Initial commit
This commit is contained in:
32
figs/numerical.fig/Makefile
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32
figs/numerical.fig/Makefile
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PROJECTNAME=bosegas convexity
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DATS=erho.dat ddrhoe.dat
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PDFS=$(addsuffix .pdf, $(PROJECTNAME))
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TEXS=$(addsuffix .tikz.tex, $(PROJECTNAME))
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all: $(PDFS)
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$(PDFS): $(DATS)
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gnuplot $(patsubst %.pdf, %.gnuplot, $@) > $(patsubst %.pdf, %.tikz.tex, $@)
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pdflatex -jobname $(basename $@) -file-line-error $(patsubst %.pdf, %.tikz.tex, $@)
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erho.dat:
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julia ./simpleq/main.jl -p "tolerance=1e-14;order=100" energy_rho > $@
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ddrhoe.dat:
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julia ./simpleq/main.jl -p "tolerance=1e-14;order=100" convexity > $@
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install: $(PDFS)
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cp $^ $(INSTALLDIR)/
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clean-aux:
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rm -f $(addsuffix .tikz.tex, $(PROJECTNAME))
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rm -f $(addsuffix .aux, $(PROJECTNAME))
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rm -f $(addsuffix .log, $(PROJECTNAME))
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clean-dat:
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rm -f $(DATS)
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clean-tex:
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rm -f $(PDFS)
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clean: clean-aux clean-tex
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34
figs/numerical.fig/bosegas.gnuplot
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34
figs/numerical.fig/bosegas.gnuplot
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@ -0,0 +1,34 @@
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set ylabel "$\\displaystyle\\frac{e}{4\\pi\\rho}$" norotate offset -1,0
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set xlabel "$\\rho$"
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set xtics 1e-6, 100, 100
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set xtics add ("$10^{-6}$" 0.000001, "$10^{-4}$" 0.0001, "$10^{-2}$" 0.01, "$1$" 1.0, "$10^2$" 100)
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unset mxtics
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set ytics 0.6, 0.1
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set mytics 2
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set xrange [0.000001:100]
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set yrange [0.6:1.05]
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# default output canvas size: 12.5cm x 8.75cm
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set term lua tikz size 8,6 standalone
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set key off
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# set linestyle
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set style line 1 linetype rgbcolor "#4169E1" linewidth 3
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set style line 2 linetype rgbcolor "#DC143C" linewidth 3
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set style line 3 linetype rgbcolor "#32CD32" linewidth 3
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set style line 4 linetype rgbcolor "#4B0082" linewidth 3
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set style line 5 linetype rgbcolor "#DAA520" linewidth 3
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set pointsize 1
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set logscale x
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plot "erho.dat" using 1:(0.95*$2/$1/(4*pi)):(1.05*$2/$1/(4*pi)) with filledcurves linetype rgbcolor "#DDDDDD", \
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"erho.dat" using 1:($2/$1/(4*pi)) with lines linestyle 1, \
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"holzmann_2019-09-28.dat" using 1:($2/$1/(4*pi)) with points linestyle 2
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30
figs/numerical.fig/convexity.gnuplot
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30
figs/numerical.fig/convexity.gnuplot
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@ -0,0 +1,30 @@
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set ylabel "$\\frac1{4\\pi}\\partial_\\rho^2(e\\rho)$" offset -4,0 norotate
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set xlabel "$\\rho$"
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set xtics 1e-6, 100, 100
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set xtics add ("$10^{-6}$" 0.000001, "$10^{-4}$" 0.0001, "$10^{-2}$" 0.01, "$1$" 1.0, "$10^2$" 100)
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unset mxtics
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set xrange [0.000001:100]
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set ytics 1.2, 0.2
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set mytics 2
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set yrange [1.2:2.1]
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# default output canvas size: 12.5cm x 8.75cm
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set term lua tikz size 8,6 standalone
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# no key
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set key off
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# set linestyle
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set style line 1 linetype rgbcolor "#4169E1" linewidth 3
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set style line 2 linetype rgbcolor "#DC143C" linewidth 3
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set style line 3 linetype rgbcolor "#000000" linewidth 1
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set style line 4 linetype rgbcolor "#4B0082" linewidth 3
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set style line 5 linetype rgbcolor "#DAA520" linewidth 3
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set pointsize 0.6
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set logscale x
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plot "ddrhoe.dat" using 1:($2/(4*pi)) with lines linestyle 1
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9
figs/numerical.fig/holzmann_2019-09-28.dat
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9
figs/numerical.fig/holzmann_2019-09-28.dat
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## data from M. Holzmann, 2019-09-22
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# rho E0 E0+dE0
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1e-4 8.3500e-4 8.3600e-4
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1e-3 9.1340e-3 9.1350e-3
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1e-2 1.0609e-1 1.0610e-1
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1e-1 1.1930e+0 1.1940e+0
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1e-0 1.2445e+1 1.2446e+1
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1e+1 1.2544e+2 1.2545e+2
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28
figs/numerical.fig/simpleq/integration.jl
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28
figs/numerical.fig/simpleq/integration.jl
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# approximate \int_a^b f using Gauss-Legendre quadratures
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function integrate_legendre(f,a,b,weights)
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out=0
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for i in 1:length(weights[1])
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out+=(b-a)/2*weights[2][i]*f((b-a)/2*weights[1][i]+(b+a)/2)
