Victor Timofeevich Zhukov

Parallel Multigrid Method for Solving Elliptic Equations

The proposed algorithm represents an efficient parallel implementation of the Fedorenko multigrid method and is intended for solving three-dimensional elliptic equations. Scalability is provided by the use of the Chebyshev iterations for solution of the coarsest grid equations and for construction of the smoothing procedures. The calculation results are given: they confirm the efficiency of the algorithm and scalability of the parallel code.

Multigrid method for elliptic equations with anisotropic discontinuous coefficients

For difference elliptic equations, an algorithm based on Fedorenko’s multigrid method is constructed. The algorithm is intended for solving three-dimensional boundary value problems for equations with anisotropic discontinuous coefficients on parallel computers. Numerical results confirming the performance and parallel efficiency of the multigrid algorithm are presented. These qualities are ensured by using, as a multigrid triad, the standard Chebyshev iteration for coarsest grid equations, Chebyshev-type smoothing explicit iterative procedures, and intergrid transfer operators in problem-dependent form.

A computational technology for constructing the optimal shape of a power plant blade assembly taking into account structural constraints

A computational technology for constructing the optimal shape of a power plant three-dimensional blade assembly is presented. The shape of the blade assembly is optimized to improve the power characteristics of the blade assembly taking into account structural constraints. The computational technology is a unified chain of algorithms beginning with constructing a CAD model of the assembly, generation of a computational grid, simulation of the flow around the assembly using OpenFoam, and finally the animated stereo visualization of the power plant operation. The visual representation of the results in all phases is required for debugging, verification, and control. The proposed technology provides a basis for finding the optimal shape of the blade assembly by varying its key geometric parameters. Practical results of the simulation are discussed.

Multigrid method for anisotropic diffusion equations based on adaptive Chebyshev smoothers

We propose an efficient multigrid algorithm for solving anisotropic elliptic difference equations. The algorithm is based on using Chebyshev’s explicit iterations at smoothing stages and in solving coarse-grid equations. We have developed a procedure for adapting smoothers to anisotropy and present examples, which show that adaptation improves the efficiency of the multigrid method and scalability of the parallel code.

On the solution of evolution equations based on multigrid and explicit iterative methods

Two schemes for solving initial–boundary value problems for three-dimensional parabolic equations are studied. One is implicit and is solved using the multigrid method, while the other is explicit iterative and is based on optimal properties of the Chebyshev polynomials. In the explicit iterative scheme, the number of iteration steps and the iteration parameters are chosen as based on the approximation and stability conditions, rather than on the optimization of iteration convergence to the solution of the implicit scheme. The features of the multigrid scheme include the implementation of the intergrid transfer operators for the case of discontinuous coefficients in the equation and the adaptation of the smoothing procedure to the spectrum of the difference operators. The results produced by these schemes as applied to model problems with anisotropic discontinuous coefficients are compared.