Highly stable parallel Volterra Runge-Kutta methods

This paper provides a class of quartic spline-on-spline interpolation S[k](x) on the function f(x), k = 0(1)4. Under the end conditions selected appropriately, we show that when the knots are equally spaced, with spacing h, then the order of accuracy of the spline-on-spline S[0](x), S[1](x) approximations f(x), f′(x) are O(h5), respectively, and the order of accuracy of the spline-on-spline S[k](x) approximations f(k)(x) are O(h4), k = 2, 3, 4.