A Finite Difference Parallel Method For Solving Parabolic Equations

In this paper,a new parallel format is constructed for parabolic equa-tions. Introducing the construction of the iterative sequence for the JGS it-erative method of elliptic equations,and then designing a new parallel forma for parabolic equations,giving to prove the stability and to analyze truncation error.From the physical point of view,the solution of non-resident posed prob-lems u(x,y,t) should tend to the solution of resident posed problems u(x,y) with t increasing,and no matter what's the initial value.To promote the idea,we construct the parallel difference scheme of parabolic equations by the paral-lel iterative solution of elliptic equations. And re-constructing a new format based on in this format.Considering the linear equation Au= f, (0.0.1)Let W={1,2,…, n} be an index set.For a given positive integer p,we define a simple partition: W= R1UR2U…Rp, (0.0.2) with Ri={ni-1+1, ni-1+2,…, ni} and ni(i=1,2,…,p) satisfying 0= n0< n1< n2<…np= n. With this index set partition,we write A, u and f in the block forms We write A and submatrix Aii (i=1,2,…,p)into the matrix sums D-1A=I—L—U,(Di-1)Aii=Ii—Li—Ui (0.0.3) where,respectively,D and Di are the diagonal matrices of A and Aii,I and Ii are two identity matrices,L and U,as well as Li和Ui are strictly lower and strictly upper triangular matrices.We then write L=B+N,U=C+M (0.0.4)其中B=diag (L1,L2,…Lp),C=(U1,U2,…Up), by using this notation,the JGS iterates uk can be expressed by uk+1=(I—B)-1(U+N)uk+(I—B)-1D-1f (0.0.5)Considering the one-dimensional elliptic equations Discreting equations Au=f,A is a coefficient matrix u is a vector.Dividing equally the interval[0,1)to be N copies,let h=l/N,then where,ui is a approximation where u(x) is on the point xi,fi=f(xi),plusing boundary conditions into the following equations Then noting And then JGS iterative is uk+1=(I—B)-1(U+N)uk+(I—B)-1D-1f, (0.0.8) then uk+1-uk=Buk+1—uk+(U+N)uk+D-1f, (0.0.9) Where,I,B,U,N,D-1 are(0.0.7)the partitions of the matrix C.According to the method of the JGS iterative sequence construction,here giving the elliptic equations (0.0.6) corresponding to the difference scheme of the parabolic equationNow consider the difference approximation of the boundary value problem (0.0.10).In order to space step h and the time step tau will be set for solution area is divided into mesh, (xi,tk)isthenode,where,xi=ih, i= 0,1,…,N,h=1/N,tk=kτ, k=0,1,….The point(xi,tk)abbreviated as (i,k).uik=u(i,k)means that the solution of the problem (0.0.12)u(x,t)is the value in the discrete points(i,k).For a given positive integer p,let L=(N-1)/p. So the entire range was di-vided into p sub-interval(or p sections),each has L points.According to JGS iteration scheme, we can see the construction of the corresponding difference scheme,as follows:For some i0,considering the calculation of the various points (i0+i,k+ 1) (i=1,2,…, L), knowing that the two endpoints (i(0)+1, k+1) and (i(0)+L,k+1) respectively use the classical explicit format (0.0.11) and the classical implicit scheme (0.0.12),and use the Saul'yev non-symmetrical format (0.0.13) in the inner point (i(0)+i, k+1) (i= 2,3, cdots, L—1). obtaining the following sub-paragraph format ,that isMaking use of JGS format, designing the following alternative meth-ods:let N—1 = Lp, N, L is a positive integer.In the same odd time step to calculate the points is divided into p sections.And from the left to the right,as follows: The classical explicit on the left border point—Saul'yev (0.0.13) non-symmetric form in the interior-point—The classical implicit point on the right border point in the next time level (even layer),remaining p sections, and from the left to the right,as follows:The classical implicit point on the left border point—Saul'yev (0.0.14) non-symmetric form in the interior-point—The classical ex-plicit on the right border point and The growth matrix of the new alternate algorithm (0.0.15) is In order to discuss its stability, we need to introduce the following Kellogg Lemma (see [2])Lemma 0.1 Setp> 0,if C is non-negative real matrix,that is, to meet the C+CT is non-negative definite,then there (pI+C)-1 exists,andLemma 0.2 Under the conditions of Lemma (0.1)Theorem 0.1 That the formula (0.0.15) had described the method is absolutely stable.Prove:We known the growth matrix of the formula (0.0.15) is T = (I+ rG2)(-1)(I-rG1)(I+rG1)(-1) (I-rG2), clearly, GL(i),(?)L(i),i= 1,2,…,p.are both non-negative definite matrices,thenG1+G1T,G2+G2T are also non-negative definite matrices, according to Kellogg Lemma,it's easy to get the following estimates therefore, we have the statement holds (0.0.15) had described the method that is absolutely stable. Theorem 0.2 The truncation error of the formula (0.0.15) is O(τ+h2)Prove: We known the format structure of the formula (0.0.15) is that in the same odd time level,the left boundary points use the classical ex-plicit method,the interior points use the Saul'yev (0.0.13) non-symmetric method,the right boundary points use the classical implicit method;in the even time next layer,the left boundary points use the classical implicit method,the interior points use the Saul'yev (0.0.14) non-symmetric method,the right boundary points use the classical explicit method.that is, the left boundary points are explicit and implicit methods alternating,the interior points are two Saul'yev non-symmetric methods alternating,the right points are implicit and explicit methods alternating.Using the Taylor expansion,in the left points,the truncation error is the in the interior points, the truncation error of the right points is Comprehensive availably,the truncation error of the formula (0.0.15) is O(τ+h2). comparing to the JGS,it's easy to see the method is better.