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end
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return out
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end
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# \int f*g where g is sampled at the Legendre nodes
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function integrate_legendre_sampled(f,g,a,b,weights)
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out=0
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for i in 1:length(weights[1])
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out+=(b-a)/2*weights[2][i]*f((b-a)/2*weights[1][i]+(b+a)/2)*g[i]
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end
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return out
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end
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# approximate \int_a^b f/sqrt((b-x)(x-a)) using Gauss-Chebyshev quadratures
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function integrate_chebyshev(f,a,b,N)
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out=0
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for i in 1:N
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out=out+pi/N*f((b-a)/2*cos((2*i-1)/(2*N)*pi)+(b+a)/2)
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end
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return out
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end
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55
figs/numerical.fig/simpleq/iteration.jl
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55
figs/numerical.fig/simpleq/iteration.jl
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# \hat u_n
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function hatun_iteration(e,order,d,v,maxiter)
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# gauss legendre weights
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weights=gausslegendre(order)
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# init V and Eta
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(V,V0,Eta,Eta0)=init_veta(weights,d,v)
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# init u and rho
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u=Array{Array{Complex{Float64}}}(undef,maxiter+1)
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u[1]=zeros(Complex{Float64},order)
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rho=zeros(Complex{Float64},maxiter+1)
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# iterate
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for n in 1:maxiter
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u[n+1]=A(e,weights,Eta)\b(u[n],e,rho[n],V)
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rho[n+1]=rhon(u[n+1],e,weights,V0,Eta0)
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end
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return (u,rho)
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end
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# A
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function A(e,weights,Eta)
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N=length(weights[1])
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out=zeros(Complex{Float64},N,N)
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for i in 1:N
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k=(1-weights[1][i])/(1+weights[1][i])
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out[i,i]=k^2+4*e
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for j in 1:N
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y=(weights[1][j]+1)/2
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out[i,j]+=weights[2][j]*(1-y)*Eta[i][j]/(2*(2*pi)^3*y^3)
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end
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end
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return out
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end
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# b
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function b(u,e,rho,V)
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out=zeros(Complex{Float64},length(V))
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for i in 1:length(V)
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out[i]=V[i]+2*e*rho*u[i]^2
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end
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return out
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end
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# rho_n
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function rhon(u,e,weights,V0,Eta0)
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S=V0
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for i in 1:length(weights[1])
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y=(weights[1][i]+1)/2
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S+=-weights[2][i]*(1-y)*u[i]*Eta0[i]/(2*(2*pi)^3*y^3)
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end
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return 2*e/S
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end
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192
figs/numerical.fig/simpleq/main.jl
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192
figs/numerical.fig/simpleq/main.jl
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@ -0,0 +1,192 @@
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using FastGaussQuadrature
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using Printf
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using LinearAlgebra
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include("tools.jl")
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include("integration.jl")
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include("simpleq.jl")
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include("simpleq-newton.jl")
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include("iteration.jl")
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function main()
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## defaults
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tolerance=1e-14
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order=100
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maxiter=21
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rho=1e-6
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e=1e-4
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d=3
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v=v_exp3d
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a0=a0_exp3d
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# plot range when plotting in x
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xmin=0
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xmax=100
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nx=100
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# read cli arguments
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(params,command)=read_args(ARGS)
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# read params
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if params!=""
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for param in split(params,";")
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terms=split(param,"=")
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if length(terms)!=2
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print(stderr,"error: could not read parameter '",param,"'.\n")
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exit(-1)
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end
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lhs=terms[1]
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rhs=terms[2]
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if lhs=="rho"
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rho=parse(Float64,rhs)
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elseif lhs=="e"
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e=parse(Float64,rhs)
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elseif lhs=="tolerance"
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tolerance=parse(Float64,rhs)
