A Finite Difference Parallel Scheme For Fourth Order Parabolic Equations

In many science fields, a lot of phenomena are described by parabolic equation or parabolic equation system. Hence, numerically solving parabolic-partial differential equation by finite difference method has important theoretical significance and application value.Due to the appearance and the increasing availability of vector computers and parallel computers, traditional finite difference methods are being tested by large scale computation, parallelism, and accuracy of all kinds of methods must be compared under the parallel circumstance. By now. the paper existed mostly consider second order parabolic equations. As to construction of parallel finite difference schemes of fourth order parabolic equations is seldom. But in many practical problems, valid parallel finite difference method of fourth order parabolic equations is more necessary.50\Ye will pay much attention to constructing applicable and valid parallel finile difference method on fourth order parabolic equations. Motivated by the results existed, we mainly adopt three-level alternating to achieve parallel finite difference computing of fourth order parabolic equation. As to the stability of this scheme is absolutely stable, the truncation error can achieve O(Af + /r). Our main results can be stated as follows.1. Four asymmetric schemesIn the chapter 2 section 1, we consider the initial-boundary value problem fourth order parabolic equations .As to the stability of explicit scheme, we have the condition r - , where8 but implicit scheme is absolutely stable.With theorem of mean value, we can achieve asymmetric schemes,Theorem 1 The. stability of sc.hciii.fK (2) and (4) olic e(|iiati()ii as follows.\\e have Four Points Group(6)We can get explicit scheme at (/ - 1. n + 1), ('/, n + 1). (/' + 1. ?+ 1) and (v + 2./) + 1). then \ve can achive truncation errors at these points all are- /,'). Theorem 3 Truncation errors of Four Points Group at these points allSo. truncation error of GE scheme is better than asymmetric schemes (2), (3), (4) and (5) in essence.We define N = im + 3. the number of interior points is 4n 4- 2 . So the52her of four points group is in. and two points are put one side. If they arc right side . \ve call this scheme is GEH. if they arc left side. \ve call it is GEL. \\e have the theorem:Theorem 4 Wlu-u r . GER .sr/tcmc. is stable..3. Alternating Group Explicit (AGE) schemeIn the chapter 2 section 3, \ve use GER and GEL schemes to construct AGE scheme of fourth order parabolic equation as follows.If \ve use the GER and GEL schemes alternately on the odd t hue level and rhe even time level, we get theAGE (Alternating Group Explicit) method.AGE can have the following algorithm.V have the following theoremTheorem 5 AGE scheme is absolutely stable.4. Alternating Segment Explicit-Implicit (ASE-I) schemeIn the chapter 2 section 4. we use SE-I scheme scheme to construct ASE-I scheme of fourth order parabolic equation as follows.\Ye want to have value on this points (i() + ; + 1) (/ = 1.2, L). At (+ 1. it + 1). (/0 + 2,y; + 1)./(, + L - 2,n + l)and(/+ L - l,n + 1), we ap|)ly as\'mmetric schemes (2), (5). (3) and (4), and implicit scheme at(/',) + /. // + 1) (/' = 1. 2. I), then \ve get. SE-I scheme.We consider fourth order parabolic equation and construct ASE-I as follow. For positive integer N and L. we assume 1 = XL and L > 5. We divide the points on domain into N segment along :;:-direction. On the same odd time level, the computation schemes of N segment are arranged to be "Explicit scheme- SE-I scheme1- Explicit scheme from left, to right orderly. On the next time level, i.e. . the even time level, we alternately use the schemes on each segment of the odd time level: change explicit, schemes into...