Two classes of internally -stable generalized Runge-Kutta processes which remain consistent with an inaccurate Jacobian

Authors:
J. D. Day and D. N. P. Murthy

Journal:
Math. Comp. **39** (1982), 491-509

MSC:
Primary 65L20

DOI:
https://doi.org/10.1090/S0025-5718-1982-0669642-X

MathSciNet review:
669642

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Generalized Runge-Kutta Processes for stiff systems of ordinary differential equations usually require an accurate evaluation of a Jacobian at every step. However, it is possible to derive processes which are Internally *S*-stable when an accurate Jacobian is used but still remain consistent and highly stable if an approximate Jacobian is used. It is shown that these processes require at least as many function evaluations as an explicit Runge-Kutta process of the same order, and second and third order processes are developed. A second class of Generalized Runge-Kutta is introduced which requires that the Jacobian be evaluated accurately less than once every step. A third order process of this class is developed, and all three methods contain an error estimator similar to those of Fehlberg or England.

**[1]**R. H. Allen & C. Pottle, "Stable integration methods for electronic circuit analysis with widely separated time constant,"*Proc. Sixth Allerton Conf. on Circuit and System Theory*, IEEE, New York, 1968, pp. 311-320.**[2]**S. S. Artem′ev,*Construction of semi-implicit Runge-Kutta methods*, Dokl. Akad. Nauk SSSR**228**(1976), no. 4, 776–778 (Russian). MR**0413494****[3]**S. S. Artem'ev & G. V. Demidov, "A stable method for the solution of the Cauchy problem for stiff systems of ordinary differential equations,"*Proc.*6*th IF1P Conf. on Optimization Techniques*, Springer-Verlag, New York, 1975, pp. 270-274.**[4]**T. D. Bui,*On an 𝐿-stable method for stiff differential equations*, Information Processing Lett.**6**(1977), no. 5, 158–161. MR**0451733**, https://doi.org/10.1016/0020-0190(77)90014-x**[5]**T. D. Bui and S. S. Ghaderpanah,*Modified Richardson extrapolation scheme for error estimate in implicit Runge-Kutta procedures for stiff systems of ordinary differential equations*, Proceedings of the Seventh Manitoba Conference on Numerical Mathematics and Computing (Univ. Manitoba, Winnipeg, Man., 1977) Congress. Numer., XX, Utilitas Math., Winnipeg, Man., 1978, pp. 251–268. MR**535013****[6]**J. C. Butcher,*Coefficients for the study of Runge-Kutta integration processes*, J. Austral. Math. Soc.**3**(1963), 185–201. MR**0152129****[7]**J. C. Butcher,*Implicit Runge-Kutta processes*, Math. Comp.**18**(1964), 50–64. MR**159424**, https://doi.org/10.1090/S0025-5718-1964-0159424-9**[8]**D. A. Calahan, "A stable, accurate method of numerical integration for nonlinear systems,"*Proc. IEEE*, v. 56, 1968, p. 744.**[9]**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**, https://doi.org/10.1145/321958.321966**[10]**J. D. Day,*On Generalized Runge Kutta Methods*, Ph. D. Thesis, Dept. of Mechanical Engineering, Univ. of Queensland, 1980.**[11]**B. L. Ehle and J. D. Lawson,*Generalized Runge-Kutta processes for stiff initial-value problems*, J. Inst. Math. Appl.**16**(1975), no. 1, 11–21. MR**391524****[12]**Eduard Eitelberg,*Numerical simulation of stiff systems with a diagonal splitting method*, Math. Comput. Simulation**21**(1979), no. 1, 109–115. MR**532612**, https://doi.org/10.1016/0378-4754(79)90110-1**[13]**R. England,*Error estimates for Runge-Kutta type solutions to systems of ordinary differential equations*, Comput. J.**12**(1969/70), 166–170. MR**242377**, https://doi.org/10.1093/comjnl/12.2.166**[14]**W. H. Enright, T. E. Hull & B. Lindberg, "Comparing numerical methods for stiff systems of ODE's,"*BIT*, v. 15, 1975, pp. 10-48.**[15]**E. Fehlberg,*Classical Fifth-, Sixth-, Seventh- and Eighth-Order Runge Kutta Formulas With Step Size Control*, NASA Tech. Report R-287, 1968.**[16]**E. Fehlberg,*Low Order Classical Runge Kutta Formulas With Step Size Control and Their Application to Some Heat Transfer Problems*, NASA Tech. Report R-315, 1969.**[17]**George E. Forsythe, Michael A. Malcolm, and Cleve B. Moler,*Computer methods for mathematical computations*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1977. Prentice-Hall Series in Automatic Computation. MR**0458783****[18]**C. W. Gear, "Algorithm 407, DIFSUB for solution of ordinary differential equations,"*Comm. ACM*,v. 14, 1971, pp. 185-190.**[19]**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**, https://doi.org/10.1093/comjnl/12.2.183**[20]**P. J. van der Houwen,*Explicit and Semi-Implicit Runge Kutta Formulas for the Integration of Stiff Equations*, Report TW132, Mathematisch Centrum, Amsterdam, 1972.**[21]**P. J. van der Houwen,*Construction of integration formulas for initial value problems*, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. North-Holland Series in Applied Mathematics and Mechanics, Vol. 19. MR**0519726****[22]**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****[23]**J. Douglas Lawson,*Generalized Runge-Kutta processes for stable systems with large Lipschitz constants*, SIAM J. Numer. Anal.**4**(1967), 372–380. MR**221759**, https://doi.org/10.1137/0704033**[24]**Syvert P. Nørsett and Arne Wolfbrandt,*Order conditions for Rosenbrock type methods*, Numer. Math.**32**(1979), no. 1, 1–15. MR**525633**, https://doi.org/10.1007/BF01397646**[25]**A. Prothero and A. Robinson,*On the stability and accuracy of one-step methods for solving stiff systems of ordinary differential equations*, Math. Comp.**28**(1974), 145–162. MR**331793**, https://doi.org/10.1090/S0025-5718-1974-0331793-2**[26]**H. H. Rosenbrock,*Some general implicit processes for the numerical solution of differential equations*, Comput. J.**5**(1962/63), 329–330. MR**155434**, https://doi.org/10.1093/comjnl/5.4.329**[27]**Trond Steihaug and Arne Wolfbrandt,*An attempt to avoid exact Jacobian and nonlinear equations in the numerical solution of stiff differential equations*, Math. Comp.**33**(1979), no. 146, 521–534. MR**521273**, https://doi.org/10.1090/S0025-5718-1979-0521273-8**[28]**J. G. Verwer,*Generalized Linear Multistep methods*II:*Numerical applications*, Report NW 12/75, Mathematisch Centrum, Amsterdam, 1975.**[29]**J. G. Verwer,*𝑆-stability properties for generalized Runge-Kutta methods*, Numer. Math.**27**(1976/77), no. 4, 359–370. MR**438722**, https://doi.org/10.1007/BF01399599

Retrieve articles in *Mathematics of Computation*
with MSC:
65L20

Retrieve articles in all journals with MSC: 65L20

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1982-0669642-X

Keywords:
Generalized Runge-Kutta procedure,
semi-implicit Runge-Kutta procedure,
approximate Jacobian,
stiff differential equations,
*L*-stability,
*A*-stability,
*S*-stability,
Internal *S*-stability

Article copyright:
© Copyright 1982
American Mathematical Society