zbMATH — the first resource for mathematics

Stability of the Peaceman-Rachford approximation. (English) Zbl 0920.47021
Let \(A\) and \(B\) be selfadjoint bounded from below operators in a Hilbert space. The author investigates stability of approximation (in the operator norm) of \(e^{-t(A+B)}\) by the Peaceman-Rachford formula \[ p(t/2n)^n \to e^{-t(A+B)}, \] where \(p(t) = (1+tA)^{-1}(1-tB)(1+tB)^{-1}(1-tA)\).
The approximation is stable if \(\| p(t)\| \leq 1+O(t)\). Sufficient conditions for stability are given; they involve conditions on the commutators of \(\sqrt A\) and \(\sqrt B\).

47B25 Linear symmetric and selfadjoint operators (unbounded)
Full Text: DOI
[1] Chernoff, P.R., Note on product formulas for semigroups of operators, J. funct. anal., 2, 238-242, (1968) · Zbl 0157.21501
[2] B. O. Dia, Université Claude Bernard-Lyon 1, 1996
[3] Dia, B.O.; Schatzman, M., Estimations sur la formule de strang, C.R. acad. sci. Paris, 320, 775-779, (1995) · Zbl 0827.47034
[4] Dia, B.O.; Schatzman, M., Commutateurs de certains semi-groupes holomorphes et applications aux directions alternées, Math. modél. an. num. (M2AN), 30, 343-383, (1996) · Zbl 0853.47024
[5] B. O. Dia, M. Schatzman, An estimate on the transfer operator, J. Funct. Anal. · Zbl 0919.47031
[6] Douglas, J., On the numerical integration of ∂^2ux22uy2ut, J. SIAM, 3, 42-65, (1955)
[7] Kato, T., Trotter’s formula for an arbitrary pair of self-adjoint contraction semigroups, Topics in functional analysis, (1978), Academic Press New York, p. 185-195
[8] Gilkey, P., Invariance theory, the heat equation, and the atiyah – singer index theorem, (1984), Publish or Peris Wilmington · Zbl 0565.58035
[9] Hormander, L., The analysis of partial differential operators, III, (1985), Springer-Verlag Berlin
[10] Kreiss, H.O., Numerical methods for solving time-dependent problems for partial differential equations, (1978), Presses Univ. Montréal Montréal
[11] Marchuk, G.I., Splitting and alternating direction methods, (), 197-467 · Zbl 0875.65049
[12] Peaceman, D.W.; Rachford, H.H., The numerical solution of parabolic and elliptic differential equations, J. SIAM, 3, 42-65, (1955) · Zbl 0067.35801
[13] Trotter, H., On the product of semigroups of operators, Proc. amer. math. soc., 10, 545-551, (1959) · Zbl 0099.10401
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.