Hermite–Birkhoff spline Quasi-Interpolation with application as dense output for Gauss–Legendre and Gauss–Lobatto Runge–Kutta schemes

Spline Quasi-Interpolation (QI) of even degree 2R on general partitions is introduced, where derivative information up to order R≥1 at the spline breakpoints is required and maximal convergence order can be proved. Relying on the B-spline basis with possible multiple inner knots, a family of quasi-interpolating splines with smoothness of order R is associated with each R≥1, since there is the possibility of using different local sequences of breakpoints to define each QI spline coefficient. By using suitable finite differences approximations of the necessary discrete derivative information, each QI spline in this family can be associated also with a twin approximant belonging to the same spline space, but requiring just function information at the breakpoints. Among possible different applications of the introduced QI scheme, a smooth continuous extension of the numerical solution of Gauss-Lobatto and Gauss-Legendre Runge-Kutta methods is here considered. When R>1, such extension is based on the use of the variant of the QI scheme with derivative approximation which preserves the approximation power of the original Runge-Kutta scheme.