A Divergence-Free Method for Solving the Incompressible Navier–Stokes Equations on Non-uniform Grids and Its Symbolic-Numeric Implementation

  • Evgenii V. VorozhtsovEmail author
  • Vasily P. Shapeev
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11661)


To increase the accuracy of computations by the method of collocations and least squares (CLS) a generalization of this method is proposed for the case of a non-uniform logically rectangular grid. The main work formulas of the CLS method on non-uniform grid, including the formulas implementing the prolongation operator on a non-uniform grid at the use of a multigrid complex are obtained with the aid of the computer algebra system (CAS) Mathematica. The proposed method has been applied for the numerical solution of two-dimensional stationary Navier–Stokes equations governing the laminar flows of viscous incompressible fluids. On a smooth test solution, the application of a non-uniform grid has enabled a 47-fold reduction of the solution error in comparison with the uniform grid case. At the solution of the problem involving singularities – the lid-driven cavity flow – the error of the solution obtained by the CLS method was reduced by the factors from 2.65 to 3.05 depending on the Reynolds number value.


Non-uniform grids Logically rectangular grids Navier–Stokes equations Krylov subspaces Multigrid Preconditioners Method of collocations and least squares 


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Authors and Affiliations

  1. 1.Khristianovich Institute of Theoretical and Applied Mechanics of the Siberian Branch of the Russian Academy of SciencesNovosibirskRussia
  2. 2.Novosibirsk National Research UniversityNovosibirskRussia

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