Derivative-free high-order uniformly accurate schemes for highly oscillatory systems

Abstract

In this paper we address the computational aspects of uniformly accurate numerical methods for solving highly oscillatory evolution equations. In particular, we introduce an approximation strategy that allows the construction of arbitrary high-order methods using solely the right-hand side of the differential equation. No derivative of the vector field is required, while uniform accuracy is retained. The strategy is then applied to two different formulations of the problem, namely the two-scale and the micro–macro formulations. Numerical experiments on the Hénon–Heiles system, as well as on the Klein–Gordon equation and a Vlasov-type problem, confirm the validity of the new strategy.

HOODESolver.jl: A Julia package for highly oscillatory problems

“…Chartier et al, 2015P. Chartier et al, , 2020Crouseilles et al, 2017); its development has been motivated by these research needs and it has already been used in some papers (Philippe Chartier et al, 2021). HOODESolver.jl provides software implementations of several theoretical ideas contained in the recent literature around the so-called two-scale method.…”

“…HOODESolver.jl provides software implementations of several theoretical ideas contained in the recent literature around the so-called two-scale method. In particular, a very recent extension proposed in (Philippe Chartier et al, 2021) enables to reach high order accuracy. The implementation focuses on a multistep method (namely Adams-Bashforth method) coupled with a spectral method for the discretization of the additional variable representing the fast scale.…”

A uniformly accurate numerical method for a class of dissipative systems

“…The theory developed therein is however of no relevance for the construction of micro-macro decompositions as it relies heavily on trees and associated elementary differentials which can hardly be computed in practice. Our approach actually shares more similarities with the one introduced for highly-oscillatory problems in [CLMV19] and later modified to become amenable for actual computations at any order [CLMZ20]. As a matter of fact, the technical arguments that sustain decomposition (1.4) are essentially adapted from [CCMM15] in a way that will be fully explained in Section 3.…”

Multiscale Modeling of Sorption Kinetics

“…This restriction becomes prohibitive for small values of ε. We will follow the strategy adopted in [5,6,7,12,13] although in the different contexts of Vlasov-Poisson equations, Klein-Gordon and nonlinear Schr 'odinger equations, to obtain a robust scheme that is able to deal with a large range of ε ∈ (0, 1] (being small or not), since our goal is to obtain a numerical scheme that is uniformly accurate in ε.…”