Numerical solution of the Poisson equation in polar coordinates by the method of collocations and least residuals

E. V. Vorozhtsov ; V. P. Shapeev

Modelirovanie i analiz informacionnyh sistem, Tome 22 (2015) no. 5, pp. 648-664.

Résumé

A version of the method of collocations and least residuals is proposed for the numerical solution of the Poisson equation in polar coordinates on non-uniform grids. By introducing general curvilinear coordinates the original Poisson equation is reduced to the Beltrami equation. A uniform grid is used in curvilinear coordinates. The grid non-uniformity in the plane of the original polar coordinates is ensured with the aid of functions which control the grid stretching and entering the formulas of the passage from polar coordinates to the curvilinear ones. The method was verified on two test problems having exact analytic solutions. The examples of numerical computations show that if the radial coordinate axis origin lies outside the computational region, the proposed method has the second order of accuracy. If the computational region contains the singularity, the application of a non-uniform grid along the radial coordinate enables an increase in the numerical solution accuracy by factors from 1.7 to 5 in comparison with the uniform grid case at the same number of grid nodes.

Détail

Mots-clés : Poisson equation
Keywords: polar coordinates, the method of collocations and least residuals.

Bibliographie

[1] Peskin C., “Numerical analysis of blood flow in the heart”, J. Comput. Phys., 25 (1977), 220–252 | DOI | MR | Zbl

[2] Gibou F., Fedkiw R., Cheng L.-T., Kang M., “A second-order-accurate symmetric discretization of the Poisson equation on irregular domains”, J. Comput. Phys., 176 (2002), 205–227 | DOI | MR | Zbl

[3] Gibou F., Fedkiw R., “A fourth order accurate discretization for the laplace and heat equations on arbitrary domains, with applications to the stefan problem”, J. Comput. Phys., 202 (2005), 577–601 | DOI | MR | Zbl

[4] Johansen H., Colella P., “A cartesian grid embedded boundary method for poisson's equation on irregular domains”, J. Comput. Phys., 147 (1998), 60–85 | DOI | MR | Zbl

[5] Shapeev A. V., Shapeev V. P., “Difference schemes of increased order of accuracy for solving elliptical equations in domain with curvilinear boundary”, Computat. Math. and Math. Phys., 40:2 (2000), 213–221 | MR | Zbl

[6] Belyaev V. V., Shapeev V. P., “Metod kollokatsiy i naimenshih kvadratov na adaptivnyh setkah v oblasti s krivolineynoi granitsey”, Vychislitelnye tehnologii, 5:4 (2000), 12–21 | MR

[7] Popinet S., “Gerris: a tree-based adaptive solver for the incompressible euler equations in complex geometries”, J. Comput. Phys., 190 (2003), 572–600 | DOI | MR | Zbl

[8] Chen H., Min C., Gibou F., “A supra-convergent finite difference scheme for the Poisson and heat equations on irregular domains and non-graded adaptive Cartesian grids”, J. Sci. Computing, 31:1/2 (2007), 19–60 | DOI | MR | Zbl

[9] Swartztrauber P. N., Sweet R. A., “The direct solution of the discrete Poisson equation on a disc”, SIAM J. Numer. Anal., 10 (1973), 900–907 | DOI | MR

[10] Lai M.-C., Lin W.-W., Wang W., “A fast spectral/difference method without pole conditions for Poisson-type equations in cylindrical and spherical geometries”, IMA J. Numer. Anal., 22:4 (2002), 537–548 | DOI | MR | Zbl

[11] Lai M.-C., “A simple compact fourth-order Poisson solver on polar geometry”, J. Comput. Phys., 182 (2002), 337–345 | DOI | Zbl

[12] Vorozhtsov E. V., “Analytic and Numerical Investigation of Gas Flow in a Casing with Rotating Disc”, Vychisl. Metody Programm., 10:2 (2009), 162–176 (in Russian)

