Validation of results of collocation methods for ODEs with the CADNA library

Abstract

This paper shows the efficiency of collocation methods for integrating ordinary differential system when they are performed with an efficient round-off error analysis method. In a first part collocation methods are briefly presented and some of their properties are recalled. Particularly, it is shown that they can be used as parallel across time methods. Then the CESTAC method for round-off error analysis, stochastic arithmetic and some of their properties are also recalled. Examples of linear and nonlinear initial values problems are solved with collocation methods, first using the usual floating-point arithmetic and then using the CADNA library which automatically implements the CESTAC method. It is clearly shown that the encountered problems, i.e., the choice of the mesh in the integrating interval and the termination criterion for stopping the iterative process may be solved thanks to the CADNA library. The obtained results highlight that collocation methods considered as parallel methods across time and performed with the CADNA library are an efficient tool for solving ODEs.