A note on the Rosenbrock procedure

Bui, T. D.

doi:10.1090/S0025-5718-1979-0528050-2

A note on the Rosenbrock procedure
HTML articles powered by AMS MathViewer

by T. D. Bui PDF

Math. Comp. 33 (1979), 971-975 Request permission

Abstract:

To be useful for extremely stiff systems of ordinary differential equations, A-stability and a maximally damped condition as $\lambda h \to - \infty$ (i.e., L-stability) are desirable. This paper investigates the condition of L-stability for a class of Runge-Kutta methods known as the Rosenbrock procedure. This procedure requires only one computation of a Jacobian matrix per step of integration, L-stable Rosenbrock methods up to order four are derived.

References

Roger Alexander, Diagonally implicit Runge-Kutta methods for stiff o.d.e.’s, SIAM J. Numer. Anal. 14 (1977), no. 6, 1006–1021. MR 458890, DOI 10.1137/0714068
Theodore A. Bickart, An efficient solution process for implicit Runge-Kutta methods, SIAM J. Numer. Anal. 14 (1977), no. 6, 1022–1027. MR 458893, DOI 10.1137/0714069
J. C. Butcher, Implicit Runge-Kutta processes, Math. Comp. 18 (1964), 50–64. MR 159424, DOI 10.1090/S0025-5718-1964-0159424-9
J. C. Butcher, On the implementation of implicit Runge-Kutta methods, Nordisk Tidskr. Informationsbehandling (BIT) 16 (1976), no. 3, 237–240. MR 488746, DOI 10.1007/bf01932265
T. D. Bui, On an $L$-stable method for stiff differential equations, Information Processing Lett. 6 (1977), no. 5, 158–161. MR 451733, DOI 10.1016/0020-0190(77)90014-x
T. D. Bui, Some $A$-stable and $L$-stable methods for the numerical integration of stiff ordinary differential equations, J. Assoc. Comput. Mach. 26 (1979), no. 3, 483–493. MR 535267, DOI 10.1145/322139.322147
J. R. Cash, Semi-implicit Runge-Kutta procedures with error estimates for the numerical integration of stiff systems of ordinary differential equations, J. Assoc. Comput. Mach. 23 (1976), no. 3, 455–460. MR 471312, DOI 10.1145/321958.321966
Germund G. Dahlquist, A special stability problem for linear multistep methods, Nordisk Tidskr. Informationsbehandling (BIT) 3 (1963), 27–43. MR 170477, DOI 10.1007/bf01963532
Byron L. Ehle, High order $A$-stable methods for the numerical solution of systems of D.E.’s, Nordisk Tidskr. Informationsbehandling (BIT) 8 (1968), 276–278. MR 239762, DOI 10.1007/bf01933437
C. F. Haines, Implicit integration processes with error estimate for the numerical solution of differential equations, Comput. J. 12 (1969/70), 183–187. MR 246513, DOI 10.1093/comjnl/12.2.183
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

Semi-Explicit Runge-Kutta Methods

Syvert P. Nørsett and Arne Wolfbrandt, Attainable order of rational approximations to the exponential function with only real poles, Nordisk Tidskr. Informationsbehandling (BIT) 17 (1977), no. 2, 200–208. MR 447900, DOI 10.1007/bf01932291
H. H. Rosenbrock, Some general implicit processes for the numerical solution of differential equations, Comput. J. 5 (1962/63), 329–330. MR 155434, DOI 10.1093/comjnl/5.4.329