Operator-splitting Method And Its Application For Soluting Parabolic Equations

In this paper, I introduced one kind of parallel interactive methods for solving parabolic equations: Operator-splitting method. And to use the instance that One-dimensional diffusion equation forms difference scheme that can be used for calculating with truncation-error of high-precision and absolute stability to introduce the specific application of Operator-splitting method in the solution of parabolic equation.We also proved the stability and analyzed the Convergence.The difference scheme of solving parabolic equations is one classical problem of solving partial differential equations. The implicit scheme of One-dimensional parabolic equations is absolutely stabilized, and the amount of calculation by method of elimination is not large. For higher-dimension parabolic equations, though the explicit scheme is simply to calculate, the condition for absolute stability is too harsh. And then though the implicit scheme is absolutely stabilized, the calculating needs to solve system of linear algebraic equations with a higher order. So we have constructed one absolutely stabilized difference scheme which can decompose each floor to several One-dimensional implicit scheme. For boundary value problem of two-dimensional equation of heat conduction,Peaceman and Rachford proposed a absolutely stabilized scheme with second order accuracy in time horizon and in space horizon in 1955,which overcame the harsh control of stability of classical explicit scheme, and solved the problem of solving scalar Pentadiagonal Linear System of equations and it was fit for parallel computing.With the development of alternating direction iterative scheme,some former Soviet Union developped another kind algorithm of solving higher-dimensional problems of Mathematical Physics:Operator-splitting method. The basic thought was to split one method into several simple ones, and the complex problem of Mathematical Physics was splitted to some simple ones which is easily resoved.Operator-splitting method for solving problem of Mathematical Physics can be classified to stationary Problems of Mathematical Physics and Nontationary Problems of Mathematical Physics by whether the problem have relation to time.Operator-splitting method for solving stationary Problems of Mathematical Physics,such as linear algebraic equation:A is amatrix,φand f is vectors To set up A > 0,soλ(A) > 0. The formation of iteration just including one parameter listed as:B is positive definite matrix,A > 0.A was splitted into:A =(?) A_i,A_i >0,j= 0,1,2,…,m,m > 2.To set upÏ„_j =Ï„> 0 and B of formation ofiteration (0-4) as: B = (?) (I +σA_i),m≥2.Many Stationary Problems of Mathematical Physics can be regarded as the progressive state of Nontationary Problems of Mathematical Physics while T→∞.Consider the following evolution problem of mathematical physics:Where A > 0,and the solutionφas well as the functions f and g are assumed sufficiently smooth and certain boundary conditions. Replace A^j by the approximating difference operaror denoted by A^j,and we also denote as A. Then we can obtain the system of linear algebraic equations. The Stabilization Method:f^j = f (t_tj+(?)) The difference scheme is absolutely stable ang approximates the problem with second-order accuracy inÏ„.The Predictor-Corrector Method: First, using the first-order approximation scheme with a comparatively large "degree" of stability.After that we write the second-order scheme (corrector) on the interval.The difference scheme is also absolutely stable and approximates the problem with second-order accuracy inÏ„.The Component-by-ComponentSplitting-Up Method:A = A(t),A >0,A =(?)A_i,A_i > 0(i = 1,2,…,m),m≥2.While t∈[t_j,t_j+1],approximate these matrices in the form:For General Approach to Component- by-ComponentSplitting-Up Method, we use the two-cycle symmetrical method:Two-cycle symmetrical method approximates are absolutely stable and approximates the problem with an accuracy up toÏ„².What is more, it can be simply popularized to solving the problem of quasilinear problem.Consider One-dimensional parabolic initial boundary value problem:One class of Saul'yel non-symmetrical format of Solving (0-1): Among this format,r = (?),the Saul'yel non-symmetrical format is absolutely stable.But the format (0-0-2)and (0-0-3) are with truncation error up to O(Δt/Δx +Δt +Δx²) in (i, k + 1/2),But the using of a separate Saul'yel non-symmetrical format, truncation error includ O(Δt/Δx +Δt +Δx²).WhileΔtandΔx tend to 0 simultaneously, the error term for the infinite can not guarantee to be a small amount and so was generally less accurate. For this reason,under normal circumstances we use them alternately at different times layers,or we use them combined at the same time layer to improve the calculation accuracy,and also we constructed a variety of parallel iterative difference scheme.(1):AGE scheme : unique combination of non-symmetric at point at(i,k + (?)) and (i + 1,k + (?)):The phase error of group Explicit scheme (0-0-4)can be O(Δt +Δx),which clear O(Δt/Δx),so it improve the phase error entitativly than the using of single format.Using group explicit scheme along the x increase or decrease of discrete points on the (i, k+1)(i = 1,2,…,m - 1) of continuous point format is group explicit method.While m - 1 = 2n - 1,we still calculate one point near the boundary using of non-symmetric form separately, and where there is a single point about the location , we construction GEL methods or GER methods separately. Each GE unit can be parellel computed. So that operator A can be splitted into operators for easily parallel computing. But algorithm has stability conditions, in order to improve the stability , we use GER and GEL alternately at different time layers, and we named this method Alternating Group Explicit method, AGE method for short.It is absolutely stable.