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elseif lhs=="order"
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order=parse(Int64,rhs)
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elseif lhs=="maxiter"
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maxiter=parse(Int64,rhs)
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elseif lhs=="xmin"
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xmin=parse(Float64,rhs)
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elseif lhs=="xmax"
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xmax=parse(Float64,rhs)
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elseif lhs=="nx"
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nx=parse(Int64,rhs)
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else
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print(stderr,"unrecognized parameter '",lhs,"'.\n")
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exit(-1)
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end
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end
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end
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## run command
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# e(rho)
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if command=="energy_rho"
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energy_rho(order,a0,d,v,maxiter,tolerance)
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# d^2(rho*e(rho))
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elseif command=="convexity"
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ddrhoe(order,a0,d,v,maxiter,tolerance)
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# u_n(x)
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elseif command=="iteration"
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iteration_ux(order,e,a0,d,v,maxiter)
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else
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print(stderr,"unrecognized command '",command,"'.\n")
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exit(-1)
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end
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end
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# read cli arguments
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function read_args(ARGS)
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# flag
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flag=""
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# output strings
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params=""
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command=""
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# loop over arguments
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for arg in ARGS
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# argument that starts with a dash
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if arg[1]=='-'
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# go through characters after dash
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for char in arg[2:length(arg)]
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# set params
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if char=='p'
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# raise flag
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flag="params"
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else
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print_usage()
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exit(-1)
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end
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end
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# arguments that do not start with a dash
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else
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if flag=="params"
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params=arg
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else
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command=arg
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end
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# reset flag
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flag=""
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end
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end
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if command==""
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print_usage()
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exit(-1)
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end
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return (params,command)
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end
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# usage
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function print_usage()
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print(stderr,"usage: simpleq [-p params] <command>\n")
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end
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# exponential potential in 3 dimensions
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function v_exp3d(k)
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return 8*pi/(1+k^2)^2
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end
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a0_exp3d=1.254356410591064819505409291110046864031181245635821944528
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# compute energy as a function of rho
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function energy_rho(order,a0,d,v,maxiter,tolerance)
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minlrho=-6
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maxlrho=2
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nlrho=100
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for j in 0:nlrho-1
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rho=10^(minlrho+(maxlrho-minlrho)*j/nlrho)
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# linear spacing
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#rho=10.0^minlrho+(10.0^maxlrho-10.0^minlrho)*j/nlrho
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(u,E)=hatu_newton(order,rho,a0,d,v,maxiter,tolerance)
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@printf("% .8e % .8e\n",rho,real(E))
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end
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end
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# compute \partial^2(e\rho) as a function of rho
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function ddrhoe(order,a0,d,v,maxiter,tolerance)
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minlrho=-6
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maxlrho=2
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nlrho=100
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for j in 0:nlrho-1
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rho=10^(minlrho+(maxlrho-minlrho)*j/nlrho)
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# interval
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drho=rho*1.01
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(u,E)=hatu_newton(order,rho,a0,d,v,maxiter,tolerance)
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(up,Ep)=hatu_newton(order,rho+drho,a0,d,v,maxiter,tolerance)
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(um,Em)=hatu_newton(order,rho-drho,a0,d,v,maxiter,tolerance)
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@printf("% .8e % .8e\n",rho,real((rho+drho)*Ep+(rho-drho)*Em-2*rho*E)/drho^2)
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end
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end
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# compute \int u_n(x) at every step
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function iteration_ux(order,e,a0,d,v,maxiter)
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(u,rho)=hatun_iteration(e,order,d,v,maxiter)
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weights=gausslegendre(order)
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intun=0.
