Contribution to the asymptotic estimation of the global error of one-step numerical integration methods: application to power system simulation.

Summary This work comes from an industrial problem of validation of numerical solutions of ordinary differential equations modeling electrical networks. The chosen approach is the asymptotic estimation of the global error. Four techniques are studied: the Richardson estimator (RS), the Zadunaisky estimator (ZD), the integration of the variational equation (EV) and the calculation of a global correction (SC). The relative order of convergence of these techniques is defined by the speed of convergence of the ratio between the estimator and the error when the discretization is refined. When this ratio is strictly positive, the estimator is said to be valid. We give details on the order of convergence of the SC estimator according to the order of the integration method it uses. To the ZD variants, we add an independent one using the Modified Equation. For variable step integration, the question remained whether ZD and SC retained their order of convergence with respect to the user tolerance. By restricting the type of local error control, we can answer in the affirmative. We further show that some Runge-Kutta methods require weaker assumptions to ensure the validity of this estimator. Numerical tests complete this analysis. They show a negative effect of the arithmetic error on some of these estimation techniques. When the global error reaches its minimum value, contrary to RS, the SC and ZD estimators underestimate it. Finally, an integrator is proposed which avoids the a priori specification of an integration path for complex time equations. Based on a local error control, it allows to automatically bypass isolated singularities.