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Multistep methods using higher derivatives and damping at infinity

Author: Rolf Jeltsch
Journal: Math. Comp. 31 (1977), 124-138
MSC: Primary 65L05
MathSciNet review: 0428716
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Abstract: Linear multistep methods using higher derivatives are discussed. The order of damping at infinity which measures the stability behavior of a k-step method for large h is introduced, A-stable methods with positive damping order are most suitable for stiff problems. A method for computing the damping order is given. Necessary and sufficient conditions for A-stability, A $ A(\alpha )$-stability and stiff stability are presented. A new A-stable two-step method of order 4 with damping order 1 is found and numerical results are given.

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  • [1] Lars V. Ahlfors, Complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1978. An introduction to the theory of analytic functions of one complex variable; International Series in Pure and Applied Mathematics. MR 510197
  • [2] Owe Axelsson, A class of 𝐴-stable methods, Nordisk Tidskr. Informationsbehandling (BIT) 9 (1969), 185–199. MR 0255059
  • [3] Garrett Birkhoff and Richard S. Varga, Discretization errors for well-set Cauchy problems. I, J. Math. and Phys. 44 (1965), 1–23. MR 0179952
  • [4] James L. Blue and Hermann K. Gummel, Rational approximations to matrix exponential for systems of stiff differential equations, J. Computational Phys. 5 (1970), 70–83. MR 0255060
  • [5] D. A. CALAHAN, "Numerical solution of linear systems with widely separated time constants," Proc. IEEE, Nov., 1967, pp. 2016-2017.
  • [6] J. C. Cavendish, W. E. Culham, and R. S. Varga, A comparison of Crank-Nicolson and Chebyshev rational methods for numerically solving linear parabolic equations, J. Computational Phys. 10 (1972), 354–368. MR 0324919
  • [7] F. H. CHIPMAN, Numerical Solution of Initial Value Problems Using A-Stable Runge-Kutta Processes, Research Report CSRR 2042, Dept. of A.A.C.S., Univ. of Waterloo, 1971.
  • [8] Colin W. Cryer, On the instability of high order backward-difference multistep methods, Nordisk Tidskr. Informations behandling (BIT) 12 (1972), 17–25. MR 0311112
  • [9] Germund Dahlquist, Convergence and stability in the numerical integration of ordinary differential equations, Math. Scand. 4 (1956), 33–53. MR 0080998,
  • [10] Germund Dahlquist, Stability and error bounds in the numerical integration of ordinary differential equations, Kungl. Tekn. Högsk. Handl. Stockholm. No. 130 (1959), 87. MR 0102921
  • [11] Germund G. Dahlquist, A special stability problem for linear multistep methods, Nordisk Tidskr. Informations-Behandling 3 (1963), 27–43. MR 0170477
  • [12] James W. Daniel and Ramon E. Moore, Computation and theory in ordinary differential equations, W. H. Freeman and Co., San Francisco, Calif., 1970 (German). MR 0267765
  • [13] E. J. Davison, A high-order Crank-Nicholson technique for solving differential equations, Comput. J. 10 (1967), 195–197. MR 0214290,
  • [14] B. L. EHLE, On Padé Approximations to the Exponential Function and A-Stable Methods for the Numerical Solution of Initial Value Problems, Research Report CSRR 2010, Dept. of A.A.C.S., Univ. of Waterloo, 1969.
  • [15] M. E. ENGELI, Symbal, Summary and Examples, Fides, Union Fiduciaire, Zürich, 1970.
  • [16] W. ENRIGHT, Studies in the Numerical Solution of Stiff Ordinary Differential Equations, Technical Report No. 46, Dept. of C. S., Univ. of Toronto, 1972.
  • [17] Tomlinson Fort, Finite Differences and Difference Equations in the Real Domain, Oxford, at the Clarendon Press, 1948. MR 0024567
  • [18] C. W. Gear, The automatic integration of stiff ordinary differential equations., Information Processing 68 (Proc. IFIP Congress, Edinburgh, 1968) North-Holland, Amsterdam, 1969, pp. 187–193. MR 0260180
  • [19] C. William Gear, Numerical initial value problems in ordinary differential equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971. MR 0315898