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for n in 2:maxiter+1
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# compute \hat u_n(0)=1/(2*rho_n)+rho_{n-1}/2*\hat u_{n-1}(0)^2
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intun=real(1/(2*rho[n])+rho[n-1]/2*intun^2)
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@printf("%2d % .15e\n",n-1,abs(intun-1/rho[maxiter+1]))
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end
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end
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main()
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95
figs/numerical.fig/simpleq/simpleq-newton.jl
Normal file
95
figs/numerical.fig/simpleq/simpleq-newton.jl
Normal file
@ -0,0 +1,95 @@
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# \hat u(k) computed using Newton algorithm
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function hatu_newton(order,rho,a0,d,v,maxiter,tolerance)
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# compute gaussian quadrature weights
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weights=gausslegendre(order)
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||||
# initialize V and Eta
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(V,V0,Eta,Eta0)=init_veta(weights,d,v)
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# initialize u, V and Eta
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u=zeros(Complex{Float64},order)
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for j in 1:order
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||||
# transformed k
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||||
k=(1-weights[1][j])/(1+weights[1][j])
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||||
if d==2
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u[j]=2*pi*a0*rho/k
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||||
elseif d==3
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||||
u[j]=4*pi*a0*rho/k^2
|
||||
end
|
||||
end
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||||
# iterate
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||||
for i in 1:maxiter-1
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new=u-inv(dXi(u,V,V0,Eta,Eta0,weights,rho,d))*Xi(u,V,V0,Eta,Eta0,weights,rho,d)
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||||
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if(norm(new-u)/norm(u)<tolerance)
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u=new
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||||
break
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||||
end
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u=new
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||||
end
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||||
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||||
return (u,en(u,V0,Eta0,rho,weights,d)*rho/2)
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||||
end
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||||
|
||||
# Xi(u)=0 is equivalent to the linear equation
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function Xi(u,V,V0,Eta,Eta0,weights,rho,d)
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order=length(weights[1])
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# init
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||||
out=zeros(Complex{Float64},order)
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|
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# compute E before running the loop
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||||
E=en(u,V0,Eta0,rho,weights,d)
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for i in 1:order
|
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# k_i
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k=(1-weights[1][i])/(1+weights[1][i])
|
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# S_i
