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Part of the book series: Springer Series in Computational Physics ((SCIENTCOMP))

Abstract

Until recently, numerical methods for solving fluid-flow problems have been dominated by finite-difference approximations. These methods are powerful and play a major role in problem solutions. In this chapter we attempt to present the fundamental advances and insight into these methods. We begin by discussing the concept of discrete pointwise approximation of functions.

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References

  • Adam, Y. Comput. Math. Appl. 1, 393–406 (1975).

    Article  MATH  Google Scholar 

  • Adam, Y. J. Comput. Phys. 24, 10–22 (1977).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Allen, J. S., and Cheng, S. I. Phys. Fluids 19, 37–52 (1970).

    Article  ADS  Google Scholar 

  • Anucina, N. N., Soy. Math. 5, 60–64 (1964).

    Google Scholar 

  • Beam, R. M., and Warming, R. F., J. Comput. Phys. 22, 87–110 (1976).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Beam, R. M., and Warming, R. F., AIM J. 16, 393–402 (1978).

    MATH  Google Scholar 

  • Berger, A. E., Solomon, J. M., Ciment, M., Leventhal, S. H., and Weinberg, B. C. Math. Comput. 35, 695–731 (1980).

    Article  MathSciNet  Google Scholar 

  • Berezin, Yu. A., Kovenja, V. M., and Yanenko, N. N. Numerical Mathematics in Continuum Mechanics (in Russian), Vol. 3, pp. 3–18, A. N. SSSR Siberian Computer Center, Novosibirsk (1972).

    Google Scholar 

  • Berezin, Yu. A., Kovenja, V. M., and Yanenko, N. N., Lecture Notes in Physics, Vol. 35, pp. 85–90, Springer-Verlag, New York (1975a).

    Google Scholar 

  • Berezin, Yu. A., Kovenja, V. M., and Yaneko, N. N. Comput. Fluids 3, 271–281 (1975b).

    Article  MATH  Google Scholar 

  • Bontoux, P., Forestier, B., and Roux, B. J. Mec. Appl. 2, 291–316 (1978).

    MathSciNet  Google Scholar 

  • Brailovskaya, I. Yu. Soy. Phys.—Dokl. 10, 107–110 (1965).

    ADS  Google Scholar 

  • Bramble, J. H., and Hubbard, B. E. Numer. Math. 4, 312–327 (1962).

    Article  MathSciNet  Google Scholar 

  • Briley, W. R., and McDonald, H. An Implicit Numerical Method for the Multidimensional Compressible Navier—Stokes Equations, United Aircraft Research Laboratory Rep. M911363–6 (Nov. 1973).

    Google Scholar 

  • Briley, W. R., and McDonald, H. Lecture Notes in Physics, Vol. 35, pp. 105–110, Springer—Verlag, New York (1975).

    Google Scholar 

  • Briley, W. R., and McDonald, H. J. Comput. Phys. 22, 372–397 (1977).

    Article  MathSciNet  ADS  Google Scholar 

  • Briley, W. R., and McDonald, H. J. Comput. Phys. 34, 54–73 (1980).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Cheng, S. I. A Critical Review of Numerical Solution of Navier—Stokes Equations. In Progress in Numerical Fluid Dynamics, Lecture Notes in Physics, Vol. 41, pp. 78–225 Springer—Verlag, New York (1975).

    Google Scholar 

  • Cheng, S. I., and Shubin, G. J. Comput. Phys. 28, 315–326 (1978).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Ciment, M., Leventhal, S. H., and Weinberg, B. C., J. Comput. Phys. 28, 135–166 (1978).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Collatz, L. The Numerical Treatment of Differential Equations, Springer—Verlag, New York (1966).

    Google Scholar 

  • Collatz, L. Hermitian Methods for Initial-Value Problems in Partial Differential Equations. In Topics in Numerical Analysis, Proceedings of the Royal Irish Academy Conference on Numerical Analysis, J. J. H. Miller, Ed., pp. 41–61, Academic Press, New York, (1972).

    Google Scholar 

  • Courant, R., Friedrichs, K. O., and Lewy, M. Math. Ann. 100, 32–76 (1928). (English translation) IBM J. Res. Devel. 11, 215–234 (1967).

