index.md

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 Chartier2020, Chartier2015 and Crouseilles2013. This package is inspired by the Differential Equations package SciML.

References

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