Comparison between Borel-Padé summation and factorial series, as time integration methods

Abstract

We compare the performance of two algorithms of computing the Borel sum of a time power series. The first one uses Padé approximants in Borel space, followed by a Laplace transform. The second is based on factorial series. These algorithms are incorporated in a numerical scheme for time integration of differential equations.

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

“…βv is the mortality rate of prey depending on the the number v of predators and δu is the birth rate of predators depending on the number of prey eaten. It is straight forward to show that system (30) possesses the first integral:…”

“…We now examine the behaviour of the schemes when the stiffness ratio varies. The stiffness ratio r is defined as the spectral condition number of the linear part of equations (30), that is…”

“…It makes use of a Padé approximation, as will be seen later, and is accordingly named Borel-Padé-Laplace algorithm (BPL). A representation of the Borel sum as an inverse factorial series also leads to an efficient algorithm [30] but will not be used.…”

Some robust integrators for large time dynamics

“…Note that at each time, the approximate solution has an analytical representation as a Laplace integral. A continued fraction representation can also be used [8].…”

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

“…The Borel-Laplace algorithm that will be discussed here results from the representation of the Borel sum as a Laplace integral. A representation as a factorial series also leads to an efficient algorithm [23] but will not be used.…”