"On combining the techniques for convergence acceleration of iteration processes during the numerical solution of Navier-Stokes equations"
Vorozhtsov E.V. and Shapeev V.P.

The problem of accelerating the iteration process of the numerical solution of boundary value problems for partial differential equations by the method of collocations and least residuals (CLR) is considered. In the CLR method it is proposed to simultaneously apply three techniques for accelerating the iteration process: a preconditioner, a multigrid algorithm, and the Krylov method. A two-parameter preconditioner is studied. It is proposed to find the optimal values of its parameters by the numerical solution of a relatively computationally inexpensive problem of minimizing the condition number of the system of linear algebraic equations for the approximate problem. The use of the found preconditioner substantially speeds up the iteration process. The individual effect of each technique as well as the effect of their combined use on accelerating the entire iteration process acceleration are analyzed. The application of the algorithm based on the Krylov subspaces gives the most significant contribution. A simultaneous combined use of all the three techniques for accelerating the iteration process of solving the boundary value problems for the two-dimensional Navier-Stokes equations reduces the CPU time of their solution by a factor of up to 160 compared to the case when no such technique is applied. The proposed combination of the above techniques for accelerating the iteration processes may also be implemented in the framework of other numerical methods for solving the partial differential equations.

Keywords: preconditioning, Krylov subspaces, multigrid algorithms, Gauss-Seidel iterations, Navier-Stokes equations, the method of collocations and least residuals.

  • Vorozhtsov E.V. – Khristianovich Institute of Theoretical and Applied Mechanics, Siberian Branch of Russian Academy of Sciences; Institutskaya ulitsa 4/1, Novosibirsk, 630090, Russia; Dr. Sci., Professor, Leading Scientist, e-mail: vorozh@itam.nsc.ru
  • Shapeev V.P. – Khristianovich Institute of Theoretical and Applied Mechanics, Siberian Branch of Russian Academy of Sciences; Institutskaya ulitsa 4/1, Novosibirsk, 630090, Russia; Dr. Sci., Professor, Principal Scientist, e-mail: vshapeev@ngs.ru; Novosibirsk National Research University, Faculty of Mechanics and Mathematics; ulitsa Pirogova 2, Novosibirsk, 630090, Russia; Dr. Sci., Professor, e-mail: vshapeev@ngs.ru