Quadrature methods for stiff ordinary differential systems

Iserles, A.

doi:10.1090/S0025-5718-1981-0595049-9

Quadrature methods for stiff ordinary differential systems
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Math. Comp. 36 (1981), 171-182 Request permission

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.

References

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
Arieh Iserles, On multivalued exponential approximations, SIAM J. Numer. Anal. 18 (1981), no. 3, 480–499. MR 615527, DOI 10.1137/0718031
R. K. Jain, Some $A$-stable methods for stiff ordinary differential equations, Math. Comp. 26 (1972), 71–77. MR 303733, DOI 10.1090/S0025-5718-1972-0303733-1
J. Douglas Lawson, Generalized Runge-Kutta processes for stable systems with large Lipschitz constants, SIAM J. Numer. Anal. 4 (1967), 372–380. MR 221759, DOI 10.1137/0704033
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
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 398110, DOI 10.1007/bf02241724
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
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 431696, DOI 10.1007/BF01399082