Abstract
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.
The research was carried out within the framework of the Program of Fundamental Scientific Research of the state academies of sciences in 2013–2020 (projects Nos. AAAA-A17-117030610134-9 and AAAA-A17-117030610136-3).
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Binzubair, H.: Efficient multigrid methods based on improved coarse grid correction techniques. Thesis. Delft Univ. of Technology, Delft, The Netherlands (2009)
Botella, O., Peyret, R.: Benchmark spectral results on the lid-driven cavity flow. Comput. Fluids 27, 421–433 (1998)
Briggs, W.L., Henson, V.E., McCormick, S.F.: A Multigrid Tutorial, 2nd edn. SIAM, Philadelphia (2000)
Fedorenko, R.P.: The speed of convergence of one iterative process. USSR Comput. Math. Math. Phys. 4(3), 227–235 (1964)
Gartling, D.K.: A test problem for outflow boundary conditions - flow over a backward-facing step. Int. J. Numer. Methods Fluids 11, 953–967 (1990)
Isaev, V.I., Shapeev, V.P.: High-accuracy versions of the collocations and least squares method for the numerical solution of the Navier-Stokes equations. Comput. Math. Math. Phys. 50, 1670–1681 (2010)
Knupp, P., Steinberg, S.: Fundamentals of Grid Generation. CRC Press, Boca Raton (1994)
Krylov, A.N.: On the numerical solution of the equation, which determines in technological questions the frequencies of small oscillations of material systems. Izv. AN SSSR, Otd. matem. i estestv. nauk 4, 491–539 (1931). (in Russian)
Kudryavtseva, I.V., Rykov, S.A., Rykov, S.V., Skobov, E.D.: Optimization Methods in the Examples in the MathCAD 15 Package. Part I, NIU ITMO, St. Petersburg (2014). (in Russian)
Shapeev, A.V., Lin, P.: An asymptotic fitting finite element method with exponential mesh refinement for accurate computation of corner eddies in viscous flows. SIAM J. Sci. Comput. 31, 1874–1900 (2009)
Shapeev, V.: Collocation and least residuals method and its applications. EPJ Web of Conferences 108, 01009. https://doi.org/10.1051/epjconf/201610801009
Shapeev, V.P., Vorozhtsov, E.V.: CAS application to the construction of the collocations and least residuals method for the solution of 3D Navier–Stokes equations. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2013. LNCS, vol. 8136, pp. 381–392. Springer, Cham (2013). https://doi.org/10.1007/978-3-319-02297-0_31
Shapeev, V.P., Vorozhtsov, E.V.: Symbolic-numeric implementation of the method of collocations and least squares for 3D Navier–Stokes equations. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2012. LNCS, vol. 7442, pp. 321–333. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-32973-9_27
Shapeev, V.P., Vorozhtsov, E.V.: Symbolic-numerical optimization and realization of the method of collocations and least residuals for solving the Navier–Stokes equations. In: Gerdt, V.P., Koepf, W., Seiler, W.M., Vorozhtsov, E.V. (eds.) CASC 2016. LNCS, vol. 9890, pp. 473–488. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-45641-6_30
Shapeev, V.P., Vorozhtsov, E.V.: The method of collocations and least residuals combining the integral form of collocation equations and the matching differential relations at the solution of PDEs. In: Gerdt, V.P., Koepf, W., Seiler, W.M., Vorozhtsov, E.V. (eds.) CASC 2017. LNCS, vol. 10490, pp. 346–361. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-66320-3_25
Sleptsov, A.G.: Collocation-grid solution of elliptic boundary-value problems. Modelirovanie v mekhanike 5(22), 101–126 (1991). (in Russian)
Thompson, J.F., Warsi, Z.U.A., Mastin, C.W.: Numerical Grid Generation - Foundations and Applications. Elsevier Science Publishing Co., New York (1985)
Wolfram, S.: The Mathematica Book, 5th edn. Wolfram Media, Inc., Champaign (2003)
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Vorozhtsov, E.V., Shapeev, V.P. (2019). A Divergence-Free Method for Solving the Incompressible Navier–Stokes Equations on Non-uniform Grids and Its Symbolic-Numeric Implementation. In: England, M., Koepf, W., Sadykov, T., Seiler, W., Vorozhtsov, E. (eds) Computer Algebra in Scientific Computing. CASC 2019. Lecture Notes in Computer Science(), vol 11661. Springer, Cham. https://doi.org/10.1007/978-3-030-26831-2_28
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