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

Abstract

Implicit Runge-Kutta methods with <italic>m</italic> stages and optimal order <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p equals 2 m"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> <mml:mi>m</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">p = 2m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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 <italic>m</italic> ) several optimal methods.