Commit 620d26d3 by Matthieu Schaller

### Added the short-range correction equations to the FMM pdf documentation.

parent 80417d31
 \subsection{Coupling the FMM to a mesh for periodic long-range forces} \label{ssec:mesh_summary} S(x) = \frac{e^x}{1 + e^x} \begin{align} \varphi_s(r) &= \frac{1}{r}\left[2 - 2S\left(\frac{2r}{r_s}\right)\right] \nonumber\\ &= \frac{1}{r}\left[2 - \frac{2e^{\frac{2r}{r_s}}}{1+e^{\frac{2r}{r_s}}}\right] \end{align} \begin{align} |\mathbf{f}_s(r)| &= \frac{1}{r^2}\left[\frac{4r}{r_s}S'\left(\frac{2r}{r_s}\right) - 2S\left(\frac{2r}{r_s}\right) + 2\right] \nonumber \\ &= \frac{1}{r^2}\left[\frac{4r}{r_s}\frac{e^{\frac{2r}{r_s}}}{(1+e^{\frac{2r}{r_s}})^2} - \frac{2e^{\frac{2r}{r_s}}}{1+e^{\frac{2r}{r_s}}} + 2\right] \end{align} \tilde\varphi_l(k) = \frac{1}{k^2}\left[\frac{\upi}{2}kr_s\textrm{csch}\left(\frac{\upi}{2}kr_s\right) \right] \begin{figure} \includegraphics[width=\columnwidth]{potential_short.pdf} \caption{aa} ... ... @@ -9,7 +26,14 @@ \begin{figure} \includegraphics[width=\columnwidth]{potential_long.pdf} \includegraphics[width=\columnwidth]{force_short.pdf} \caption{bb} \label{fig:fmm:force_short} \end{figure} \begin{figure} \includegraphics[width=\columnwidth]{potential_long.pdf} \caption{cc} \label{fig:fmm:potential_long} \end{figure}
 ... ... @@ -67,56 +67,42 @@ phi_newton = 1. / r phit_newton = 1. / k**2 force_newton = 1. / r**2 def smoothstep(x): #S_2(x) ret = 6*x**5 - 15*x**4 + 10*x**3 #ret = 3*x**2 - 2*x**3 ret[x < 0] = 0. ret[x > 1] = 1. return ret def my_exp(x): #return 1. + x + (x**2 / 2.) + (x**3 / 6.) + (x**4 / 24.) + (x**5 / 120.)# + (x**6 / 720.) return exp(x) return 1. + x + (x**2 / 2.) + (x**3 / 6.) + (x**4 / 24.) + (x**5 / 120.)# + (x**6 / 720.) #return exp(x) def csch(x): # hyperbolic cosecant return 1. / sinh(x) def sigmoid(x): return 1. / (1. + 1./my_exp(x)) #return x / sqrt(1. + x**2) return my_exp(x) / (my_exp(x) + 1.) def d_sigmoid(x): return my_exp(x) / ((my_exp(x) + 1)**2) def swift_corr(x): #return 2. * smoothstep(x/4. + 1./2.) - 1. #return sigmoid(4. * x) return 2 * sigmoid( 4 * x ) - 1 def d_swift_corr(x): return 2 * d_sigmoid( 4 * x ) def csch(x): # hyperbolic cosecant return 1. / sinh(x) figure() x = linspace(-4, 4, 100) plot(x, special.erf(x), '-', color=colors[0]) plot(x, swift_corr(x), '-', color=colors[1]) plot(x, x, '-', color=colors[2]) ylim(-1.1, 1.1) xlim(-4.1, 4.1) #plot(x, exp(x), '-', color=colors[0]) #plot(x, my_exp(x), '-', color=colors[1]) savefig("temp.pdf") #figure() #x = linspace(-4, 4, 100) #plot(x, special.erf(x), '-', color=colors[0]) #plot(x, swift_corr(x), '-', color=colors[1]) #plot(x, x, '-', color=colors[2]) #ylim(-1.1, 1.1) #xlim(-4.1, 4.1) #savefig("temp.pdf") # Correction in real space corr_short_gadget2 = special.erf(r / (2.*r_s)) corr_long_gadget2 = exp(-k**2*r_s**2) corr_short_swift = swift_corr(r / (2.*r_s)) #corr_long_swift = k * r_s * math.pi / (2. * sinh(0.5 * math.pi * r_s * k)) corr_long_swift = math.pi * k * r_s * csch(0.5 * math.pi * r_s * k) / 2. eta_short_gadget2 = special.erfc(r / 2.*r_s) + (r / (r_s * math.sqrt(math.pi))) * exp(-r**2 / (4.*r_s**2)) eta_short_swift = (2. * r * d_swift_corr(r / (2.