## The numerical analysis of implicit Runge-Kutta methods for a certain nonlinear integro-differential equation

HTML articles powered by AMS MathViewer

- by Wei Yuan and Tao Tang PDF
- Math. Comp.
**54**(1990), 155-168 Request permission

## Abstract:

In this paper and in an earlier 1987 paper, the mathematical theory and numerical methods for the nonlinear integro-differential equation \[ \begin {array}{*{20}{c}} {u’(t) + p(t)u(t) + \int _0^t {k(t} ,s)u(t - s)u(s) ds = q(t),\quad 0 \leq t \leq T,} \hfill \\ {u(0) = {u_0}} \hfill \\ \end {array} \] are considered. Equations of this type occur as model equations for describing turbulent diffusion. Previously, the existence and uniqueness properties of the solutions of the model equation were solved completely, and a class of implicit Runge-Kutta methods with*m*stages for the approximate solution of the model equation was introduced. In this paper, we give a further numerical analysis of these methods. It is proved that the implicit Runge-Kutta methods with

*n*stages are of optimal approximation order $p = 2m$. Some computational examples are given.

## References

- M. Aguilar and H. Brunner,
*Collocation methods for second-order Volterra integro-differential equations*, Appl. Numer. Math.**4**(1988), no. 6, 455–470. MR**964297**, DOI 10.1016/0168-9274(88)90009-8
C. T. H. Baker, - 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,
*Implicit Runge-Kutta methods of optimal order for Volterra integro-differential equations*, Math. Comp.**42**(1984), no. 165, 95–109. MR**725986**, DOI 10.1090/S0025-5718-1984-0725986-6 - 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 - 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** - Ll. G. Chambers,
*Integral equations: a short course*, International Textbook Co., Ltd., London, 1976. MR**0438057** - S. H. Chang and J. T. Day,
*On the numerical solution of a certain nonlinear integro-differential equation*, J. Comput. Phys.**26**(1978), no. 2, 162–168. MR**484431**, DOI 10.1016/0021-9991(78)90088-8 - 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** - Jennifer A. Dixon,
*A nonlinear weakly singular Volterra integro-differential equation arising from a reaction-diffusion study in a small cell*, J. Comput. Appl. Math.**18**(1987), no. 3, 289–305. MR**898770**, DOI 10.1016/0377-0427(87)90003-3 - Charles M. Elliott and Sean McKee,
*On the numerical solution of an integro-differential equation arising from wave-power hydraulics*, BIT**21**(1981), no. 3, 318–325. MR**640931**, DOI 10.1007/BF01941466 - 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
C. Lubich, Diploma Thesis, University of Innsbruck, 1981.
—, - 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 - S. McKee,
*The analysis of a variable step, variable coefficient linear multistep method for solving a singular integro-differential equation arising from the diffusion of*, J. Inst. Math. Appl.**23**(1979), no. 3, 373–388. MR**533231**, DOI 10.1093/imamat/23.3.373 - 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**, DOI 10.1093/imamat/8.2.235
A. S. Monin and A. M. Yaglom, - Andrea Prosperetti,
*A numerical method for the solution of certain classes of nonlinear Volterra integro-differential and integral equations*, Internat. J. Numer. Methods Engrg.**11**(1977), no. 3, 431–438. MR**431764**, DOI 10.1002/nme.1620110304 - Thomas L. Saaty,
*Modern nonlinear equations*, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1967. MR**0218160** - T. Tang and Wei Yuan,
*The further study of a certain nonlinear integro-differential equation*, J. Comput. Phys.**72**(1987), no. 2, 486–497. MR**913215**, DOI 10.1016/0021-9991(87)90095-7
B. A. Velikson, - Vito Volterra,
*Theory of functionals and of integral and integro-differential equations*, Dover Publications, Inc., New York, 1959. With a preface by G. C. Evans, a biography of Vito Volterra and a bibliography of his published works by E. Whittaker. MR**0100765**

*Initial value problems for Volterra integro-differential equations*, in Modern Numerical Methods for Ordinary Differential Equations (G. Hall and J. M. Watt, eds.), Clarendon Press, Oxford, 1976, pp. 296-307.

*Runge-Kutta theory for Volterra integro-differential equations*, Preprint No. 131, Sonderforschungsbereich 123, University of Heidelberg, 1981.

*Statistical hydromechanics*, Part 2, "Nauka", Moscow, 1967. (Russian)

*Solution of a nonlinear integro-differential equation*, U.S.S.R. Comput. Math. and Math. Phys.

**15**(1975), 256-259.

## Additional Information

- © Copyright 1990 American Mathematical Society
- Journal: Math. Comp.
**54**(1990), 155-168 - MSC: Primary 65R20
- DOI: https://doi.org/10.1090/S0025-5718-1990-0979942-6
- MathSciNet review: 979942