zbMATH — the first resource for mathematics

Commutators of certain holomorphic semigroups and applications to alternate directions. (Commutateurs de certains semi-groupes holomorphes et applications aux directions alternées.) (French) Zbl 0853.47024
Summary: Let \(A\) and \(B\) be two non necessarily commuting operators. Two high-order alternate direction formulae are defined by \[ M_1(t)= \textstyle{{4\over 3}} e^{tA/4} e^{tB/2} e^{tA/2} e^{tB/2} e^{tA/4}- \textstyle{{1\over 3}} e^{tA/2} e^{tB} e^{tA/2}, \] \[ M_2(t)= \textstyle{{2\over 3}} (e^{tA/2} e^{tB} e^{tA/2}+ e^{tB/2} e^{tA} e^{tB/2})- \textstyle{{1\over 6}} (e^{tA} e^{tB}+ e^{tB} e^{tA}) \] and in the sense of formal series \[ \begin{cases} e^{t(A+ B)}- M_1(t/n)^n= O(n^{- 4})\\ e^{t(A+ B)}- M_2(t/n)^n= O(n^{- 3})\end{cases}\tag{\(*\)} \] If \(a\), \(a_0\), \(b\) and \(b_0\) are strictly positive and infinitely differentiable from \(\mathbb{T}^2= (\mathbb{R}/\mathbb{Z})^2\) to \(\mathbb{R}\), operators \(A\) and \(B\) are defined by \[ A= {\partial\over \partial x_1} \Biggl(a(x_1, x_2) {\partial\over \partial x_1}\Biggr)- a_0(x_1, x_2);\;B= {\partial\over \partial x_2} \Biggl(b(x_1, x_2) {\partial\over \partial x_2}\Biggr)- b_0(x_1, x_2). \] These two operators generate holomorphic semigroups in \(L^2(\mathbb{T}^2)\) and the following estimates hold in operator norm in \(L^2(\mathbb{T}^2)\), \[ |M_1(t)|= O(t)\quad \text{and} \quad |M_2(t)|= O(t). \] Here, there exists a constant \(c\) such that \[ |M_1(t/n)^n|\leq e^{ct}\quad \text{and} \quad |M_2(t/n)^n|\leq e^{ct}, \] which implies that formulae \((*)\) are stable.

47D06 One-parameter semigroups and linear evolution equations
47F05 General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX)
Full Text: DOI EuDML
[1] N. BOURBAKI, 1953, Espaces vectoriels topologiques, Livre V. Herman, Paris. MR54161 · Zbl 0050.10703
[2] N. BOURBAKI, 1969, Intégration, Livre VI. Herman, Paris. MR276436 · Zbl 0189.14201
[3] D. O. DIA, Analyse numérique de certains schémas de directions alternées d’ordre élevé, en préparation.
[4] T. KATO, 1966, Perturbation Theory for Linear Operators, Springer, Berlin. Zbl0148.12601 · Zbl 0148.12601
[5] F. RIESZ, 1952, Leçons d’Analyse Fonctionnelle, Akadémiai Kiadó, Budapest. Zbl0122.11205 MR55417 · Zbl 0122.11205
[6] M. SCHATZMAN, Higher order formulae for time integration, International Conference on spectral and higher order methods, Montpellier, juin 1992, à paraître dans les ’Proceedings ICOSAHOM 92’.
[7] Q. SHENG, 1989, Solving Linear Partial Differential Equations by Exponential Splitting, IMA J. Numer. Anal., 9, pp. 199-212. Zbl0676.65116 MR1000457 · Zbl 0676.65116 · doi:10.1093/imanum/9.2.199
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.