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S=V[i]-1/(rho*(2*pi)^d)*integrate_legendre_sampled(y->(1-y)/y^3,Eta[i].*u,0,1,weights)
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# X_i
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X=k^2/(2*E*rho)
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# U_i
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out[i]=u[i]-S/(2*E*(X+1))*Phi(S/(E*(X+1)^2))
|
||||
end
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||||
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||||
return out
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||||
end
|
||||
|
||||
# derivative of Xi (for Newton)
|
||||
function dXi(u,V,V0,Eta,Eta0,weights,rho,d)
|
||||
order=length(weights[1])
|
||||
|
||||
# init
|
||||
out=zeros(Complex{Float64},order,order)
|
||||
|
||||
# compute E before the loop
|
||||
E=en(u,V0,Eta0,rho,weights,d)
|
||||
|
||||
for i in 1:order
|
||||
# k_i
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||||
k=(1-weights[1][i])/(1+weights[1][i])
|
||||
# S_i
|
||||
S=V[i]-1/(rho*(2*pi)^d)*integrate_legendre_sampled(y->(1-y)/y^3,Eta[i].*u,0,1,weights)
|
||||
# X_i
|
||||
X=k^2/(2*E*rho)
|
||||
|
||||
for j in 1:order
|
||||
y=(weights[1][j]+1)/2
|
||||
dS=-1/rho*(1-y)*Eta[i][j]/(2*(2*pi)^d*y^3)*weights[2][j]
|
||||
dE=-1/rho*(1-y)*Eta0[j]/(2*(2*pi)^d*y^3)*weights[2][j]
|
||||
dX=-k^2/(2*E^2*rho)*dE
|
||||
out[i,j]=(i==j ? 1 : 0)-(dS-S*dE/E-S*dX/(X+1))/(2*E*(X+1))*Phi(S/(E*(X+1)^2))-S/(2*E^2*(X+1)^3)*(dS-S*dE/E-2*S*dX/(X+1))*dPhi(S/(E*(X+1)^2))
|
||||
end
|
||||
end
|
||||
|
||||
return out
|
||||
end
|
||||
|
||||
# energy
|
||||
function en(u,V0,Eta0,rho,weights,d)
|
||||
return V0-1/(rho*(2*pi)^d)*integrate_legendre_sampled(y->(1-y)/y^3,Eta0.*u,0,1,weights)
|
||||
end
|
||||
|
||||
40
figs/numerical.fig/simpleq/simpleq.jl
Normal file
40
figs/numerical.fig/simpleq/simpleq.jl
Normal file
@ -0,0 +1,40 @@
|
||||
# \eta
|
||||
function eta(x,t,weights,d,v)
|
||||
if d==2
|
||||
return integrate_chebyshev(y->4*((x+t)*y+abs(x-t)*(1-y))*v((x+t)*y+abs(x-t)*(1-y))/sqrt(((x+t)*y+abs(x-t)*(2-y))*((x+t)*(1+y)+abs(x-t)*(1-y))),0,1,length(weights))
|
||||
elseif d==3
|
||||
return (x>t ? 2*t/x : 2)* integrate_legendre(y->2*pi*((x+t)*y+abs(x-t)*(1-y))*v((x+t)*y+abs(x-t)*(1-y)),0,1,weights)
|
||||
end
|
||||
end
|
||||
|
||||
# initialize V and Eta
|
||||
function init_veta(weights,d,v)
|
||||
order=length(weights[1])
|
||||
V=Array{Complex{Float64}}(undef,order)
|
||||
Eta=Array{Array{Complex{Float64}}}(undef,order)
|
||||
Eta0=Array{Complex{Float64}}(undef,order)
|
||||
V0=v(0)
|
||||
for i in 1:order
|
||||
k=(1-weights[1][i])/(1+weights[1][i])
|
||||
V[i]=v(k)
|
||||
Eta[i]=Array{Complex{Float64}}(undef,order)
|
||||
for j in 1:order
|
||||
y=(weights[1][j]+1)/2
|
||||
Eta[i][j]=eta(k,(1-y)/y,weights,d,v)
|
||||
end
|
||||
y=(weights[1][i]+1)/2
|
||||
Eta0[i]=eta(0,(1-y)/y,weights,d,v)
|
||||
end
|
||||
return(V,V0,Eta,Eta0)
|
||||
end
|
||||
|
||||
# inverse Fourier transform
|
||||
function u_x(x,u,weights,d)
|
||||
order=length(weights[1])
|
||||
if d==2
|
||||
out=integrate_legendre_sampled(y->(1-y)/y^3*besselj(0,x*(1-y)/y)/(2*pi),u,0,1,weights)
|
||||
elseif d==3
|
||||
out=integrate_legendre_sampled(y->(1-y)/y^3*sin(x*(1-y)/y)/x/(2*pi^2),u,0,1,weights)
|
||||
end
|
||||
return out
|
||||
end
|
||||
20
figs/numerical.fig/simpleq/tools.jl
Normal file
20
figs/numerical.fig/simpleq/tools.jl
Normal file
@ -0,0 +1,20 @@
|
||||
# \Phi(x)=2*(1-sqrt(1-x))/x
|
||||
function Phi(x)
|
||||
if abs(x)>1e-5
|
||||
return 2*(1-sqrt(1-x))/x
|
||||
else
|
||||
return 1+x/4+x^2/8+5*x^3/64+7*x^4/128+21*x^5/512
|
||||
end
|
||||
end
|
||||
# \partial\Phi
|
||||
function dPhi(x)
|
||||
#if abs(x-1)<1e-5
|
||||
# @printf(stderr,"warning: dPhi is singular at 1, and evaluating it at (% .8e+i% .8e)\n",real(x),imag(x))
|
||||
#end
|
||||
if abs(x)>1e-5
|
||||
return 1/(sqrt(1-x)*x)-2*(1-sqrt(1-x))/x^2
|
||||
else
|
||||
return 1/4+x/4+15*x^2/64+7*x^3/32+105*x^4/512+99*x^5/512
|
||||
end
|
||||
end
|
||||
|
||||
Reference in New Issue
Block a user