    Google Scholar 

  • Courant, R., Isaacson, E., and Rees, M. Commun. Pure Appl. Math. 5, 243–255 (1952).

    Article  MathSciNet  MATH  Google Scholar 

  • Daube, O., and Ta Phuoc Loc, J. Mec. 17, 651–678 (1978).

    ADS  MATH  Google Scholar 

  • Dennis, S. C. R., and Chang, G. -Z. Phys. Fluids 12, Suppl. II, II. 83–1I. 93 (1969).

    Google Scholar 

  • Douglas, J., and Gunn, J. E. Numer. Math. 6, 428–453 (1964).

    Article  MathSciNet  MATH  Google Scholar 

  • Elsaesser, E. Etude de Méthodes Hermitiennes pour la Résolution des Equations de Navier—Stokes pour un Fluide Incompressible en Régime Stationnaire, Thèse Doct. 3’’’’ Cycle, Université Pierre et Marie Curie, Paris (1980).

    Google Scholar 

  • Elsaesser, E., and Peyret, R. Méthodes Hermitiennes pour la Résolution Numérique des Equations de Navier—Stokes. In Méthodes Numériques dans les Sciences de l’Ingénieur, E. Absi and R. Glowinski, Eds., pp. 249–258, Dunod, Paris (1979).

    Google Scholar 

  • Fairweather, G., and Mitchell, A. R. SIAM J. Numer. Anal. 4, 163–170 (1967).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Garabedian, P. R., Math. Tables Aids Comput. 10, 183–185 (1956).

    Article  MathSciNet  MATH  Google Scholar 

  • Ghia, K. N., Shin, C. T., and Ghia, U. Use of Spline Approximation for Higher-Order Accurate Solutions of Navier—Stokes Equations in Primitive Variables. In Proc. 4th AIAA Computational Fluid Dynamics Conference, pp. 284–291, Williamsburg, VA. (1979).

    Google Scholar 

  • Godunov, S. K., and Ryabenski, V. S. The Theory of Difference Schemes, North-Holland, Amsterdam (1964).

    Google Scholar 

  • Gourlay, A. R., and Morris, J. L. Math. Comput. 22, 715–720 (1968).

    Article  MathSciNet  MATH  Google Scholar 

  • Gourlay, A. R., and Morris, J. L. J. Comput. Phys. 5, 229–243 (1970).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Gustaffson, B. Math. Comput. 29, 396–406 (1975).

    Article  MathSciNet  Google Scholar 

  • Harten, A., Hyman, J. M., and Lax, P. D. Commun. Pure Appt. Math. 29, 297–322 (1976).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Hirsh, R. S. J. Comput. Phys. 19, 90–109 (1975).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Hirt, C. W., J. Comput. Phys. 2, 339–355 (1968).

    Article  ADS  MATH  Google Scholar 

  • Hollanders, H., and Peyret, R. Rech. Aérosp. No. 1981–4, 287–294 (1981).

    Google Scholar 

  • Isaacson, E. and Keller, H. B. Analysis of Numerical Methods, Wiley and Sons, New York (1966).

    MATH  Google Scholar 

  • Khosla, P. M., and Rubin, S. G. Comput. Fluids 2, 207–209 (1974).

    Article  MATH  Google Scholar 

  • Krause, E., Hirschel, E. H., and Kordulla, W. Comput. Fluids 4, 77–92 (1976).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Kreiss, H. O. Math. Comput. 26, 605–624 (1972).

    Article  MathSciNet  MATH  Google Scholar 

  • Kreiss, H. O., and Oliger, J. Methods for the Approximate Solution of Time-Dependent Problems. GARP Publ. series No. 10, Global Atmospheric Research Program (1973).

    Google Scholar 

  • Laval, P. C. R. Acad. Sci. B, 204, 239–242 (1980).

    MathSciNet  Google Scholar 

  • Laval, P. C. R. Acad. Sci. B, 221, 125–128 (1981a).

    MathSciNet  ADS  Google Scholar 

  • Laval, P. Schémas Explicites de Désintégration du Second Ordre pour la Résolution des Problémes Hyperboliques Non Linéaires: Théorie et Applications aux Ecoulements Transsoniques. ONERA, Note Technique No. 1981–10 (1981b).