*r_s)) / r_s) - 2*sigmoid(2*r / (r_s)) + 2. eta_short_swift = 4. * (r / r_s) * d_sigmoid(2. * r / r_s) - 2. * sigmoid(2 * r / r_s) + 2. # Corection in Fourier space corr_long_gadget2 = exp(-k**2*r_s**2) corr_long_swift = math.pi * k * r_s * csch(0.5 * math.pi * r_s * k) / 2. # Shortrange term2 # Shortrange term phi_short_gadget2 = (1. / r ) * (1. - corr_short_gadget2) phi_short_swift = (1. / r ) * (1. - corr_short_swift) force_short_gadget2 = (1. / r**2) * eta_short_gadget2 ... ... @@ -149,18 +135,17 @@ legend(loc="upper right", frameon=True, handletextpad=0.1, handlelength=3.2, fon # Correction subplot(312, xscale="log", yscale="log") #plot(r_rs, np.ones(np.size(r)), '--', lw=1.4, color=colors[0]) 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(r_rs, np.ones(np.size(r))*0.01, 'k--', alpha=0.5, lw=0.5) plot(r_rs, np.ones(np.size(r))*0.01, 'k:', alpha=0.5, lw=0.5) plot([1., 1.], [-1e5, 1e5], 'k-', alpha=0.5, lw=0.5) yticks([1e-2, 1e-1, 1], ["$0.01$", "$0.1$", "$1$"]) xlim(1.1*r_min/r_s, 0.9*r_max/r_s) ylim(3e-3, 1.5) ylabel("$\\chi_s(r)$", labelpad=-3) #ylabel("$\\chi_s(r)$", labelpad=-3) ylabel("$\\varphi_s(r) \\times r$", labelpad=-2) # 1 - Correction subplot(313, xscale="log", yscale="log") ... ... @@ -168,12 +153,13 @@ 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) 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) #ylabel("$1 - \\chi_s(r)$", labelpad=-2) ylabel("$1 - \\varphi_s(r) \\times r$", labelpad=-2) yticks([1e-2, 1e-1, 1], ["$0.01$", "$0.1$", "$1$"]) xlabel("$r / r_s$", labelpad=-3) ... ... @@ -200,16 +186,17 @@ legend(loc="upper right", frameon=True, handletextpad=0.1, handlelength=3.2, fon # Correction subplot(312, xscale="log", yscale="log") plot(r_rs, np.ones(np.size(r)), '--', lw=1.4, color=colors[0]) plot(r_rs, eta_short_gadget2, '-', lw=1.4, color=colors[2]) plot(r_rs, eta_short_swift, '-', lw=1.4, color=colors[3]) 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) plot(r_rs, np.ones(np.size(r))*0.01, 'k:', alpha=0.5, lw=0.5) plot([1., 1.], [-1e5, 1e5], 'k-', alpha=0.5, lw=0.5) yticks([1e-2, 1e-1, 1], ["$0.01$", "$0.1$", "$1$"]) xlim(1.1*r_min/r_s, 0.9*r_max/r_s) ylim(3e-3, 1.5) ylabel("$\\eta_s(r)$", labelpad=-3) #ylabel("$\\eta_s(r)$", labelpad=-3) ylabel("$|\\mathbf{f}_s(r)|\\times r^2$", labelpad=-2) # 1 - Correction subplot(313, xscale="log", yscale="log") ... ... @@ -217,12 +204,13 @@ plot(r_rs, 1. - eta_short_gadget2, '-', lw=1.4, color=colors[2]) plot(r_rs, 1. - eta_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) 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("$1 - \\eta_s(r)$", labelpad=-2) #ylabel("$1 - \\eta_s(r)$", labelpad=-2) ylabel("$1 - |\\mathbf{f}_s(r)|\\times r^2$", labelpad=-3) yticks([1e-2, 1e-1, 1], ["$0.01$", "$0.1$", "$1$"]) xlabel("$r / r_s$", labelpad=-3) ... ... @@ -231,7 +219,7 @@ savefig("force_short.pdf") ################################################################################################## figure() subplot(211, xscale="log", yscale="log") subplot(311, xscale="log", yscale="log") # Potential plot(k_rs, phit_newton, '--', lw=1.4, label="${\\rm Newtonian}$", color=colors[0]) ... ... @@ -245,22 +233,35 @@ legend(loc="lower left", frameon=True, handletextpad=0.1, handlelength=3.2, font 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) yticks([1e-4, 1e-2, 1e0, 1e2], ["$10^{-4}$", "$10^{-2}$", "$10^{0}$", "$10^{2}$"]) subplot(212, xscale="log", yscale="log") subplot(312, 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(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) 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], ["$0.01$", "$0.1$", "$1$"]) subplot(313, xscale="log", yscale="log") plot(k_rs, 1. - phit_long_gadget2 * k**2, '-', lw=1.4, label="${\\rm Gadget}$", color=colors[2]) plot(k_rs, 1. - 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)), '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("$1 - k^2 \\times \\tilde{\\varphi_l}(k)$", labelpad=-3) yticks([1e-2, 1e-1, 1], ["$0.01$", "$0.1$", "$1$"]) xlabel("$k \\times r_s$", labelpad=0) savefig("potential_long.pdf")
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