Информационная система "Конференции"

Тезисы докладов

Численное решение дифференциальных и интегральных уравнений

The choice of smoothers in multigrid method (MGM) is discussed. Multigrid method with different smoothers for solving linear algebraic equation systems with strongly nonsymmetric matrix obtained after difference approximation of the convection-diffusion equation with dominant convection is proposed. Difficulties will be experienced with standard numerical methods for this class of problems. The following convection-diffusion problem is considered: $-frac 1{Pe}Delta u+frac 12left( {Pu_x+left( {Pu}right) _x+Qu_y+left( {Qu}% right) _y}right) =F$ Specially created triangular iterative method has been used as the smoother of multigrid method. $Bfrac{y_{n+1}-y_n}tau +Ay_n=F$ where the operator $B$ is chosen in the special form: 1)$B=E+2tau K_ell $ or $% B=E+2tau K_u$ 2)$B=alpha _iE+2K_ell $ or $B=alpha _iE+2K_u$ 3)$% B=alpha E+2K_ell $ or $B=alpha E+2K_u$ where $alpha _i,alpha ,tau >0$ are parameters, $K_ell ,K_u$ are lower and upper triangular parts of the skew-symmetric matrixes $A_1=left( {A-A^{*}}right) /2$, $A=A_0+A_1$, $A_0$ is the symmetric part of the matrix $A$. This choice of smoothers allows us to use multigrid method without restriction on the equation coefficients and grid size for the solution of arising linear algebraic equation systems and doesn't require diagonal dominant from the matrix. Some theoretical and numerical results are presented. The dependence of convergence rate of these methods on the coefficient of nonsymmetry $k={{% Pe*h}% mathord{left/ {vphantom {{Pe*h} 2}} right. kern-nulldelimiterspace}2}$ are researched.

Примечание. Тезисы докладов публикуются в авторской редакции

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