HOODESolver.jl
The objective of this Julia package is to valorize the recent developments carried out within INRIA team MINGuS on Uniformly Accurate numerical methods (UA) for highly oscillating problems. We propose to solve the following equation
\[\frac{d u(t)}{dt} = \frac{1}{\varepsilon} A u(t) + f(t, u(t)), \qquad u(t=t_{start})=u_{in}, \qquad \varepsilon\in ]0, 1], \qquad (1)\]
with
- $u : t\in [t_{start}, t_{end}] \mapsto u(t)\in \mathbb{R}^n, \quad t_{start}, t_{end}\in \mathbb{R}$,
- $u_{in}\in \mathbb{R}^n$,
- $A\in {\mathcal{M}}_{n,n}(\mathbb{R})$ is such that $\tau \mapsto \exp(\tau A)$ is $2 \pi$-periodic,
- $f : (t, u) \in \mathbb{R}\times \mathbb{R}^n \mapsto \mathbb{R}^n$.
The purpose here is to write an explanatory documentation of the Julia package containing the two-scale method (see Philippe Chartier, Mohammed Lemou, Florian Méhats, Xiaofei Zhao (2020), Philippe Chartier, Nicolas Crouseilles, Mohammed Lemou, Florian Méhats (2015) and Nicolas Crouseilles, Mohammed Lemou, Florian Méhats (2013). This package is inspired by the Differential Equations package SciML.
References
- Chartier2015
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Philippe Chartier, Nicolas Crouseilles, Mohammed Lemou, Florian Méhats, Uniformly accurate numerical schemes for highly oscillatory Klein-Gordon and nonlinear Schroedinger equations, Numerische Mathematik, 129(2), 211-250, 2015.
- Chartier2020
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Philippe Chartier, Mohammed Lemou, Florian Méhats, Xiaofei Zhao, Derivative-free High-order Uniformly Accurate Schemes for Highly-oscillatory Systems, submitted, 2020.
- Crouseilles2013
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Nicolas Crouseilles, Mohammed Lemou, Florian Méhats, Asymptotic Preserving schemes for highly oscillatory Vlasov–Poisson equations, Journal of Computational Physics, 248, 287 - 308, 2013.
- Cox2002
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S.M. Cox, P.C. Matthews, Exponential Time Differencing for Stiff Systems, Journal of Computational Physics, 176(2), 430 - 455, 2002.