NSDERungeKutta.jl

A Julia package implementing Runge-Kutta methods.

Installation

This package is compatible with Julia v1.10 and above. From the Julia REPL,

]add https://github.com/giancarloantonucci/NSDERungeKutta.jl

Read the documentation for a complete overview of this package.

Usage

Let's say that we want to solve the simple gravity pendulum problem using the midpoint method. Here is one way to do it with NSDERungeKutta.jl:

using NSDERungeKutta
f(u, t) = [u[2]; -sin(u[1])]
u0 = [0.0; π/4]
tspan = (0.0, 2π/3 * √9.81)
problem = IVP(f, u0, tspan)
solver = Midpoint(h = 1e-2)
solution = solve(problem, solver)

We can plot the obtained solution by extracting its fields u and t, e.g. using the convenient macro @↓ u, t = solution from ArrowMacros.jl. Alternatively, we can use the available predefined recipes:

using Plots, LaTeXStrings
gr(fontfamily = "Computer Modern", framestyle = :box, label = "", tickdirection = :out)
p₁ = plot(solution, xlabel = L"t", label = [L"\theta" L"\omega"])
p₂ = phaseplot(solution, variables = (1, 2), xlabel = L"\theta", ylabel = L"\omega")
plot(size = (900, 450), p₁, p₂, left_margin = 3Plots.mm, bottom_margin = 3Plots.mm)

For convenience, this package re-exports all the ODE problems defined in NSDEBase.jl, e.g. SimplePendulum for the above problem.

This package has some predefined recipes to plot stability regions and order stars too:

p₁ = stabilityf(RK4(), xlabel = L"\Re(z)", ylabel = L"\Im(z)", colour = :blues, resolution = 500)
p₂ = orderstarf(RK4(), xlabel = L"\Re(z)", ylabel = L"\Im(z)", colour = :blues, resolution = 500)
plot(size = (1000, 400), p₁, p₂, left_margin = 5Plots.mm, bottom_margin = 5Plots.mm)

Methods

This package currently supports the following methods:

Explicit:

• Euler/ExplicitEuler, Heun2, Midpoint/ExplicitMidpoint, Ralston2, Heun3, RungeKutta3/RK3, Ralston3, SSPRK3, Ralston4, RungeKutta4/RK4, Rule38, Butcher5, KuttaNystrom5, Butcher6, Butcher7,
• (Embedded) HeunEuler, BogackiShampine, Fehlberg45, DormandPrince54, Verner65, Fehlberg78.

Diagonally Implicit:

• BackwardEuler/ImplicitEuler, ImplicitMidpoint/GaussLegendre2, SDIRK2, LobattoIII2, CrankNicolson/LobattoIIIA2, SDIRK3, RadauI3, RadauII3, SDIRK4, LobattoIII4.

Implicit:

• LobattoIIIC2, RadauIA3, RadauIIA3, GaussLegendre4, LobattoIIIA4, LobattoIIIB4, LobattoIIIC4, RadauI5, RadauIA5, RadauII5, RadauIIA5, GaussLegendre6.