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 [1], [2] and [3]. This package is inspired by the Differential Equations package SciML.
References
- [1]
- P. Chartier, M. Lemou, F. Méhats and X. Zhao. Derivative-free High-order Uniformly Accurate Schemes for Highly-oscillatory Systems, submitted.
- [2]
- P. Chartier, N. Crouseilles, M. Lemou and F. Méhats. Uniformly accurate numerical schemes for highly oscillatory Klein-Gordon and nonlinear Schroedinger equations. Numerische Mathematik 129, 211–250 (2015).
- [3]
- N. Crouseilles, M. Lemou and F. Méhats. Asymptotic Preserving schemes for highly oscillatory Vlasov–Poisson equations. Journal of Computational Physics 248, 287–308 (2013).
- [4]
- S. Cox and P. Matthews. Exponential Time Differencing for Stiff Systems. Journal of Computational Physics 176, 430–455 (2002).