skip to main content
article
Free Access

A Polyalgorithm for the Numerical Solution of Ordinary Differential Equations

Authors Info & Claims
Published:01 March 1975Publication History
First page image

References

  1. 1 BROWN, R.L. Recursive calculations'of corrector coefficients. ACM SIGNUM Newsletter, 8 (Oct. 1973), 12-13. Google ScholarGoogle Scholar
  2. 2 BJUREL, G., DAHLQUIST, G., LINDBERG, B., LINDE, S., AND ODEN, L. Survey of stiff ordinary differential equations. Rep. NA 70.11, Dep. of Information Processing, Royal Inst. Technology, S-100 44, Stockholm, Nov. 1972.Google ScholarGoogle Scholar
  3. 3 BRAYTON, R. K., GUSTAVSON, F. G., AND HACHT:EL, G. D. 2~k new efficient algorithm for solving differential-algebraic systems using implicit backward differentiation formulas. Proc. IEEE 60 (1972), 98-108.Google ScholarGoogle Scholar
  4. 4 CAVENDISH, J. C., CULHAM, W. E., AND VARQA, R. S. A comparison of Crank-Nicolson and Chebyshev-rational methods for numerically solving linear parabolic equations. J. Computational Phys. 10 (1972), 354-368.Google ScholarGoogle Scholar
  5. 5 CHANG, J. S., HINDMARSH, A. C., AND MADSEN, N.K. Simulation of chemical kinetics transport in the stratosphere. Rep. UCRL-74823, Lawrence Livermore Lab., U. of California, Livermore, Calif., Oct. 1973. (Also in St~fl D~fferent~al Systems, R. A. Wffioughby, Ed., Plenum Press, New York, 1974, pp. 51-65.)Google ScholarGoogle Scholar
  6. 6 CODDINGTON, E. A., AND LEVINSON, N. Theory of Ordinary Differential Equations. McGraw- Hill, New York, 1955.Google ScholarGoogle Scholar
  7. 7 CODY, W. J., MEINARDUS, G., AND VARGA, R. S. Chebyshev rational approximations to e-~ in {0, -{- ~~) and applications to heat conduction problems. J. Approximation Theory (1969), 50-65.Google ScholarGoogle Scholar
  8. 8 CnANS, P. C., AND FOX, P. A. A comparative study of computer programs for integrating differential equations. In Numerical Mathematics Computer Programs, Library One--Basic Routines for General Use, Vol. 2, Issue 2, Numerical Mathematics Program Library Project, Computer Sci. Res. Cent., Bell Telephone Lab., Inc., Murray Hill, N.J., Feb. 1969.Google ScholarGoogle Scholar
  9. 9 CURTISS, C. F., AND HmSCHFELDER, J. O. Integration of stiff equations. Proc. Nat. Acad~ Sci. U.S.A. 38 (1952), 235-243.Google ScholarGoogle Scholar
  10. 10 DICKINSON, R. P., JR. Private communication.Google ScholarGoogle Scholar
  11. 11 EHL~, B. L. A comparison of some methods for solving certain stiff ordinary differential equations. Rep. 70, Dep. of Math., U. of Victoria, Victoria, B.C., Canada, Nov. 1972.Google ScholarGoogle Scholar
  12. 12 ENRIGHT, W. H. Studies in the numerical solution of stiff differential equations. Tech. Rep. 46, Dep. of Computer Sci., U. of Toronto, Toronto, Ont., Canada, Oct. 1972. (Also Ph.D. Th., Dep. of Computer Science, U. of Toronto, 1972.) Google ScholarGoogle Scholar
  13. 13 GEAR, C. W. Numerwal Initml Value Problems ~n Ordinary Differential Equatwns. Prentice- Hall, Englewood Cliffs, N.J., 1971. Google ScholarGoogle Scholar
  14. 14 GEAa, C. W. The automatic integration of ordinary differential equations. Comm. ACM Google ScholarGoogle Scholar
  15. 15 GEAR, C.W. Algorithm 407: DIFSUB for solution of ordinary differential equations. Comm. ACM 1~, 3 (March 1971), 185-190. Google ScholarGoogle Scholar
  16. 16 GEAR, C.W. Asymptotic estimation of errors and derivatives for the numerical solution of ordinary differential equations. UIUCDCS-R-73-598, U. of Illinois, Urbana, Ill., Oct. 1973; also, in Information Processing 74 (IFIP 74), Vol. 3, North-Holland, Amsterdam, 1974, pp. 447--451.Google ScholarGoogle Scholar