  • [20] Eberhard Griepentrog, Mehrschrittverfahren zur numerischen Integration von gewöhnlichen Differentialgleichungssystemen und asymptotische Exaktheit, Wiss. Z. Humboldt-Univ. Berlin Math.-Natur. Reihe 19 (1970), 637–653 (German, with Russian, English and French summaries). MR 0321300
  • [21] Peter Henrici, Discrete variable methods in ordinary differential equations, John Wiley & Sons, Inc., New York-London, 1962. MR 0135729
  • [22] K. HENSEL & G. LANDSBERG, Theorie der algebraischen Funktionen einer Variablen, Teubner, Leipzig, 1902.
  • [23] Rolf Jeltsch, Integration of iterated integrals by multistep methods, Numer. Math. 21 (1973/74), 303–316. MR 0337013,
  • [24] Rolf Jeltsch, Multistep multiderivative methods and Hermite-Birkhoff interpolation, Proceedings of the Fifth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1975) Utilitas Math. Publ., Winnipeg, Man., 1976, pp. 417–428. Congressus Numerantium, No. XVI. MR 0436600
  • [25] J. D. Lambert, Computational methods in ordinary differential equations, John Wiley & Sons, London-New York-Sydney, 1973. Introductory Mathematics for Scientists and Engineers. MR 0423815
  • [26] John D. Lambert and Andrew R. Mitchell, On the solution of 𝑦′=𝑓(𝑥,𝑦) by a class of high accuracy difference formulae of low order, Z. Angew. Math. Phys. 13 (1962), 223–232 (English, with German summary). MR 0140188,
  • [27] J. D. LAWSON, Order Constrained Best Rational Approximation to $ \exp (x)$ on $ ( -\; \infty ,0]$. (Private communication.)
  • [28] Werner Liniger and Ralph A. Willoughby, Efficient integration methods for stiff systems of ordinary differential equations, SIAM J. Numer. Anal. 7 (1970), 47–66. MR 0260181,
  • [29] F. R. LOSCALZO, On the Use of Spline Functions for the Numerical Solution of Ordinary Differential Equations, MRC Technical Summary Report No. 869, Univ. of Wisconsin, May 1968.
  • [30] G. J. Makinson, Stable high order implicit methods for the numerical solution of systems of differential equations, Comput. J. 11 (1968/1969), 305–310. MR 0235737,
  • [31] Morris Marden, Geometry of polynomials, Second edition. Mathematical Surveys, No. 3, American Mathematical Society, Providence, R.I., 1966. MR 0225972
  • [32] Ramon E. Moore, The automatic analysis and control of error in digital computation based on the use of interval numbers, Error in Digital Computation, Vol. 1 (Proc. Advanced Sem. Conducted by Math. Res. Center, U.S. Army, Univ. Wisconsin, Madison, Wis., 1964) Wiley, New York, 1965, pp. 61–130. MR 0176614
  • [33] Michael R. Osborne, A new method for the integration of stiff systems of ordinary differential equations, Information Processing 68 (Proc. IFIP Congress, Edinburgh, 1968) North-Holland, Amsterdam, 1969, pp. 200–204. MR 0260182
  • [34] Albert Pfluger, Theorie der Riemannschen Flächen, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1957 (German). MR 0084031
  • [35] Manfred Reimer, Finite difference forms containing derivatives of higher order, SIAM J. Numer. Anal. 5 (1968), 725–738. MR 0250508,
  • [36] Hans J. Stetter, Analysis of discretization methods for ordinary differential equations, Springer-Verlag, New York-Heidelberg, 1973. Springer Tracts in Natural Philosophy, Vol. 23. MR 0426438
  • [37] W. E. THOMPSON, "Solution of linear differential equations," Comput. J., v. 10, 1968, pp. 417-418.
  • [38] R. S. VARGA, "Some results in approximation theory with applications to numerical analysis," in Numerical Solution of Partial Differential Equations, II (SYNSPADE 1970), (Proc. Sympos., Univ. of Maryland, 1970), edited by B. E. Hubbard, Academic Press, New York, 1971, pp. 623-649.
  • [39] Olof B. Widlund, A note on unconditionally stable linear multistep methods, Nordisk Tidskr. Informations-Behandling 7 (1967), 65–70. MR 0215533

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Keywords: Ordinary differential equations, linear k-step methods using higher derivatives, Obrechkoff methods, Hermite methods, A-stable, order of damping at infinity, stiffly stable, strongly A-stable, L-stable
Article copyright: © Copyright 1977 American Mathematical Society

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