The Finite Difference Parallel Algorithms For Fourth Order Parabolic Equations

In this paper, we consider fourth order parabolic equations with initial condition and boundary conditionThese equations are the linear part of many other equations. Such as K-S equation, fourth order nonlinear Cahn-Hilliard equation and so on. By now, the paper existed mostly consider second order parabolic equations, As to construction of parallel finite difference schemes of fourth order or high order parabolic equations is seldom. Valid parallel finite difference methods of high order equations mentioned above are more necessary, in particular, how to do with the high order term. We hope the result of this paper makes an essential contribution in the parallel algorithm of the equations mentioned above. This is my purpose of my paper.First, we give a group of Saul'yev asymmetric difference schemes and C-N scheme for the fourth order equations. Using the schemes, parallel alternating segment difference schemes are constructed. The matrix formsWe analyze the truncation errors by the equivalent segment schemes of three-level. On the same time level, the schemes are used symmetrically in the space direction, respectively, the signs of the terms with the parameter h are opposite. Thus the effect of terms with h in the errors can be canceled. This results in high accuracy for schemes. Theorem 1. Assume n is an even number, for any positive number r, alternating segment schemes (0-4)-(0-5) are absolutely stable.In the following part, we give general alternating difference schemes.If for some j₀, j₁ letξ_j0 = 0 andξ_j1 = 1, then some concrete alternating difference schemes with intrinsic parallelism are obtained, and the global computation domain is decoupled into some sub-segments. These general alternating difference schemes with intrinsic parallelism contain many parallel algorithms, literature[26] and the algorithms in the second chapter are the special cases of these schemes.Theorem 2. Assume 0≤ξ_j≤1,j = 0,1,2,…, J, alternating difference schemes (0-6)-(0-7) are absolutely stable, and there hold(1) for (2) when n is an even, (3) when n is an odd, where when 2(Ï„₀+Ï„₁ +…+Ï„_k) = T, the time step lengthes are variable, we consider the parallel difference scheme with variable time step lengthes.Theorem 3. Assume 0≤ξ_j≤1.j = 0,1,2,…, J, and letÏ„₀≤τ₁≤…≤τ_K, for any given constant C₀≥1,Ï„_K≤C₀Ï„₀, the parallel difference schemes with variable time step lengthes (0-8)-(0-9) are absolutely stable, and there holds...