Implicit Runge-Kutta methods of optimal order for Volterra integro-differential equations

Brunner, Hermann

doi:10.1090/S0025-5718-1984-0725986-6

Implicit Runge-Kutta methods of optimal order for Volterra integro-differential equations
HTML articles powered by AMS MathViewer

by Hermann Brunner PDF

Math. Comp. 42 (1984), 95-109 Request permission

Abstract:

Implicit Runge-Kutta methods with m stages and optimal order $p = 2m$ for the approximate solution of Volterra integro-differential equations can be viewed as fully discretized collocation methods in certain polynomial spline spaces. The choice of the quadrature formulas needed for the full discretization of the collocations is investigated, and it is shown that, in contrast to ordinary differential equations, there exist (for fixed m) several optimal methods.

References

Modern Numerical Methods for Ordinary Differential Equations

H. Brunner, On the numerical solution of nonlinear Volterra integro-differential equations, Nordisk Tidskr. Informationsbehandling (BIT) 13 (1973), 381–390. MR 331829, DOI 10.1007/bf01933399
Hermann Brunner, The application of the variation of constants formulas in the numerical analysis of integral and integro-differential equations, Utilitas Math. 19 (1981), 255–290. MR 624058
H. Brunner and J. D. Lambert, Stability of numerical methods for Volterra integro-differential equations, Computing (Arch. Elektron. Rechnen) 12 (1974), no. 1, 75–89 (English, with German summary). MR 418490, DOI 10.1007/bf02239501
Colin W. Cryer, Numerical methods for functional differential equations, Delay and functional differential equations and their applications (Proc. Conf., Park City, Utah, 1972) Academic Press, New York, 1972, pp. 17–101. MR 0388820
Alan Feldstein and John R. Sopka, Numerical methods for nonlinear Volterra integro-differential equations, SIAM J. Numer. Anal. 11 (1974), 826–846. MR 375816, DOI 10.1137/0711067
A. Ghizzetti and A. Ossicini, Quadrature formulae, Academic Press, New York, 1970. MR 0269116
A. Guillou and J. L. Soulé, La résolution numérique des problèmes différentiels aux conditions initiales par des méthodes de collocation, Rev. Française Informat. Recherche Opérationnelle 3 (1969), no. Sér. R-3, 17–44 (French). MR 0280008
Eugene Isaacson and Herbert Bishop Keller, Analysis of numerical methods, John Wiley & Sons, Inc., New York-London-Sydney, 1966. MR 0201039

Numerische Behandlung Volterra’scher Integrodifferentialgleichungen

Runge-Kutta Theory for Volterra Integrodifferential Equations

Athena Makroglou, Convergence of a block-by-block method for nonlinear Volterra integro-differential equations, Math. Comp. 35 (1980), no. 151, 783–796. MR 572856, DOI 10.1090/S0025-5718-1980-0572856-9
William L. Mocarsky, Convergence of step-by-step methods for non-linear integro-differential equations, J. Inst. Math. Appl. 8 (1971), 235–239. MR 287734
Syvert P. Nørsett, Collocation and perturbed collocation methods, Numerical analysis (Proc. 8th Biennial Conf., Univ. Dundee, Dundee, 1979), Lecture Notes in Math., vol. 773, Springer, Berlin, 1980, pp. 119–132. MR 569466
S. P. Nørsett and G. Wanner, The real-pole sandwich for rational approximations and oscillation equations, BIT 19 (1979), no. 1, 79–94. MR 530118, DOI 10.1007/BF01931224