Multistep methods using higher derivatives and damping at infinity

Author:
Rolf Jeltsch

Journal:
Math. Comp. **31** (1977), 124-138

MSC:
Primary 65L05

DOI:
https://doi.org/10.1090/S0025-5718-1977-0428716-7

MathSciNet review:
0428716

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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* -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.

**[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**, https://doi.org/10.7146/math.scand.a-10454**[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**, https://doi.org/10.1093/comjnl/10.2.195**[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**, https://doi.org/10.1007/BF01436385**[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**, https://doi.org/10.1007/BF01601084**[27]**J. D. LAWSON,*Order Constrained Best Rational Approximation to**on*. (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**, https://doi.org/10.1137/0707002**[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**, https://doi.org/10.1093/comjnl/11.3.305**[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**, https://doi.org/10.1137/0705056**[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**

Retrieve articles in *Mathematics of Computation*
with MSC:
65L05

Retrieve articles in all journals with MSC: 65L05

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1977-0428716-7

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