Borel-Laplace summation method used as time integration scheme

Abstract

Abstract.A time integration method for the resolution of ordinary and partial differential equations is proposed. The method consists in computing a formal solution as a (eventually divergent) time series. Next, the Borel resummation method is applied to deduce an (sectorial) analytical solution. The speed and spectral properties of the scheme are analyzed through some examples. Applications to fluid mechanics are presented.Résumé. On propose une méthode numérique d'intégration temporelle d'équations différentielles ou aux dérivées partielles. Cette méthode consiste d'abordà calculer une solution sous forme de série formelle, dont le rayon de convergence peutêtre nul. Ensuite, la méthode de resommation de BorelLaplace est utilisée pour déduire une solution analytique (dans un secteur) de l'équation. La rapidité et les propriétés géométriques du schéma sont analyséesà travers quelques exemples. Des applications en mécanique des fluides sont présentées.

Performance of Borel–Padé–Laplace integrator for the solution of stiff and non-stiff problems

“…For example, it generally allows much larger time steps than other explicit methods for the resolution of many problems [28]. Its ability to cross some types of singularities, its high-order symplecticity, or its high-order iso-spectrality in solving a Lax pair problem have been stated in [29]. Another advantage of BL is that decreasing or increasing the approximation order is as simple as changing the value of a parameter in the code.…”

Some robust integrators for large time dynamics

“…The theory of Borel summation can be found, for example, in [2,[19][20][21]. Some other works on BPL, as a time integrator, can be found in [7,22]. In this section, we present very briefly the Borel-Padé-Laplace algorithm, integrated into a numerical scheme.…”

Performance of Borel-Padé-Laplace integrator for the resolution of stiff and non-stiff problems

“…For example, it generally allows much bigger time steps than other explicit methods for the resolution of many problems [21]. Its ability to cross some types of singularities, its high-order symplecticity, or its high-order iso-spectrality in solving a Lax pair problem have been stated in [22]. Another advantage of BL is that lowering or raising the approximation order is as simple as changing the value of a parameter in the code.…”