Create landau_damping.jl

src/landau_damping.jl

@@ -0,0 +1,87 @@
+import Sobol
+
+
+"""
+```math
+    f(x) = \\frac{1]{2\\pi} (1 + \\alpha cos(k x)) 
+    \\exp \\big( - \\frac{ (v - \\mu)^2 }{ \\sigma^2} \\big)
+```
+"""
+struct LandauDamping
+
+    α :: Float64
+    k :: Float64
+    σ :: Float64
+
+end
+
+"""
+Input r is a random number ``\\in [0,1]``
+
+```math
+    f(x) = 1 + \\alpha cos(k x)
+```
+on some domain ``[0, 2\\pi/k]``
+
+Solve the equation ``P(x)-r=0`` with Newton’s method
+
+```math
+    x^{n+1} = x^n – (P(x)-(2\\pi r / k)/f(x) 
+```
+
+with 
+```math
+P(x) = \\int_0^x (1 + \\alpha cos(k y)) dy
+```
+```math
+P(x) = x + \\frac{\\alpha}{k} sin (k x)
+```
+"""
+function newton( ld :: LandauDamping, r)
+    x0, x1 = 0.0, 1.0
+    alpha, k = ld.α, ld.k
+    r *= 2π / k
+    while (abs(x1-x0) > 1e-12)
+        p = x0 + alpha * sin( k * x0) / k 
+        f = 1 + alpha * cos( k * x0)
+        x0, x1 = x1, x0 - (p - r) / f
+    end
+    x1
+end
+
+"""
+    sample!( landau, xp, yp, domain )
+
+Particle sampling for Landau damping initial distribution
+function. 
+- `xp` :: preallocated array containing particles positions
+- `vp` :: preallocated array containing particles velocities
+"""
+function sample!( ld :: LandauDamping, 
+                  xp :: Vector{Float64}, 
+                  vp :: Vector{Float64},
+                  domain :: Vector{Float64} )
+
+   nbpart = length(xp)
+   @assert length(xp) == length(vp)
+
+   s = SobolSeq(2)
+
+   for k=0:nbpart-1
+
+	  v = ld.σ * sqrt(-2 * log( (ld.k+0.5)/nbpart))
+      r1, r2 = Sobol.next!(s)
+      θ = r1 * 2π
+      xp[k+1] =  newton(r2)
+      vp[k+1] =  v * sin(θ)
+
+   end
+
+end
+
+#xp, vp = landau(100000);
+# -
+#
+#p = histogram([xp,vp], normalize=true, bins = 100,  layout=(2,1), lab = "draws")
+#plot!(p[1,1], x-> (1+0.1*cos(0.5*x))/4π, 0., 4π)
+#plot!(p[2,1], x-> (exp(-x^2/2))/sqrt(2π), -6, 6)