(2): Improved Alternating Group Explicit method at the boundary value of the algorithm Department: In AGE method, we still calculate one point near the boundary using of Saul'yel non-symmetric form separately while m -1 = 2n - 1,and we know that the using of Saul'yel non-symmetric form separtely enlarger initial truncation error , Thus at each of the boundary value point we use classical implicit scheme or classical explicit scheme. At the border we could as much as possible to maintain a smaller truncation error to enhance the calculation accuracy.Therefore, the general algorithm for the following improvement AGE:In close proximity to the left of the value of two points, we use the format as follows:In close proximity to the right value points, there are two designs, if there is one point, we use classical implicit scheme or classical explicit scheme; if two points,we design as follows:Because of the truncation error of Crank-Nicolson scheme is O(Δt² +Δx²),we use Crank-Nicolson scheme at around near the point boundary value.At the border we could as much as possible to maintain a smaller truncation error, so as to enhance the calculation accuracy.In close proximity to the left of the value of two points, we use the format as followsIn close proximity to the right value points, there are two designs, if there is two point, we use scheme as follows: If there is there points,we use schemes as follows:(3): ASE-I method Based on the Alternating Group Explicit scheme, we can design a more general sub-significant - Implicit (ASE-I) method. It break through the difficulties calculation of implicit scheme ,it is both parallel and stability as well as a better truncation error.What's more, it is suitable to apply on MIMD computers. Implicit structure of paragraph was designed as follows: for m - 1 = KL,K is the paragraph number,L is the numbers of points of each implicit structure of paragraph,For a certain i₀,we consider the calculation of points marked (i₀ + i,k + 1)(i = 1,2,…, L). At two 1 "endpoints" (i₀ + 1,k + 1) and (i₀ + L,k + 1),we use asymmetric Saul'yel format (0-2) and (0-3) respectively , while in the "interior point" we use classical implicit scheme.Use this paragraph,we design a general ASE-I method.When m - 1 = (?) + L,L > 3,N is an odd number, we can give one kinds of the ASE-I method that can be extended to solve the two -dimensional sub-problem of the treatment algorithm.(4):Another ASE-I algorithm, For m - 1= K L,K is the paragraph number, L is the numbers of points of each implicit structure of paragraph, For a certain i₀,we consider the calculation of points marked (i₀ + i,k + 1)(i = 1, 2,…,L).At two "endpoints" (i₀ + 1,k + 1) and (i₀ + L,k + 1),we use asymmetric Saul'yel format (0-0-2)and(0-0-3) respectively, while in the interior point we use classical implicit scheme. We named this implicit structure of paragraph as Saul'yel implicit paragraph; We designed another paragraph for implicit, for a certain i₀(i₀ + i,k +1)(i =1, 2,…,L). In the two "endpoints"(i₀ + 1,k + 1) and (i₀ + L,k + 1) ,we use the classical explicit scheme, and in the interior point we use the classical scheme, it was another implicit. We have put this paragraph referred to as classical explicit - implicit paragraph. We can design a general ASE-I method using these two implicit paragraphs.(5)Alternating sub-C-N algorithm: The approximation of problem (0-0-1)of the Crank-Nicolson scheme with second order accuracy.It is necessary for solving tridiagonal equations. Based on exploration of the Crank-Nicolson difference scheme for parallel computing, by introducting other kinds of asymmetric Saul'yel scheme, we re-constructed C-N scheme, so that the solution of differential equations can be splitted into several independent parallel solution of the smaller scale of the equationsThese scheme known as the I₁,I₂,I'₁,I'₂ of which,I_t,I₂ are semi-implicit,I'₁,I'₂ are implicit.Alternating Crank-Nicolson method of sub-algorithm is to use the four non-symmetrical format to split traditional C-N format into independent implicit paragraph that can be calculated separatly, so that the corresponding operator G is also splitted into parallel computing non-negative solution that is relatively easy operator. This we get the sub-C-N method,which is absolute stability.In this paper, we gave each parallel iterative differential format a detailed algorithm design, and its mathematical description can be unified math into the following matrix form, we also give proof of the stability. b₁,b₂ is the vector of the boundary value.Corresponding to different algorithms have different matrix form, the paper described them in detail.Finally ,I gave a brief introduction of the multi-dimensional, fourth-level format of the parallel iteration and some theoretical results,and describe the major work of future research.And at last,I have gave numerical example of several typical of parallel difference. I also analyzed the experimental data with the theoretical analysis ,which is consistent with each other well.