A Note on Leapfrog Integration
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This slide gives a brief explanation of leapfrog integration, with an example and codes in Julia provided.
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A Note on Leapfrog Integration
- 1. A Note on Leapfrog Integration By Kai Xu Last updated on 15 Dec 2016
- 2. The problem setting Suppose we we want solve the following dynamic system = = F (x) = = f(x) ≜ v but we don't have an analytical solution to x(t). Then we can approximate it using leapfrog integration. x¨ dt2 d x2 x˙ dt dx
- 3. Leapfrog algorithm The leapfrog integration algorithm iterates the following three steps, given an initial state x(0): 1. x = x + v Δt 2. a = F (x) 3. v = v + a Δt After n iterations with some time step Δt, the sequence {x } is the approximatition of x(t) on time period [0, nΔt]. i i−1 i− 2 1 i i+ 2 1 i− 2 1 i i
- 4. An example - problem setting Suppose we are given = = 2 v = = = 2x and pretent to not know the analytical solution x(t) = t , given initial state x(0) = 0. x¨ dt2 d x2 x˙ dt dx 2
- 5. An example - result
- 6. An example - Julia code using Plots; pyplot() F(x) = 2; f(x) = 2 * x; x(t) = t ^ 2 ts = 0:0.1:2; xs = map(t -> x(t), ts) plot(ts, xs, lab="analytical", legendfont = font(13)) xᵢ = 0; dt = 0.1; v = f(xᵢ); ts = []; xs = [] for n in 1:20 xᵢ = xᵢ + v * dt a = F(xᵢ) v = v + a * dt push!(ts, (n-1) * dt); push!(xs, xᵢ) end plot!(ts, xs, lab="leapfrog", legendfont = font(13))