[13] Sleptsov A. G., “Kollokatsionno-setochnoe reshenie ellipticheskikh kraevykh zadach”, Modelirovanie v mekhanike, 5(22):2 (1991), 101–126 | MR | Zbl

[14] Verzicco R., Orlandi P., “A finite-difference scheme for three-dimensional incompressible flows in cylindrical coordinates”, J. Comput. Phys., 123:2 (1996), 402–414 | DOI | MR | Zbl

[15] Thompson J. F., Warsi Z. U. A., Mastin C. W., Numerical Grid Generation: Foundations and Applications, North-Holland, New York, 1985, 483 pp. | MR | Zbl

[16] Knupp P., Steinberg S., Fundamentals of Grid Generation, CRC Press, Boca Raton, 1994, 286 pp. | MR | Zbl

[17] Borges L., Daripa P., “A fast parallel algorithm for the Poisson equation on a disk”, J. Comput. Phys., 169 (2001), 151–192 | DOI | MR | Zbl

[18] Ray R. K., Kalita J. C., “A transformation-free HOC scheme for incompressible viscous flows on nonuniform polar grids”, Int. J. Numer. Methods in Fluids, 62 (2010), 683–708 | MR | Zbl

[19] Semin L. G., Sleptsov A. G., Shapeev V. P., “Metod kollokatsiy–naimenshih kvadratov dlia uravneniy Stoksa”, Vychislitelnye tehnologii, 1:2 (1996), 90–98 (in Russian) | MR | Zbl

[20] 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”, Computat. Math. and Math. Phys., 50:10 (2010), 1670–1681 | DOI | MR | Zbl

[21] Isaev V. I., Shapeev V. P., “High-order accurate collocations and least squares method for solving the Navier–Stokes equations”, Dokl. Math., 85 (2012), 71–74 | DOI | MR | Zbl

[22] 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”, CASC 2013, LNCS, 8136, eds. Gerdt V. P., Koepf W., Mayr E. W., Vorozhtsov E. V., Springer, Heidelberg, 2013, 381–392 | MR | Zbl

[23] Shapeev V. P., Vorozhtsov E. V., Isaev V. I., Idimeshev S. V., “The Method of Collocations and Least Residuals for Three-dimensional Navier–Stokes Equations”, Vychisl. Metody Programm., 14:3 (2013), 306–322 (in Russian)

[24] Shapeev V. P., Vorozhtsov E. V., “Application of Computer Algebra Systems to the Construction of the Collocations and Least Residuals Method for Solving the 3D Navier–Stokes Equations”, Modeling and Analysis of Information Systems, 21:5 (2014), 131–147 (in Russian)

[25] Shapeev V. P., Vorozhtsov E. V., “CAS application to the construction of the collocations and least residuals method for the solution of the Burgers and Korteweg–de Vries–Burgers equations”, CASC 2014, LNCS, 8660, eds. Gerdt V. P., Koepf W., Mayr E. W., Vorozhtsov E. V., Springer, Heidelberg, 2014, 432–446 | Zbl

[26] Isaev V. I., Shapeev V. P., Eremin S. A., “Issledovanie svoistv metoda kollokatsii i naimenshih kvadratov resheniya kraevyh zadach dlia uravneniya Puassona i uravneniy Navie–Stoksa”, Vychislitelnye tehnologii, 12:3 (2007), 53–70 (in Russian) | Zbl

[27] Ferziger J. H., Perić M., Computational Methods for Fluid Dynamics, 3rd Edition, Springer-Verlag, Berlin, 2002, 423 pp. | MR | Zbl

[28] Schwarz H. A., “Über einen Grenzübergang durch alternierendes Verfahren”, Vierteljahrs schrift der naturforschenden Gesellschaft in Zürich, 15 (1870), 272–286

[29] Golub G. H., Van Loan C. F., Matrix Computations, 3rd Edition, Johns Hopkins University Press, Baltimore, 1996, 694 pp. | MR | Zbl