound. Comm. Math. Phys. 217(3), 489–502 (2001). Erratum: Comm. Math. Phys. 254(1), 255 (2005) · Zbl 0996.47046
[8] Jahnke T. and Lubich C. (2000). Error bounds for exponential operator splittings. BIT 40(4): 735–744 · Zbl 0972.65061
[9] Kufner A. (1985). Weighted Sobolev Spaces. Wiley, New York · Zbl 0567.46009
[10] Kühnemund F. and Wacker M. (2001). Commutator conditions implying the convergence of the Lie–Trotter products. Proc. Am. Math. Soc. 129(12): 3569–3582 · Zbl 0987.34061
[11] McLachlan R.I. and Quispel G.R.W. (2002). Splitting methods. Acta Numer. 11: 341–434 · Zbl 1105.65341
[12] Neidhardt H. and Zagrebnov V.A. (1999). Trotter–Kato product formula and operator-norm convergence. Comm. Math. Phys. 205(1): 129–159 · Zbl 0949.47019
[13] von Neumann J. (1951). Eine Spektraltheorie für allgemeine Operatoren eines unitären Raumes. Math. Nachr. 4: 258–281 · Zbl 0042.12301
[14] Ostermann A. (2002). Stability of W-methods with applications to operator splitting and to geometric theory. Appl. Numer. Math. 42(1–3): 353–366 · Zbl 1007.65063
[15] Pazy A. (1983). Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York · Zbl 0516.47023
[16] Rasmussen S. (1972). Non-linear semi-groups, evolution equations and productintegral representations. Various publications series, Aarhus University 20: 1–89
[17] Schatzman M. (1999). Stability of the Peaceman–Rachford approximation. J. Funct. Anal. 162(1): 219–255 · Zbl 0920.47021
[18] Triebel H. (1978). Interpolation Theory, Function Spaces, Differential Operators. North-Holland, Amsterdam · Zbl 0387.46032
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.