Stability results for one-step discretized collocation methods in the numerical treatment of Volterra integral equations

Authors:
M. R. Crisci, E. Russo and A. Vecchio

Journal:
Math. Comp. **58** (1992), 119-134

MSC:
Primary 65R20; Secondary 45D05, 45L05

MathSciNet review:
1106963

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Abstract: This paper is concerned with the stability analysis of the discretized collocation method for the second-kind Volterra integral equation with degenerate kernel. A fixed-order recurrence relation with variable coefficients is derived, and local stability conditions are given independent of the discretization. Local stability and stability with respect to an isolated perturbation of some methods are proved. The reliability of the derived stability conditions is shown by numerical experiments.

**[1]**Siamak Amini,*On the stability of Volterra integral equations with separable kernels*, Appl. Anal.**24**(1987), no. 4, 241–251. MR**907340**, 10.1080/00036818708839667**[2]**Siamak Amini, Christopher T. H. Baker, Pieter J. van der Houwen, and Paul H. M. Wolkenfelt,*Stability analysis of numerical methods for Volterra integral equations with polynomial convolution kernels*, J. Integral Equations**5**(1983), no. 1, 73–92. MR**683033****[3]**Christopher T. H. Baker,*Structure of recurrence relations in the study of stability in the numerical treatment of Volterra integral and integro-differential equations*, J. Integral Equations**2**(1980), no. 1, 11–29. MR**559015****[4]**J. G. Blom and H. Brunner,*The numerical solution of nonlinear Volterra integral equations of the second kind by collocation and iterated collocation methods*, SIAM J. Sci. Statist. Comput.**8**(1987), no. 5, 806–830. MR**902744**, 10.1137/0908068**[5]**J. M. Bownds and J. M. Cushing,*Some stability criteria for linear systems of Volterra integral equations*, Funkcial. Ekvac.**15**(1972), 101–117. MR**0328520****[6]**Hermann Brunner,*Superconvergence in collocation and implicit Runge-Kutta methods for Volterra-type integral equations of the second kind*, Numerical treatment of integral equations (Workshop, Math. Res. Inst., Oberwolfach, 1979) Internat. Ser. Numer. Math., vol. 53, Birkhäuser, Basel-Boston, Mass., 1980, pp. 54–72. MR**590441****[7]**H. Brunner,*Collocation methods for one-dimensional Fredholm and Volterra integral equations*, The state of the art in numerical analysis (Birmingham, 1986) Inst. Math. Appl. Conf. Ser. New Ser., vol. 9, Oxford Univ. Press, New York, 1987, pp. 563–600. MR**921678****[8]**Hermann Brunner and Syvert P. Nørsett,*Superconvergence of collocation methods for Volterra and Abel integral equations of the second kind*, Numer. Math.**36**(1980/81), no. 4, 347–358. MR**614853**, 10.1007/BF01395951**[9]**H. Brunner and P. J. van der Houwen,*The numerical solution of Volterra equations*, CWI Monographs, vol. 3, North-Holland Publishing Co., Amsterdam, 1986. MR**871871****[10]**M. R. Crisci, E. Russo, and A. Vecchio,*On the stability of the one-step exact collocation methods for the numerical solution of the second kind Volterra integral equation*, BIT**29**(1989), no. 2, 258–269. MR**997533**, 10.1007/BF01952681**[11]**Maria Rosaria Crisci, Elvira Russo, and Antonia Vecchio,*On the stability of the one-step exact collocation method for the second kind Volterra integral equation with degenerate kernel*, Computing**40**(1988), no. 4, 315–328 (English, with German summary). MR**969649**, 10.1007/BF02276915**[12]**K. Dekker and J. G. Verwer,*Stability of Runge-Kutta methods for stiff nonlinear differential equations*, CWI Monographs, vol. 2, North-Holland Publishing Co., Amsterdam, 1984. MR**774402****[13]**F. de Hoog and R. Weiss,*Implicit Runge-Kutta methods for second kind Volterra integral equations*, Numer. Math.**23**(1974/75), 199–213. MR**0373349****[14]**P. J. van der Houwen,*Stability results for discrete Volterra equations*, Report NW 149/83, Centre for Mathematics and Computer Science, Amsterdam, 1983.**[15]**P. J. van der Houwen and P. H. M. Wolkenfelt,*On the stability of multistep formulas for Volterra integral equations of the second kind*, Computing**24**(1980), no. 4, 341–347 (English, with German summary). MR**620232**, 10.1007/BF02237819

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1992-1106963-7

Keywords:
Volterra integral equations,
degenerate kernel,
stability,
discrete collocation methods,
Runge-Kutta methods

Article copyright:
© Copyright 1992
American Mathematical Society