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+{
+ "cells": [
+ {
+ "cell_type": "code",
+ "execution_count": 17,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "using Plots, LinearAlgebra, Statistics, BenchmarkTools\n",
+ "using HermiteGF\n",
+ "using SparseArrays"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {
+ "slideshow": {
+ "slide_type": "slide"
+ }
+ },
+ "source": [
+ "# Vlasov-Poisson equation\n",
+ "We consider the dimensionless Vlasov-Poisson equation for one species\n",
+ "with a neutralizing background.\n",
+ "\n",
+ "$$ \n",
+ "\\frac{\\partial f}{\\partial t}+ v\\cdot \\nabla_x f + E(t,x) \\cdot \\nabla_v f = 0, \\\\\n",
+ "- \\Delta \\phi = 1 - \\rho, E = - \\nabla \\phi \\\\\n",
+ "\\rho(t,x) = \\int f(t,x,v)dv.\n",
+ "$$\n",
+ "\n",
+ "- [Vlasov Equation - Wikipedia](https://en.wikipedia.org/wiki/Vlasov_equation)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 18,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "SparseMatrixCSC{Float64,Int64}"
+ ]
+ },
+ "execution_count": 18,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "nx = 64\n",
+ "matrix = spdiagm(-1 => ones(Float64,nx-2),\n",
+ " 0 => -2*ones(Float64,nx),\n",
+ " 1 => ones(Float64,nx-2))\n",
+ "\n",
+ "typeof(matrix)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 19,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "struct UniformMesh\n",
+ " nx :: Int64\n",
+ " dx :: Float64\n",
+ " x :: Vector{Float64}\n",
+ " function UniformMesh( xmin, xmax, nx)\n",
+ " dx = (xmax - xmin)/ nx\n",
+ " x = range(xmin, stop=xmax, length=nx+1)[1:end-1]\n",
+ " new( nx, dx, x)\n",
+ " end\n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 21,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "struct ChebyshevMesh\n",
+ " nx :: Int64\n",
+ " x :: Vector{Float64}\n",
+ " dx :: Vector{Float64}\n",
+ " function ChebyshevMesh( nodes::HermiteGF.NodesType)\n",
+ " dx =[ x1 - x0 for (x1,x0) in zip(nodes.xk[2:end],nodes[1:end-1]) ]\n",
+ " x = nodes.xk\n",
+ " new( nx, dx, x)\n",
+ " end\n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "$$\n",
+ "f_i'' \\approx \\frac{2 \\Big[ f_{i+1}\n",
+ "- \\big(1+\\frac{h_i}{h_{i-1}}\\big) f_i\n",
+ "+\\frac{h_i}{h_{i-1}}f_{i-1} \n",
+ " \\Big]}\n",
+ "{ h_i h_{i-1} (1+\\frac{h_i}{h_{i-1}})}\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "## Julia implementation of solving Poisson on a square grid using Chebyshev collocation\n",
+ "# julia> load(\"logg.jl\")\n",
+ "# julia> loggProfile(10000)\n",
+ "# error: 3.310154100518143e-7\n",
+ "# Average time (us): 132.39421844482422\n",
+ "\n",
+ "function logg(m)\n",
+ " (D,xb) = cheb(m+1) # Chebyshev differentiation matrix and nodes\n",
+ " D = 2*D; xb = 0.5 .+ 0.5*xb # Remap to interval [0,1]\n",
+ " D2 = -(D*D)[2:end-1,2:end-1] # Negative 1D Laplacian, with Dirichlet BCs\n",
+ " xd = xb[2:end-1] # Remove Dirichlet BCs\n",
+ " sinx = sin.(pi*xd'')\n",
+ " xexact = kron(sinx', sinx) # sin(pi*x)*sin(pi*y)\n",
+ " b = 2*pi^2*xexact\n",
+ " # Compute action of inverse of L using sum factorization\n",
+ " # See Lynch, Rice, and Thomas (1964)\n",
+ " (Lambda, R) = eig(D2)\n",
+ " e = ones(m)\n",
+ " Diag = kron(Lambda,e) + kron(e,Lambda)\n",
+ " iDiag = 1/Diag\n",
+ " Rinv = inv(R)\n",
+ " # L = kron(D2,eye(m)) + kron(eye(m),D2)\n",
+ " # Linv = kron(R,R) * diagm(reshape(iDiag,m^2,1)) * kron(Rinv,Rinv)\n",
+ " x = R * (iDiag .* (Rinv*b*Rinv')) * R' # x = Linv * b (with reshaping)\n",
