Convergence of the Variable-step Variable-order 3-stage Hermite–birkhoff Ode/dde Solver of Order 5 to 15

The ordinary and delay differential equations (ODEs/DDEs) solver HB515DDE is based on the discrete hybrid variable-step variable-order 3-stage Hermite–Birkhoff ODE solver of (consistency) order 5 to 15. The current version of the solver can handle ODEs and DDEs with statedependent, non-vanishing, small, vanishing and asymptotically vanishing delays. Delayed values are computed using Hermite interpolation and small delays are dealt with using extrapolation. Discontinuities in DDEs are located by a bisection method. HB515DDE has proved itself superior to many other known DDE solver as Matlab’s ddesd. This article presents the theory behind HB515DDE including convergence proofs for the ODE and DDE parts separately. It is shown that the method is convergent of order 5 when seen as an ODE method and of order 3 when seen as a DDE solver under some assumptions. Résumé Le solveur d’équations différentielles avec ou sans retard, HB515DDE, provient d’un solveur hybride à pas et ordre variables du type Hermite– Birkhoff à 3 étages d’ordre 5 à 15. HB515DDE peut résoudre des ODEs et DDEs avec retards spatio-variables, infiniment petits ou non, et asymptotiquement infiniment petits. On calcule les retards au moyen d’interpolation d’Hermite. HB515DDE est supérieur à plusieurs solveurs de DDEs connus, tel ddesd de Matlab. On présente la théorie de HB515DDE et l’on démontre sa convergence d’ordre 5 pour les ODEs et 3 pour les DDEs.