Higher order alternate directions methods

✍ Scribed by Michelle Schatzman

• Publisher: Elsevier Science
• Year: 1994
• Tongue: English
• Weight: 318 KB
• Volume: 116
• Category: Article
• ISSN: 0045-7825
• DOI: 10.1016/s0045-7825(94)80026-x

✦ Synopsis

Alternate direction methods are based on the Lie formula (eA/"ea/n)" -e '~+s = O(l/n) for square matrices A and B. A more precise formula is known: (eA/2"eS/"e~/2")" -e a+a = O(1/n2). The search for more accurate product formulae is negative, in the following sense: there is no integer k and no choice of nonnegative real numbers aj and ,8 i, for 1 ~< j ~< k such that (e"'~/"e a~s/" . .. e~kA/"e#kS/")" --e A+s, is of order 3, unless the commutators [A, [A, B] ] or [B, [B, A] ] are linearly dependent. Replacing exponentials by approximation of exponentials does not improve the situation.

However, linear combinations of product formulae can be constructed; they lead to s~able numerical schemes of order 3 and more.

Applications to domain decomposition methods are given, as well as some numerical experiments.