+ " norm(reshape(x - xexact,m^2,1)) # L2 norm of the vector\n",
+ "end\n",
+ "\n",
+ "function cheb(N)\n",
+ " x = cos.(pi*(0:N)/N)\n",
+ " c = ones(N+1)\n",
+ " c[1] = 2\n",
+ " c[end] = 2\n",
+ " for i in 2:2:N+1\n",
+ " c[i] = - c[i]\n",
+ " end\n",
+ " X = repeat(x,1,N+1)\n",
+ " dX = X .- X'\n",
+ " D = (c*(1/c')) ./ (dX+eye(N+1)) # off-diagonal entries\n",
+ " D = D - diagm(sum(D,dims=2)) # diagonal entries\n",
+ " (D, x)\n",
+ "end\n",
+ "\n"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 36,
+ "metadata": {},
+ "outputs": [
+ {
+ "ename": "MethodError",
+ "evalue": "MethodError: no method matching /(::Int64, ::Adjoint{Float64,Array{Float64,1}})\nClosest candidates are:\n /(::Union{Int128, Int16, Int32, Int64, Int8, UInt128, UInt16, UInt32, UInt64, UInt8}, !Matched::Union{Int128, Int16, Int32, Int64, Int8, UInt128, UInt16, UInt32, UInt64, UInt8}) at int.jl:59\n /(::Union{Int16, Int32, Int64, Int8, UInt16, UInt32, UInt64, UInt8}, !Matched::BigInt) at gmp.jl:466\n /(::T<:Integer, !Matched::T<:Integer) where T<:Integer at int.jl:57\n ...",
+ "output_type": "error",
+ "traceback": [
+ "MethodError: no method matching /(::Int64, ::Adjoint{Float64,Array{Float64,1}})\nClosest candidates are:\n /(::Union{Int128, Int16, Int32, Int64, Int8, UInt128, UInt16, UInt32, UInt64, UInt8}, !Matched::Union{Int128, Int16, Int32, Int64, Int8, UInt128, UInt16, UInt32, UInt64, UInt8}) at int.jl:59\n /(::Union{Int16, Int32, Int64, Int8, UInt16, UInt32, UInt64, UInt8}, !Matched::BigInt) at gmp.jl:466\n /(::T<:Integer, !Matched::T<:Integer) where T<:Integer at int.jl:57\n ...",
+ "",
+ "Stacktrace:",
+ " [1] cheb(::Int64) at ./In[34]:33",
+ " [2] logg(::Int64) at ./In[34]:8",
+ " [3] top-level scope at util.jl:156",
+ " [4] top-level scope at In[36]:1"
+ ]
+ }
+ ],
+ "source": [
+ "@time logg(7)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 4,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "struct Poisson\n",
+ " \n",
+ " nx :: Int64\n",
+ " dx :: Float64\n",
+ " Φ :: Array{Float64,1}\n",
+ " matrix :: SparseMatrixCSC{Float64,Int64}\n",
+ " \n",
+ " function Poisson( meshx )\n",
+ " nx = meshx.nx\n",
+ " dx = meshx.dx\n",
+ " Φ = zeros(Float64,nx)\n",
+ " matrix = spdiagm(-1 => -ones(Float64,nx-2),\n",
+ " 0 => +2*ones(Float64,nx),\n",
+ " 1 => -ones(Float64,nx-2))\n",
+ " new( nx, dx, Φ, matrix)\n",
+ "\n",
+ " end\n",
+ " \n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 10,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "function (p::Poisson)( rho::Array{Float64,1} )\n",
+ " p.Φ .= p.matrix \\ rho\n",
+ " (circshift(p.Φ, 1) - circshift(p.Φ, -1)) ./ (2*p.dx)\n",
+ "end "
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 11,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "compute_rho"
+ ]
+ },
+ "execution_count": 11,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "\"\"\"\n",
+ "Compute charge density\n",
+ "ρ(x,t) = ∫ f(x,v,t) dv\n",
+ "\"\"\"\n",
+ "function compute_rho(meshv::UniformMesh, \n",
+ " f::Array{Complex{Float64},2})\n",
+ " \n",
+ " dv = meshv.dx\n",
+ " rho = dv * vec(sum(real(f), dims=2))\n",
+ " rho .- mean(rho)\n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Landau Damping\n",
+ "\n",
+ "[Landau damping - Wikipedia](https://en.wikipedia.org/wiki/Landau_damping)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 12,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "landau (generic function with 1 method)"
+ ]
+ },