    Google Scholar 

  • Lax, P. D. Commun. Pure Appl. Math. 7, 159–193 (1954).

    Article  MathSciNet  MATH  Google Scholar 

  • Lax, P. D. Commun. Pure Appl. Math. 10, 537–566 (1957).

    Article  MathSciNet  MATH  Google Scholar 

  • Lax, P. D., and Wendroff, B. Commun. Pure Appl. Math. 13, 217–237 (1960).

    Article  MathSciNet  MATH  Google Scholar 

  • Lecointe, Y., and Piquet, J. On the Numerical Solution of Some Types of Unsteady Incompressible Viscous Flow. Second International Conference on Numerical Methods in Laminar and Turbulent Flow, July 13–16, 1981, Venice, Italy.

    Google Scholar 

  • Lerat, A. C. R. Acad. Sci. A 288, 1033–1036 (1979).

    MathSciNet  MATH  Google Scholar 

  • Lerat, A. Sur le Calcul des Solutions Faibles des Systémes Hyperboliques de Lois de Conservation a l’Aide de Schémas aux Différences. ONERA Publ. No. 1981–1 (1981).

    Google Scholar 

  • Lerat, A., and Peyret, R. C. R. Acad. Sci. A 276, 759–762 (1973a).

    MathSciNet  MATH  Google Scholar 

  • Lerat, A., and Peyret, R. C. R. Acad. Sci. A 277, 363–366 (1973b).

    MathSciNet  MATH  Google Scholar 

  • Lerat, A. and Peyret, R., Comput. Fluids 2, 35–52 (1974a).

    Article  MathSciNet  MATH  Google Scholar 

  • Lerat, A. and Peyret, R. Lecture Notes in Physics, Vol. 35, pp. 251–256, Springer—Verlag, New York (1974b).

    Google Scholar 

  • Lerat, A., and Peyret, R. Rech. Aerosp. 1975–2, 61–79 (1975).

    MathSciNet  Google Scholar 

  • Lerat, A., and Sides, T. Proceedings of the Conference on Numerical Methods in Aerodynamic Fluid Dynamics, Univ. Reading, March (1981).

    Google Scholar 

  • Lindemuth, J., and Killeen, J. J. Comput Phys. 13, 181–208 (1973).

    Google Scholar 

  • MacCormack, R. W. The Effect of Viscosity in Hypervelocity Impact Cratering. AIAA paper No. 69–354 (1969).

    Google Scholar 

  • MacCormack, R. W. In Lecture Notes in Physics, Vol. 8, pp. 151–163, Springer—Verlag, New York (1971).

    Google Scholar 

  • Magnus, R., and Yoshihara, H. AIAA J. 13, 1622–1628 (1975).

    Article  ADS  Google Scholar 

  • Mehta, U. B. Dynamic Stall of an Oscillating Airfoil. AGARD Conference Proceedings No. 227, Unsteady Aerodynamics, Ottawa, Canada, pp. 231–2332 (1977).

    Google Scholar 

  • McGuire, G. R., and Morris, J. L. J. Inst. Math. Applic. 10, 150–165 (1972).

    Article  MathSciNet  MATH  Google Scholar 

  • McGuire, G. R., and Morris, J. L. J. Comput. Phys. 11, 531–549 (1973).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Mitchell, A. R. Computational Methods in Partial Differential Equations. Wiley, New York (1969).

    MATH  Google Scholar 

  • Moretti, G. Proceedings of 1974 Heat Transfer and Fluid Mechanics Institute, Stanford Press, Stanford (1974).

    Google Scholar 

  • Neron, M. Etude et Application d’une Méthode Implicite de Résolution des Équations de Navier—Stokes pur un Fluide Compressible, en Coordonneés Cylindriques. Thése Doct.-Ingénieur, Université Pierre et Marie Curie, Paris (1981).

    Google Scholar 

  • Oleinik, O. A. Am. Math. Soc. Trans.,Ser. 2, No. 26, pp. 95–172. [Translation of Usp. Math. Nauk 12, 3–73 (1957).]

    Google Scholar 

  • Orszag, S. A., and Tang, C.-M. J. Fluid Mech. 90, 129–143 (1979).

    Article  ADS  Google Scholar 

  • Peaceman, D. W., and Rachford, H. H. J. Soc. Indus. Appl. Math. 3, 28–41 (1955).

