Difference Scheme And Parallel Computation For A Class High Order Diffusion Equations

In many science fields, a lot of phenomena is described by parabolic equation or parabolic system. Hence, numerically solving parabolic partial differential equation by finite difference method is significant in theory and application. As to construction of parallel finite difference schemes of sixth order and exceeding sixth order parabolic equations,existing work is seldom. As tothe stability of explicit scheme of sixth order diffusion equation, we have thecondition r < 1/32A;, where r = h and t are step sizes in space and in timerespectively. In order to solve implicit scheme of sixth order, we have to solve 7's diagonal matrix , so valid parallel finite difference method for solving sixth order parabolic equations is more necessary.In this paper ,we will pay much attention to constructing practical and effective finite difference method for sixth order parabolic equations. Our mainresults can be stated as follows.1. Finite difference schemes for sixth order linear parabolic equationsWe consider the initial-boundary value problem of sixth order parabolicequations as follows.which boundary conditions areand initial condition is(3). We have explicit scheme and implicit scheme as follows 1.1 Explicit Schemewhich boundary conditions are(5)1.2 Implicit Schemewhich boundary conditions areThen we have the following result.Theorem 1 When r < 1/32k(r =T/h6), explicit scheme is stable, and itstruncation error is Rnj=O(r + h2); implicit scheme is absolutely stable, and its truncation error is also Rnj = O(r + h2).2. Finite difference schemes for sixth order nonlinear parabolic equationsIn this section, we construct schemes of sixth order nonlinear parabolic equation as follows.When H(s) = s, uA(u) is A(u) ,we can achieve 2.1 Explicit Scheme2.2 Implicit Scheme(9)Theorem 2 If H(s)andA(u) are smooth enough and bounded, truncation errors of explicit scheme and implicit scheme are Rnj = O(t + h2).3. Finite difference parallel schemes for sixth order parabolic equationsIn this section , we construct finite difference parallel schemes for sixth order parabolic equations as follows. where A; is a integer; X = Dh,where D is a positive integer, and X < min. We will refer to points (xj,tn) as boundary points if i = 0 or N, or if n = 0. Similarly, we refer to them as interface points and n > 0. Otherwise, they are interior points.In advancing the solution from time level t = tn-1 to t = tn, one should first computes the values of U at the interface, which requires to simutaneously use the explicit scheme at the interface points. After the interface values have been computed, the implicit computation on the subdomains can be done in parallel.3.1 Finite difference parallel schemes for sixth order linear parabolic equationsWhen are interface points, we have the equationwhen (xj,tn) is interior point or boundary point, we have the equation3.2 Finite difference parallel schemes for sixth order nonlinear parabolicequationsWhen are interface points, we have the equationwhen (xj,tn) is interior point or boundary point, we have the equationIn this paper.we achieve explicit scheme and implicit scheme for sixth order diffusion equation, and improve one domain decomposition algorithm with good stability and high accuracy.