Block implicit one-step methods

Watanabe, Daniel S.

doi:10.1090/S0025-5718-1978-0494959-0

Block implicit one-step methods
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by Daniel S. Watanabe PDF

Math. Comp. 32 (1978), 405-414 Request permission

Abstract:

A new class of block implicit one-step methods for ordinary differential equations is presented. The methods are based on quadrature and generate function values at nonmesh points through Hermite interpolation. A general convergence theorem for block implicit methods is given, and the stability of the new class of methods is analyzed. The class contains A-stable, stiffly stable, strongly A-stable, and strongly stiffly stable methods. Numerical results demonstrating the efficiency and effectiveness of a particular block method are presented.

References

George D. Andria, George D. Byrne, and David R. Hill, Natural spline block implicit methods, Nordisk Tidskr. Informationsbehandling (BIT) 13 (1973), 131–144. MR 323110, DOI 10.1007/bf01933485

Mathematical Software

C. G. Broyden, A new method of solving nonlinear simultaneous equations, Comput. J. 12 (1969/70), 94–99. MR 245197, DOI 10.1093/comjnl/12.1.94
J. C. Butcher, Implicit Runge-Kutta processes, Math. Comp. 18 (1964), 50–64. MR 159424, DOI 10.1090/S0025-5718-1964-0159424-9

Numerical Solution of Initial Value Problems Using A-Stable Runge-Kutta Processes

On Padé Approximations to the Exponential Function and A-Stable Methods for the Numerical Solution of Initial Value Problems

C. William Gear, Numerical initial value problems in ordinary differential equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971. MR 0315898

J. Reine Angew. Math.

Bernie L. Hulme, Discrete Galerkin and related one-step methods for ordinary differential equations, Math. Comp. 26 (1972), 881–891. MR 315899, DOI 10.1090/S0025-5718-1972-0315899-8
J. Barkley Rosser, A Runge-Kutta for all seasons, SIAM Rev. 9 (1967), 417–452. MR 219242, DOI 10.1137/1009069
L. F. Shampine and H. A. Watts, Block implicit one-step methods, Math. Comp. 23 (1969), 731–740. MR 264854, DOI 10.1090/S0025-5718-1969-0264854-5
Hans J. Stetter, Economical global error estimation, Stiff differential systems (Proc. Internat. Sympos., Wildbad, Germany, 1973) IBM Res. Sympos. Ser., Plenum, New York, 1974, pp. 245–258. MR 0405863
H. A. Watts and L. F. Shampine, $A$-stable block implicit one-step methods, Nordisk Tidskr. Informationsbehandling (BIT) 12 (1972), 252–266. MR 307483, DOI 10.1007/bf01932819
Jack Williams and Frank de Hoog, A class of $A$-stable advanced multistep methods, Math. Comp. 28 (1974), 163–177. MR 356519, DOI 10.1090/S0025-5718-1974-0356519-8