    Article  MathSciNet  MATH  Google Scholar 

  • Peyret, R. C. R. Acad. Sci. A 272, 1274–1277 (1971).

    MATH  Google Scholar 

  • Peyret, R. Résolution Numérique des Systèmes Hyperboliques. Application à la Dynamique des Gaz. ONERA, Publication No. 1977–5 (1977).

    Google Scholar 

  • Peyret, R. C. R. Acad. Sci. A 286, 59–62 (1978a).

    MathSciNet  ADS  MATH  Google Scholar 

  • Peyret, R. A Hermitian Finite-Difference Method for the Solution of the Navier—Stokes Equations. In Proceedings of the First International Conference on Numerical Methods in Laminar and Turbulent Flow, 43–54, Pentech Press, Plymouth, UK (1978b).

    Google Scholar 

  • Peyret, R. and Viviand, H. Computation of Viscous Compressible Flows Based on the Navier—Stokes Equations. AGARDograph No. 212 (1975).

    Google Scholar 

  • Raviart, P. A. Méthodes d’ Éléments Finis en Mécanique des Fluides. Ecole d’Eté d’Analyse Numérique, EDF, INRIA, CEA (1979).

    Google Scholar 

  • Richtmyer, R. D. A Survey of Difference Methods for Non-Steady Gas Dynamics. NCAR, Tech. Note 63–2, Boulder, CO (1962).

    Google Scholar 

  • Richtmyer, R. D. and Morton, K. W. Difference Methods for Initial Value Problems, Interscience, New York (1967).

    MATH  Google Scholar 

  • Roache, P. J. Computational Fluid Dynamics, Hermosa Publishers, Albuquerque, NM (1972).

    MATH  Google Scholar 

  • Rubin, E. L., and Burstein, S. Z. J. Comput. Phys. 2, 178–196 (1967).

    Article  ADS  MATH  Google Scholar 

  • Rubin, S. G., and Khosla, P. K. J. Comput. Phys. 24, 217–246 (1977).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Shokin, Y. I., Proc. Steklov Inst. Math. 122, 67–86 (1973).

    Google Scholar 

  • Shokin, Y. I. In Lecture Notes in Physics, Vol. 59, pp. 410–414, Springer—Verlag, New York (1976).

    Google Scholar 

  • Steger, J. L. and Warming, R. F. J. Comput. Phys. 40, 263–293 (1981).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Strang, G. SIAM J. Numerical Analysis, 5, 506–517 (1968).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Ta Phuoc Loc, J. Mec. 14, 109–134 (1975).

    ADS  MATH  Google Scholar 

  • Ta Phuoc Loc, J. Fluid Mech. 100, 111–128 (1980).

    Article  ADS  MATH  Google Scholar 

  • Thommen, H. U., Z. Angew. Math. Phys. 17, 369–384 (1966).

    Article  MathSciNet  MATH  Google Scholar 

  • Varga, R. S. Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, NJ (1962).

    Google Scholar 

  • Veldman, A. E. P. Comput. Fluids 1, 251–271 (1973).

    Article  MathSciNet  MATH  Google Scholar 

  • Von Neumann, J, and Richtmyer, R. D. J. Appl. Phys. 21, 232–257 (1950).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Warming, R. F., and Beam, R. M. SIAM-AMS Proc. 11, 85–129 (1978).

    MathSciNet  Google Scholar 

  • Warming, R. F., and Hyett, B. J. J. Comput. Phys. 14, 159–179 (1974).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Warming, R. F., Kutler, P., and Lomax, H. AIAA J. 11, 189–196 (1973).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Yanenko, N. N. The Method of Fractional Steps,Springer—Verlag, New York (1971). [In French, Méthodes à Pas Fractionnaires,Armand Colin, Paris (1968).]

    Google Scholar 

  • Yanenko, N. N., and Shokin, Y. I. Phys. Fluids 12, Suppl. I.., I1. 28–11. 33 (1969).

    Google Scholar 

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Peyret, R., Taylor, T.D. (1983). Finite-Difference Methods. In: Computational Methods for Fluid Flow. Springer Series in Computational Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-85952-6_2

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  • DOI: https://doi.org/10.1007/978-3-642-85952-6_2

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