+ "execution_count": 12,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "function landau(tf::Float64, nt::Int64)\n",
+ " \n",
+ " nx, nv = 128, 256\n",
+ " xmin, xmax = 0.0, 4*pi\n",
+ " vmin, vmax = -6., 6.\n",
+ " meshx = UniformMesh(xmin, xmax, nx)\n",
+ " meshv = UniformMesh(vmin, vmax, nv)\n",
+ " x = meshx.x\n",
+ " v = meshv.x\n",
+ " dx = meshx.dx\n",
+ " \n",
+ " # Create Vlasov-Poisson simulation\n",
+ " poisson = Poisson(meshx)\n",
+ " \n",
+ " eps, kx = 0.001, 0.5\n",
+ " f = zeros(Complex{Float64},(nx,nv))\n",
+ " f .= (1.0.+eps*cos.(kx*x))/sqrt(2π) * transpose(exp.(-0.5*v.^2))\n",
+ "\n",
+ " ρ = compute_rho( meshv, f)\n",
+ " \n",
+ " e = poisson( ρ )\n",
+ " \n",
+ " ## Set time domain\n",
+ " #dt = tf / nt\n",
+ " #\n",
+ " ## Run simulation\n",
+ " #ℰ = Float64[]\n",
+ " #\n",
+ " #for it in 1:nt\n",
+ " # advection!(f, p, meshx, v, nv, 0.5*dt)\n",
+ " # rho = compute_rho(meshv, f)\n",
+ " # e = compute_e(meshx, rho)\n",
+ " # push!(ℰ, 0.5*log(sum(e.*e)*dx))\n",
+ " # transpose!(fᵗ, f)\n",
+ " # advection!(fᵗ, p, meshv, e, nx, dt)\n",
+ " # transpose!(f, fᵗ)\n",
+ " # advection!(f, p, meshx, v, nv, 0.5*dt)\n",
+ " #end\n",
+ " # \n",
+ " #ℰ\n",
+ " meshx.x, e\n",
+ "\n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 13,
+ "metadata": {},
+ "outputs": [
+ {
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+ ]
+ },
+ "execution_count": 13,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "nt = 1000\n",
+ "tf = 100.0\n",
+ "t = range(0.0, stop=tf, length=nt)\n",
+ "x, e = landau(tf, nt)\n",
+ "plot( x, e)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 9,
+ "metadata": {},
+ "outputs": [
+ {
+ "ename": "MethodError",
+ "evalue": "MethodError: no method matching adjoint(::Type{Poisson})\nClosest candidates are:\n adjoint(!Matched::Missing) at missing.jl:79\n adjoint(!Matched::Number) at number.jl:193\n adjoint(!Matched::Adjoint{#s177,#s176} where #s176<:Union{StaticArray{Tuple{N},T,1} where T where N, StaticArray{Tuple{N,M},T,2} where T where M where N} where #s177) at /Users/navaro/.julia/packages/StaticArrays/WmJnA/src/linalg.jl:78\n ...",
+ "output_type": "error",
+ "traceback": [
+ "MethodError: no method matching adjoint(::Type{Poisson})\nClosest candidates are:\n adjoint(!Matched::Missing) at missing.jl:79\n adjoint(!Matched::Number) at number.jl:193\n adjoint(!Matched::Adjoint{#s177,#s176} where #s176<:Union{StaticArray{Tuple{N},T,1} where T where N, StaticArray{Tuple{N,M},T,2} where T where M where N} where #s177) at /Users/navaro/.julia/packages/StaticArrays/WmJnA/src/linalg.jl:78\n ...",
+ "",
+ "Stacktrace:",
+ " [1] \\(::Type, ::Array{Float64,1}) at ./operators.jl:536",
+ " [2] (::Poisson)(::Array{Float64,1}) at ./In[5]:2",
+ " [3] landau(::Float64, ::Int64) at ./In[7]:21",
+ " [4] top-level scope at util.jl:156",
+ " [5] top-level scope at In[9]:6"
+ ]
+ }
+ ],
+ "source": [
+ "using Profile\n",
+ "\n",
+ "nt = 1000\n",
+ "tf = 100.0\n",
+ "t = range(0.0, stop=tf, length=nt)\n",
+ "@time nrj = landau(tf, nt)\n",
+ "plot( t, nrj; label = \"E\")\n",
+ "plot!(t, -0.1533*t.-5.50; label=\"-0.1533t.-5.5\")"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "using ProfileView\n",
+ "ProfileView.v"
+ ]
+ }
+ ],
+ "metadata": {
+ "kernelspec": {
+ "display_name": "Julia 1.0.1",
+ "language": "julia",
+ "name": "julia-1.0"
+ },
+ "language_info": {
+ "file_extension": ".jl",
+ "mimetype": "application/julia",
+ "name": "julia",
+ "version": "1.0.1"
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}