ABSTRACT

We here describe, in some detail, the actual implementation of HBVMs. In fact, in order to exploit all the potentialities of the methods, they must be carefully and efficiently coded.

As was seen in Section 3.2, a HBVMpk, sq method is a k-stage implicit Runge-Kutta method of order 2s. In view of the result presented in Theorem 3.4 on page 91, k is usually chosen (arbitrarily) larger than s, thus it is not convenient to implement a HBVM the way it is done for a classical RungeKutta method, since the discrete problem generated by the computation of the stages has (block) dimension k. We shall instead compute the s coefficients γˆj of the corresponding polynomial approximation (see (3.22)–(3.24) on page 89)

upchq “ y0 ` h s´1ÿ j“0

Pjpxqdx γˆj , c P r0, 1s.