  17. 17 GEAa, C. W., AND TU, K.-W. The effect of variable mesh size on the stability of multistep methods. UIUCDCS-R-73-570, U. of Illinois, Urbana, Ill., April 1973. (Also in SIAM J. Numer. Anal. 11 (1974), 1025--1043.)Google ScholarGoogle Scholar
  18. 18 GEAR, C. W., AND WATANABE, D.S. Stability and convergence of variable order multistep methods. SIAM J. Numer. Anal. 11 (1974), 1044-1058.Google ScholarGoogle Scholar
  19. 19 GELINAS, R. J. Diurnal kinetic modelling. UCRL-75373, Lawrence Livermore Lab., U. of California, Livermore, Calif., Jan. 1974. (To appear in Proc. of the IAMAP/IAPSO Conf., Melbourne, Australia, Jan. 14-25, 1974 )Google ScholarGoogle Scholar
  20. 20 HALE, J.K. Ordinary Dlfferentml Equations. Wiley-Interscience, New York, 1969.Google ScholarGoogle Scholar
  21. 21 Hv.sR:ci, P. Discrete Variable Methods in Ordinary Differential Equatwns. Wiley, New York, 1962.Google ScholarGoogle Scholar
  22. 22 HINDMARSH, A. C. GEAR: ordinary differential equation system solver. UCID-30001, Rev. 2, Lawrence Livermore Lab., U. of California, Livermore, Calif., Aug. 1972.Google ScholarGoogle Scholar
  23. 23 HINDMARSH, A. C. The construction of mathematical software, part III: the control of error in the GEAR package for ordinary differential equations. UCID-30050, Pt. 3, Lawrence Livermore Lab, U. of Cahfornia, Livermore, Calif., Aug. 1972.Google ScholarGoogle Scholar
  24. 24 HINDMAaSH, A. C. Linear multistep methods for ordinary differential equations: method formulations, stability, and the methods of Nordsieck and Gear. UCRL-51186, Rev. 1, Lawrence Livermore Lab., U. of California, Livermore, Calif., March 1972.Google ScholarGoogle Scholar
  25. 25 HIm)MARSH, A. C. GEARB: solution of ordinary differential equations having banded Jacobian. UCID-30059, Lawrence Livermore Lab., U. of California, Livermore, Calif., May 1973.Google ScholarGoogle Scholar
  26. 26 HULL, T. E., ENRIGHT, W. H., FELLEN~ B. M., AND SEDGWICK, A. :E. Comparing numerical methods for ordinary differential equations. SIAM J. Namer. Anal. 9 (1972), 603-637.Google ScholarGoogle Scholar
  27. 27 KROGH, F.T. A variable step variable order multistep method for the numerical solution of ordinary differential equations. In Information Processing 68 (Proc. IFIP 68), Vol. I, A. J. H. Morrell, Ed., North-Holland, Amsterdam, 1969, pp. 194-199.Google ScholarGoogle Scholar
  28. 28 KaoGE, F.T. Changing stepsize in the integration of differential equations using modified divided differences. Proc. of the Conf. on the Numerical Solutmn of Ordinary Dlfferentml Equatmns, U. of Texas at Austin, Oct. 19--20, 1972, D. G. Bett~s, Ed., Lecture Notes in Mathematics, Vol. 362, Springer-Verlag, New York, 1974, pp. 22-71.Google ScholarGoogle Scholar
  29. 29 LAPIDUS, L., AND S:EINFELD, j. H. Numerical Solution of Ordinary Dzfferential Equations. Academic Press, New York, 1971.Google ScholarGoogle Scholar
  30. 30 LINIGER, W., AND WILLOUGHBY, R. Efficient integration methods for stiff systems of ordinary differential equations. SIAM J. Numer. Anal. 7 (1970), 47-66.Google ScholarGoogle Scholar
  31. 31 MADSEN, N. K, AND SINCOVEC, R. F. The numerical method of lines for the solution of nonhnear partmi differential equations. UCRL-75142, Lawrence Livermore Lab., U. of Califorma, Livermore, Calif., Sept. 1973.Google ScholarGoogle Scholar
  32. 32 M)~KELA, M. On a generalized interpolation approach to the numerical integration of ordinary dlfferentml equations. A z~,. Acad. Sc~. Fenn~cae, Ser. A, I, 503 (1973), 1-43.Google ScholarGoogle Scholar
  33. 33 MILNE, W.E. A note on the numerical integration of differential equations. J. Res. Nat. Bur. Stand. ~3 (1949), 537-542.Google ScholarGoogle Scholar
  34. 34 MILNE, W.E. Numerical Solution of D~fferential Equatwns, Wiley, New York, 1953.Google ScholarGoogle Scholar
