Convergence and Stability Analysis for Modified Runge—Kutta Methods in the Numerical Treatment of Second-Kind Volterra Integral Equations

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Abstract
In this paper modified and conventional Runge—Kutta methods for second kind Volterra integral equations are discussed in a uniform way. The modification presented takes into account the residual of the previous step with the aim of improving the stability behaviour. A general convergence theorem is given which establishes that the modified methods may lose one order of accuracy. Furthermore, the stability behaviour of the methods is analysed and explicit stability results are derived. It transpires that every A-stable Runge—Kutta method for ordinary differential equations generates mixed methods which can be made A-stable by a suitable modification.