First version of the documentation of the MM scheme. Still need to work out...

First version of the documentation of the MM scheme. Still need to work out the correct analytical expression of the truncation formula in Fourier space.

.gitignore

@@ -102,6 +102,8 @@ theory/paper_pasc/pasc_paper.pdf
 theory/Multipoles/fmm.pdf
 theory/Multipoles/fmm_standalone.pdf
 theory/Multipoles/potential.pdf
+theory/Multipoles/potential_long.pdf
+theory/Multipoles/potential_short.pdf
 
 m4/libtool.m4
 m4/ltoptions.m4

theory/Multipoles/mesh_summary.tex

 \subsection{Coupling the FMM to a mesh for periodic long-range forces}
 \label{ssec:mesh_summary}
+
+\begin{figure}
+\includegraphics[width=\columnwidth]{potential_short.pdf}
+\caption{aa}
+\label{fig:fmm:potential_short}
+\end{figure}
+
+
+\begin{figure}
+\includegraphics[width=\columnwidth]{potential_long.pdf}
+\caption{bb}
+\label{fig:fmm:potential_long}
+\end{figure}

theory/Multipoles/plot_mesh.py

+###############################################################################
+ # This file is part of SWIFT.
+ # Copyright (c) 2016  Matthieu Schaller (matthieu.schaller@durham.ac.uk)
+ # 
+ # This program is free software: you can redistribute it and/or modify
+ # it under the terms of the GNU Lesser General Public License as published
+ # by the Free Software Foundation, either version 3 of the License, or
+ # (at your option) any later version.
+ # 
+ # This program is distributed in the hope that it will be useful,
+ # but WITHOUT ANY WARRANTY; without even the implied warranty of
+ # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+ # GNU General Public License for more details.
+ # 
+ # You should have received a copy of the GNU Lesser General Public License
+ # along with this program.  If not, see <http://www.gnu.org/licenses/>.
+ # 
+ ##############################################################################
+import matplotlib
+matplotlib.use("Agg")
+from pylab import *
+from scipy import integrate
+from scipy import special
+from scipy.optimize import curve_fit
+from scipy.optimize import fsolve
+from matplotlib.font_manager import FontProperties
+import numpy
+import math
+
+params = {'axes.labelsize': 9,
+'axes.titlesize': 10,
+'font.size': 10,
+'legend.fontsize': 10,
+'xtick.labelsize': 8,
+'ytick.labelsize': 8,
+'text.usetex': True,
+'figure.figsize' : (3.15,3.15),
+'figure.subplot.left'    : 0.12,
+'figure.subplot.right'   : 0.99  ,
+'figure.subplot.bottom'  : 0.09  ,
+'figure.subplot.top'     : 0.99  ,
+'figure.subplot.wspace'  : 0.  ,
+'figure.subplot.hspace'  : 0.  ,
+'lines.markersize' : 6,
+'lines.linewidth' : 3.,
+'text.latex.unicode': True
+}
+rcParams.update(params)
+rc('font',**{'family':'sans-serif','sans-serif':['Times']})
+colors=['#4477AA', '#CC6677', '#DDCC77', '#117733']
+
+
+# Parameters
+r_s = 2.
+r_min = 1e-2
+r_max = 1.5e2
+
+# Radius
+r = logspace(log10(r_min), log10(r_max), 401)
+r_rs = r / r_s
+
+k = logspace(log10(r_min/r_s**2), log10(r_max/r_s**2), 401)
+k_rs = k * r_s
+
+# Newtonian solution
+phi_newton = 1. / r
+phit_newton = 1. / k**2
+
+def smoothstep(x): #S_2(x)
+    ret = 6*x**5 - 15*x**4 + 10*x**3#3*x**2 - 2*x**3
+    ret[x < 0] = 0.
+    ret[x > 1] = 1.
+    return ret
+    
+# Correcction in real space
+corr_short_gadget2 = special.erf(r / (2.*r_s))
+corr_long_gadget2 = exp(-k**2*r_s**2)
+corr_short_swift = smoothstep(r / (2.*r_s))
+corr_long_swift = 0.5 * (90 * r_s * k * np.cos(k * r_s)**2 + (r_s**2 * k**2 - 3) * np.sin(2*r_s*k))/ (r_s**5 * k**7) 
+
+# Shortrange term
+phi_short_gadget2 = (1.  / r ) * (1. - corr_short_gadget2)
+phi_short_swift = (1.  / r ) * (1. - corr_short_swift)
+
+# Long-range term
