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""" | ||
runge_kutta_integration(f::Function, x0::Real, y0::Real, h::Real, x_stop::Real) | ||
Numerically solve initial value problems of the form ``y' = f(x, y)`` to find ``y(x)`` using the Runge-Kutta 4th order integration scheme. | ||
Starting with the differential equation ``\\frac{\\mathrm{d}y}{\\mathrm{d}x} = y' = f(x, y)`` and the initial condition ``y(x_0) = y_0``, each step calculates 4 approximations of the gradient | ||
```math | ||
\\begin{align*} | ||
k_1 &= f(x_n, y_n),\\\\ | ||
k_2 &= f(x_n + \\frac{h}{2}, y_n + k_1\\frac{h}{2}),\\\\ | ||
k_3 &= f(x_n + \\frac{h}{2}, y_n + k_2\\frac{h}{2}),\\\\ | ||
k_4 &= f(x_n + h, y_n + k_3h), | ||
\\end{align*} | ||
``` | ||
and uses the weighted average of them, | ||
```math | ||
\\bar{k} = \\frac{k_1 + 2k_2 + 2k_3 + k_4}{6}, | ||
``` | ||
to find the approximate value of ``y(x_{n+1})`` and update ``x`` and ``y`` accordingly | ||
```math | ||
\\begin{align*} | ||
x &\\rightarrow x_{n+1} = x_n + h\\\\ | ||
y &\\rightarrow y_{n+1} = y_n + h\\bar{k}. | ||
\\end{align*} | ||
``` | ||
# Arguments: | ||
- `f`: The function ``y' = f(x, y)`` to solve for ``y(x)``. | ||
- `x0`: The starting value of x. | ||
- `y0`: The starting value of y. | ||
- `h`: The step size, should be >0. | ||
- `x_stop`: The final value of x to solve to, i.e. integrate over the range `[x0, x_stop]` | ||
# Examples | ||
```julia | ||
julia> # If you have a constant slope of y' = 1, the analytic solution is y = x | ||
julia> runge_kutta_integration((x, y)->1, 0, 0, 1, 3) | ||
([0.0, 1.0, 2.0, 3.0], [0.0, 1.0, 2.0, 3.0]) | ||
julia> # Consider solving y' = exp(x), which has the analytic solution y = exp(x). | ||
julia> runge_kutta_integration((x, y)->exp(x), 0, 1., 0.1, 0.1) | ||
([0.0, 0.1], [1.0, 1.105170921726329]) | ||
julia> exp.([0.0, 0.1]) | ||
2-element Vector{Float64}: | ||
1.0 | ||
1.1051709180756477 | ||
``` | ||
# References | ||
- [https://en.wikipedia.org/wiki/Initial_value_problem](https://en.wikipedia.org/wiki/Initial_value_problem) | ||
- [https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods](https://en.wikipedia.org/wiki/Initial_value_problem) | ||
# Contributors: | ||
- [E-W-Jones](https://github.com/E-W-Jones) | ||
""" | ||
function runge_kutta_integration(f::Function, x0::Real, y0::Real, h::Real, x_stop::Real) | ||
h > 0 || throw(DomainError(h, "The step size `h` should be >0.")) | ||
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x = Float64(x0) | ||
y = Float64(y0) | ||
output_x = [x] | ||
output_y = [y] | ||
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while x < x_stop | ||
k1 = f(x, y) | ||
k2 = f(x + h/2, y + k1*h/2) | ||
k3 = f(x + h/2, y + k2*h/2) | ||
k4 = f(x + h , y + k3*h ) | ||
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y += h * (k1 + 2k2 + 2k3 + k4) / 6 | ||
x += h | ||
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push!(output_x, x) | ||
push!(output_y, y) | ||
end | ||
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return output_x, output_y | ||
end |
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