Application of Computer Algebra Systems to the Construction of the Collocations and Least Residuals Method for Solving the 3D Navier–Stokes Equations

Abstract

The method of collocations and least residuals (CLR), which was proposed previously for the numerical solution of two-dimensional Navier–Stokes equations governing the stationary flows of a viscous incompressible fluid, is extended here for the three-dimensional case. The solution is sought in the implemented version of the method in the form of an expansion in the basis solenoidal functions. At all stages of the CLR method construction, a computer algebra system (CAS) is applied for the derivation and verification of the formulas of the method and for their translation into arithmetic operators of the Fortran language. For accelerating the convergence of iterations a sufficiently universal algorithm is proposed, which is simple in its implementation and is based on the use of the Krylov’s subspaces. The obtained computational formulas of the CLR method were verified on the exact analytic solution of a test problem. Comparisons with the published numerical results of solving the benchmark problem of the 3D driven cubic cavity flow show that the accuracy of the results obtained by the CLR method corresponds to the known high-accuracy solutions.

Keywords

• трехмерные уравнения навье–стокса
• метод коллокаций и наименьших невязок
• компьютерная алгебра
• течение в кубической каверне
• подпространства крылова