LSODA.jl: the LSODA algorithm for solving ordinary differential equations
Introduction
LSODA.jl is a Julia package that interfaces to the liblsoda library, developped by Simon Frost (@sdwfrost), thereby providing a way to use the LSODA algorithm from Linda Petzold and Alan Hindmarsh from Julia. Clang.jl has been used to write the library and Sundials.jl was a inspiring source.
Installation
To install this package, run the command Pkg.clone("https://github.com/rveltz/LSODA.jl.git")
Example
We first need to load the library.
```example 1 using LSODA
We next need to define a function that provides the derivatives `ydot` given the time, `t`, initial conditions, `y`, and optionally some additional data, `data`. Note that this function modifies `ydot` in-place and returns `nothing`.
```example 1
function rhs!(t, y, ydot, data)
ydot[1]=1.0E4 * y[2] * y[3] - .04E0 * y[1]
ydot[3]=3.0E7 * y[2] * y[2]
ydot[2]=-ydot[1] - ydot[3]
nothing
end
The model can be solved by providing an initial condition for the state variables, and a time span over which to simulate.
```example 1 y0 = [1.,0.,0.] tspan = [0., 0.4] res = lsoda(rhs!, y0, tspan, reltol= 1e-4, abstol = Vector([1.e-6,1.e-10,1.e-6]))
This should give the following.
```example 1
at t = 4.0000e-01 y= 9.851712e-01 3.386380e-05 1.479493e-02
at t = 4.0000e+00 y= 9.055333e-01 2.240655e-05 9.444430e-02
at t = 4.0000e+01 y= 7.158403e-01 9.186334e-06 2.841505e-01
at t = 4.0000e+02 y= 4.505250e-01 3.222964e-06 5.494717e-01
at t = 4.0000e+03 y= 1.831976e-01 8.941774e-07 8.168016e-01
at t = 4.0000e+04 y= 3.898729e-02 1.621940e-07 9.610125e-01
at t = 4.0000e+05 y= 4.936362e-03 1.984221e-08 9.950636e-01
at t = 4.0000e+06 y= 5.161832e-04 2.065786e-09 9.994838e-01
at t = 4.0000e+07 y= 5.179811e-05 2.072030e-10 9.999482e-01
at t = 4.0000e+08 y= 5.283524e-06 2.113420e-11 9.999947e-01
at t = 4.0000e+09 y= 4.658945e-07 1.863579e-12 9.999995e-01
at t = 4.0000e+10 y= 1.423392e-08 5.693574e-14 1.000000e+00
Application programming interface
Functions
#
LSODA.lsoda
— Type.
lsoda(f::Function, y0::Vector{Float64}, tspan::Vector{Float64}; userdata::Any=nothing, reltol::Union{Float64,Vector}=1e-4, abstol::Union{Float64,Vector}=1e-10)
Solves a set of ordinary differential equations using the LSODA algorithm. The vector field encoded in an inplace f::Function needs to have the self-explanatory arguments f(t, y, ydot, data)
#
LSODA.lsoda_evolve!
— Function.
lsoda_evolve!(ctx::lsoda_context_t,y::Vector{Float64},tspan::Vector{Float64})
Solves a set of ordinary differential equations using the LSODA algorithm and the context variable ctx. This avoid re-allocating ctx. You have to be carefull to remember the current time or this function will return an error.