Quadrature methods for stiff ordinary differential systems

Author:
A. Iserles

Journal:
Math. Comp. **36** (1981), 171-182

MSC:
Primary 65L05

MathSciNet review:
595049

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The quadrature methods are based upon a substitution of an explicit *A*-stable first approximation into a generalized convolution formula. They are *A*-stable, explicit, and of arbitrarily high order.

The generalized convolution formula is derived and its order-raising properties examined. Two families of explicit *A*-stable first approximations are developed, generalizing results of Lawson and Nørsett. Various aspects of the numerical implementation are discussed.

Numerical results supplement the paper and exemplify the various merits and weaknesses of the quadrature methods.

**[1]**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****[2]**Arieh Iserles,*On multivalued exponential approximations*, SIAM J. Numer. Anal.**18**(1981), no. 3, 480–499. MR**615527**, 10.1137/0718031**[3]**R. K. Jain,*Some 𝐴-stable methods for stiff ordinary differential equations*, Math. Comp.**26**(1972), 71–77. MR**0303733**, 10.1090/S0025-5718-1972-0303733-1**[4]**J. Douglas Lawson,*Generalized Runge-Kutta processes for stable systems with large Lipschitz constants*, SIAM J. Numer. Anal.**4**(1967), 372–380. MR**0221759****[5]**W. Liniger,*High-order 𝐴-stable averaging algorithms for stiff differential systems*, Numerical methods for differential systems, Academic Press, New York, 1976, pp. 1–23. MR**0468186****[6]**G. Meister,*Über die Integration von Differentialgleichungssystemen 1. Ordnung mit exponentiell angepassten numerischen Methoden*, Computing (Arch. Elektron. Rechnen)**13**(1974), no. 3-4, 327–352. MR**0398110****[7]**Syvert P. Nørsett,*An 𝐴-stable modification of the Adams-Bashforth methods*, Conf. on Numerical Solution of Differential Equations (Dundee, 1969) Springer, Berlin, 1969, pp. 214–219. MR**0267771****[8]**Pedro E. Zadunaisky,*On the estimation of errors propagated in the numerical integration of ordinary differential equations*, Numer. Math.**27**(1976/77), no. 1, 21–39. MR**0431696**

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

Retrieve articles in all journals with MSC: 65L05

Additional Information

DOI:
http://dx.doi.org/10.1090/S0025-5718-1981-0595049-9

Article copyright:
© Copyright 1981
American Mathematical Society