HOODESolver.jl

A Julia package for solving numerically highly-oscillatory ODE problems

This Julia package implements Uniformly Accurate numerical methods (UA) for highly oscillating problems. We propose to solve the following equation:

$$\frac{d u(t)}{dt} = \frac{1}{\varepsilon} A u(t) + f(t, u(t)), \qquad u(t=t_{start})=u_{in}, \qquad \varepsilon \in ]0, 1],$$

with

$u : t \in [t_{start}, t_{end}] \mapsto u(t) \in \mathbb{R}^n, t_{start}, t_{end} \in \mathbb{R}, \qquad u_{in}\in \mathbb{R}^n$,

$A\in {\mathcal{M}}_{n,n}(\mathbb{R})\quad$ is such that $\quad \tau \mapsto e^{\tau A}\quad$ is $2 \pi$-periodic, $\quad f : (t, u) \in \mathbb{R} \times \mathbb{R}^n \mapsto \mathbb{R}^n$.

HOODESolver.jl is a registered package and can be installed using the Julia package manager. From the Julia REPL, enter Pkg mode (by pressing ])

julia>]
(@v1.5) pkg> add HOODESolver

The following is an example with the system of Hénon-Heiles. Please see the documentation for further usage, tutorials, and api reference.

using HOODESolver
using Plots

epsilon= 0.0001

A = [ 0 0 1 0 ; 
      0 0 0 0 ; 
     -1 0 0 0 ; 
      0 0 0 0 ]

f1 = LinearHOODEOperator( epsilon, A)

f2 = (u,p,t) ->  [ 0, u[4], 2*u[1]*u[2], -u[2] - u[1]^2 + u[2]^2 ] 

tspan = (0.0, 3.0)

u0 = [0.55, 0.12, 0.03, 0.89]
prob = SplitODEProblem(f1, f2, u0, tspan)

solve the defined problem

sol = solve(prob, HOODEAB()) 
plot(sol) 

For support with using HOODESolver.jl, please open an issue describing the problem and steps to reproduce it.

Here's an outline of the workflow you should use if you want to make contributions to this package.

1. Fork this repository
2. Make a new branch on your fork, named after whatever changes you'll be making
3. Apply your code changes to the branch on your fork
4. When you're done, submit a PR to HOODESolver.jl to merge your fork into master branch.

This package is licensed under the MIT Expat license. See LICENSE for more information.