  35. 35 M~LNE-THOMSON, L.M. The Calculus of Finite D,fferences. Macmillan, London, 1933.Google ScholarGoogle Scholar
  36. 36 NORDSIECK, A. On the numerical integration of ordinary differential equations. Math.{ Computation 16 (1962), 22-49.Google ScholarGoogle Scholar
  37. 37 ODEH, F., AND LINIGER, W. A note on the unconditional fixed-h stability of linear multistep formulae Computing 7 (1971), 240-253.Google ScholarGoogle Scholar
  38. 38 PIOTROWSKY, P. Stability, consistency and convergence of variable K-step methods. In Conf. on the Nun~erical Solution of Differential Equations, J. L. Morris, Ed., Lecture Notes in Mathematics, Vol. 109, Sprmger-Verlag, New York, 1969, pp. 221-227.Google ScholarGoogle Scholar
  39. 39 RICE, J. R. On the construction of polyalgorithms for automatic numerical analysis. In Interactwe Systems for Experimental Applied Mathematzcs, M. Klerer and J. Reinfelds, Eds., Academic Press, New York, 1968, pp. 301-313.Google ScholarGoogle Scholar
  40. 40 RUBN~.R-PETERSON, T. An efficient algorithm using backward time-scaled differences for solving stiff differential-algebraic systems. Tech. U. of Denmark, 2800 Lyngby. Sept. 1973. Presented at the Seminar in Numerical Analysis, Royal Inst. of Technology, Stockholm, Oct. 10-12, 1973.Google ScholarGoogle Scholar
  41. 41 SCHECTER, R.S. The Variatzonal Method in Engineering. McGraw-Hill, New York, 1967.{Google ScholarGoogle Scholar
  42. 42 SEDGWICK, A. An efficient variable order variable step Adams method. Tech. Rep. 53, Dep. of Computer Sci, U. of Toronto, Toronto, Ont., Canada, May 1973. (Also Ph.D. Th., Dep. of Computer Sci., U. of Toronto, 1973.) Google ScholarGoogle Scholar
  43. 43 SHAMPINE, L. F., AND GORDON, M.K. Computer Solution of Ordinary D~fferential Equations: Initial Value Problems. In press, Freeman, San Francisco, Calif.Google ScholarGoogle Scholar
  44. 44 SHAMPINE, L. F., AND GORDON, M.K. Local error and variable order Adams codes. A ppl. Math. Computation, to appear.Google ScholarGoogle Scholar
  45. 45 SINCOVEC, R.F. Private communication.Google ScholarGoogle Scholar
  46. 46 STETTER, I-I. J. Asymptotic expansions for the error of discretization algorithms for nonlinear{ functional equations. Numer. Math. 7 (1965), 18-31.Google ScholarGoogle Scholar
  47. 47 Tu, K.-W. Stability and convergence of general multistep and multivalue methods with variable step size. UIUCDCS-R-72-526, U. of Illinois, Urbana, Ill., July 1972. (Also Ph.D. Th., Dep. of Math., U. of Ilhnois, 1972.)Google ScholarGoogle Scholar

Index Terms

  1. A Polyalgorithm for the Numerical Solution of Ordinary Differential Equations

      Recommendations

      Comments

      Login options

      Check if you have access through your login credentials or your institution to get full access on this article.

      Sign in

      Full Access

      • Published in

        cover image ACM Transactions on Mathematical Software
        ACM Transactions on Mathematical Software  Volume 1, Issue 1
        March 1975
        96 pages
        ISSN:0098-3500
        EISSN:1557-7295
        DOI:10.1145/355626
        Issue’s Table of Contents

        Copyright © 1975 ACM

        Publisher

        Association for Computing Machinery

        New York, NY, United States

        Publication History

        • Published: 1 March 1975
        Published in toms Volume 1, Issue 1

        Permissions

        Request permissions about this article.

        Request Permissions

        Check for updates

        Qualifiers

        • article

      PDF Format

      View or Download as a PDF file.

      PDF

      eReader

      View online with eReader.

      eReader