Vlasov-Poisson

We consider the dimensionless Vlasov-Poisson equation for one species with a neutralizing background.

\[ \frac{∂f}{∂t}+ v⋅∇_x f + E(t,x) ⋅ ∇_v f = 0, \\ - Δϕ = 1 - ρ, E = - ∇ ϕ \\ ρ(t,x) = ∫ f(t,x,v) dv\]

using Plots, LinearAlgebra
import Splittings:UniformMesh, BSpline
import Splittings:@Strang
import Splittings

function push_t!(f, mesh1, v, n2, dt)
    Splittings.advection!(f, mesh1, v, n2, dt, BSpline(5))
end

push_t! (generic function with 1 method)

function push_v!(f, fᵗ, mesh1, mesh2, nrj, dt)
    rho = Splittings.compute_rho(mesh2, f)
    e   = Splittings.compute_e(mesh1, rho)
    push!(nrj, 0.5*log(sum(e.*e)*mesh1.step))
    transpose!(fᵗ, f)
    Splittings.advection!(fᵗ, mesh2, e, mesh1.length, dt, BSpline(5))
    transpose!(f, fᵗ)
end

push_v! (generic function with 1 method)

function landau(tf, nt)

  n1, n2 = 32, 64
  x1min, x1max = 0.0, 4π
  x2min, x2max = -6., 6.
  mesh1 = UniformMesh(x1min, x1max, n1; endpoint=false)
  mesh2 = UniformMesh(x2min, x2max, n2; endpoint=false)
  x = mesh1.points
  v = mesh2.points

  ϵ, kx = 0.001, 0.5
  f = zeros(Complex{Float64},(n1,n2))
  f .= (1.0.+ϵ*cos.(kx*x))/sqrt(2π) * transpose(exp.(-0.5*v.^2))
  fᵗ = zeros(Complex{Float64},(n2,n1))

  dt = tf / nt

  nrj = Float64[]

  for it in 1:nt
      @Strang( push_t!(f, mesh1, v, n2, dt),
               push_v!(f, fᵗ, mesh1, mesh2, nrj, dt))
  end

  nrj

end

landau (generic function with 1 method)

nt = 500
tf = 50.0
t  = range(0.0, stop=tf, length=nt)
@time nrj = landau(tf, nt)
plot( t, nrj)
plot!(t, -0.1533*t.-5.50)

  5.731592 seconds (9.43 M allocations: 528.477 MiB, 4.46% gc time)

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