Информационная система "Конференции"

Тезисы докладов

Вычислительная алгебра

When parabolic equations are solve numerically, the fractional step methods and methods of weak approximation are rarely used because derivative along the normal from temperature function on a border does not correspond to direction of heat flow. Hence, problems of taking into account of boundary conditions and constructing absolutely stable implicit finite difference scheme on each time interval are arisen. Nowadays to solve systems of linear algebraic equations with sparse special high order matrix the iterative methods are applied. Its general advantages are high practice economy and broad abilities to construct self-adapting algorithms for solving different boundary problems. In this paper an iterative algorithm applied to 3D anisotropic time dependent heat transfer boundary problem is considered. The rate of convergence of this one is investigate, the comparative analysis with explicit Buleev method EXIF-53 is carried on. Let's consider a boundary problem begin{center} begin{equation} rho cdot c_pcdot frac{partial ,T}{partial ,t}=k_{11}cdot frac{% partial ^{,2},T}{partial ,x^2}+2cdot k_{12}cdot frac{partial ^{,2},T}{partial ,xpartial ,y}+2cdot k_{13}cdot frac{partial ^{,2},T}{partial ,xpartial ,z}+k_{22}cdot frac{partial ^{,2},T}{% partial ,y^2}+2cdot k_{23}cdot frac{partial ^{,2},T}{partial ,ypartial ,z}+k_{33}cdot frac{partial ^{,2},T}{partial ,z^2}+f,, end{equation} [ left( k_{11}cdot frac{partial ,T}{partial ,x}+k_{12}cdot frac{% partial ,T}{partial ,y}+k_{13}cdot frac{partial ,T}{partial ,z}% right) _{x=0}=gamma _1left( T-Vright) , ] [ left( k_{11}cdot frac{partial ,T}{partial ,x}+k_{12}cdot frac{% partial ,T}{partial ,y}+k_{13}cdot frac{partial ,T}{partial ,z}% right) _{x=a}=gamma _2left( V-Tright) , ] [ left( k_{12}cdot frac{partial ,T}{partial ,x}+k_{22}cdot frac{% partial ,T}{partial ,y}+k_{23}cdot frac{partial ,T}{partial ,z}% right) _{y=0}=gamma _3left( T-Vright) , ] [ left( k_{12}cdot frac{partial ,T}{partial ,x}+k_{22}cdot frac{% partial ,T}{partial ,y}+k_{23}cdot frac{partial ,T}{partial ,z}% right) _{y=b}=gamma _4left( V-Tright) , ] [ left( k_{13}cdot frac{partial ,T}{partial ,x}+k_{23}cdot frac{% partial ,T}{partial ,y}+k_{33}cdot frac{partial ,T}{partial ,z}% right) _{z=0}=gamma _5left( T-Vright) , ] [ left( k_{13}cdot frac{partial ,T}{partial ,x}+k_{23}cdot frac{% partial ,T}{partial ,y}+k_{33}cdot frac{partial ,T}{partial ,z}% right) _{z=c}=gamma _6left( V-Tright) , ] end{center} begin{equation} T,|_{t=0}=u_0(x,y,z), end{equation} where $rho$ - density, $c_p$ - heat capacity, $k_{i,j}$, $i,j=overline{1,3}$ - coefficients of thermal conductvity tensor,$T_e$ - external temperature, $gamma _i, i=overline{1,6}$ - thermal output coefficients. A finite difference scheme for (1) was written on 27-point mesh. The approximation of boundary conditions (2) were made as follows begin{eqnarray} U_{1,n,k} &=&varphi _{2,n,k}U_{2,n,k}+widetilde{varphi }% _{1,n+1,k}U_{1,n+1,k}+phi _{1,n,k+1}U_{1,n,k+1}+xi _{2,n,k}; nonumber U_{i,1,k} &=&varphi _{i,2,k}U_{i,2,k}+widetilde{varphi }% _{i+1,1,k}U_{i+1,1,k}+phi _{i,1,k+1}U_{i,1,k+1}+xi _{i,2,k}; nonumber U_{N,n,k} &=&varphi _{N-1,n,k}U_{N-1,n,k}+widetilde{varphi }% _{N,n-1,k}U_{N,n-1,k}+phi _{N,n,k-1}U_{N,n,k-1}+xi _{N-1,n,k}; nonumber U_{i,N,k} &=&varphi _{i,N-1,k}U_{i,N-1,k}+widetilde{varphi }% _{i-1,N,k}U_{i-1,N,k}+phi _{i,N,k-1}U_{i,N,k-1}+xi _{i,N-1,k}; nonumber U_{i,n,1} &=&varphi _{i,n,2}U_{i,n,2}+widetilde{varphi }% _{i+1,n,1}U_{i+1,n,1}+phi _{i,n+1,1}U_{i,n+1,1}+xi _{i,n,2}; nonumber U_{i,n,N} &=&varphi _{i,n,N-1}U_{i,n,N-1}+widetilde{varphi }% _{i-1,n,N}U_{i-1,n,N}+phi _{i,n-1,N}U_{i,n-1,N}+xi _{i,n,N-1.} nonumber end{eqnarray} To solve this linear system author proposed a generalized iterative algorithm, which have very high rate of convergance (RC) varying from $Oleft( N^{-6/7}right) ,$for isotropic materials till $Oleft( 5/8ln ^{-1}(N+1)right) $for especially anisotropic domains. Such high RC have incomplete factorization methods only, for example, explicit Buleev method EXIF-53. The practical computations approved high performance of proposed iterative process for family of time dependent heat transfer boundary problems in especially anisotropic domains. It kept a construct manner of matrix incomplete factorization method and didn't require any information about differential operator spectrum.

Примечание. Тезисы докладов публикуются в авторской редакции

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