+phi_long_gadget2 = (1.  / r ) * corr_short_gadget2
+phi_long_swift = (1.  / r ) * corr_short_swift
+phit_long_gadget2 = corr_long_gadget2 / k**2
+phit_long_swift = corr_long_swift / k**2
+
+figure()
+
+# Potential
+subplot(311, xscale="log", yscale="log")
+
+plot(r_rs, phi_newton, '--', lw=1.4, label="${\\rm Newtonian}$", color=colors[0])
+plot(r_rs, phi_short_gadget2, '-', lw=1.4, label="${\\rm Gadget}$", color=colors[2])
+plot(r_rs, phi_short_swift, '-', lw=1.4, label="${\\rm SWIFT}$", color=colors[3])
+plot([1., 1.], [1e-5, 1e5], 'k-', alpha=0.5, lw=0.5)
+
+xlim(1.1*r_min/ r_s, 0.9*r_max / r_s)
+ylim(1.1/r_max, 0.9/r_min)
+ylabel("$\\varphi_s(r)$", labelpad=-3)
+
+legend(loc="upper right", frameon=True, handletextpad=0.1, handlelength=3.2, fontsize=8)
+
+# Correction
+subplot(312, xscale="log", yscale="linear")
+#plot(r_rs, np.ones(np.size(r)), '--', lw=1.4, color=colors[0])
+plot(r_rs, 1. - corr_short_gadget2, '-', lw=1.4, color=colors[2])
+plot(r_rs, 1. - corr_short_swift, '-', lw=1.4, color=colors[3])
+plot(r_rs, np.zeros(np.size(r)), 'k--', alpha=0.5, lw=0.5)
+plot(r_rs, np.ones(np.size(r)), 'k--', alpha=0.5, lw=0.5)
+plot([1., 1.], [-1e5, 1e5], 'k-', alpha=0.5, lw=0.5)
+
+xlim(1.1*r_min/r_s, 0.9*r_max/r_s)
+ylim(-0.1, 1.1)
+ylabel("$\\chi_s(r)$", labelpad=2)
+
+subplot(313, xscale="log", yscale="log")
+
+#print corr_short_gadget2
+
+#plot(r_rs, np.abs(1. - np.ones(np.size(r))), '--', lw=1.4, color=colors[0])
+plot(r_rs, corr_short_gadget2, '-', lw=1.4, color=colors[2])
+plot(r_rs, corr_short_swift, '-', lw=1.4, color=colors[3])
+
+
+plot([1., 1.], [1e-5, 1e5], 'k-', alpha=0.5, lw=0.5)
+plot(r_rs, np.ones(np.size(r)), 'k--', alpha=0.5, lw=0.5)
+plot(r_rs, np.ones(np.size(r))*0.01, 'k--', alpha=0.5, lw=0.5)
+
+xlim(1.1*r_min/r_s, 0.9*r_max/r_s)
+ylim(3e-3, 1.5)
+ylabel("$1 - \\chi_s(r)$", labelpad=-2)
+yticks([1e-2, 1e-1, 1], ["$0.01$", "$0.1$", "$1$"])
+xlabel("$r / r_s$", labelpad=-3)
+
+#ylim(0, 0.95)
+
+savefig("potential_short.pdf")
+
+
+
+
+figure()
+subplot(211, xscale="log", yscale="log")
+
+# Potential
+plot(k_rs, phit_newton, '--', lw=1.4, label="${\\rm Newtonian}$", color=colors[0])
+plot(k_rs, phit_long_gadget2, '-', lw=1.4, label="${\\rm Gadget}$", color=colors[2])
+plot(k_rs, phit_long_swift, '-', lw=1.4, label="${\\rm SWIFT}$", color=colors[3])
+plot([1., 1.], [1e-5, 1e5], 'k-', alpha=0.5, lw=0.5)
+
+legend(loc="lower left", frameon=True, handletextpad=0.1, handlelength=3.2, fontsize=8)
+
+xlim(1.1*r_min/ r_s, 0.9*r_max / r_s)
+ylim(1.1/r_max**2, 0.9/r_min**2)
+ylabel("$\\tilde\\varphi_l(k)$", labelpad=-3)
+
+
+subplot(212, xscale="log", yscale="log")
+
+# Potential normalized
+plot(k_rs, phit_newton * k**2, '--', lw=1.4, label="${\\rm Newtonian}$", color=colors[0])
+plot(k_rs, phit_long_gadget2 * k**2, '-', lw=1.4, label="${\\rm Gadget}$", color=colors[2])
+plot(k_rs, phit_long_swift * k**2, '-', lw=1.4, label="${\\rm SWIFT}$", color=colors[3])
+plot([1., 1.], [1e-5, 1e5], 'k-', alpha=0.5, lw=0.5)
+plot(r_rs, np.ones(np.size(r))*0.01, 'k--', alpha=0.5, lw=0.5)
+
+xlim(1.1*r_min/ r_s, 0.9*r_max / r_s)
+ylim(3e-3, 1.5)
+ylabel("$k^2 \\times \\tilde\\varphi_l(k)$", labelpad=-3)
+yticks([1e-2, 1e-1, 1], ["$0.01$", "$0.1$", "$1$"])
+xlabel("$k \\times r_s$", labelpad=0)
+
+savefig("potential_long.pdf")

theory/Multipoles/run.sh

 #!/bin/bash
-if [! -e potential.pdf ]
+if [ ! -e potential.pdf ]
 then
-    echo "Generating figures..."
+    echo "Generating 1st figure..."
     python plot_potential.py
 fi
+if [ ! -e potential_short.pdf ]
+then
+    echo "Generating 2nd figures..."
+    python plot_mesh.py
+fi
 echo "Generating PDF..."
 pdflatex -jobname=fmm fmm_standalone.tex
 bibtex fmm.aux