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

Yuan, Wei; Tang, Tao

doi:10.1090/S0025-5718-1990-0979942-6

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

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