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+<html lang="en"><head><meta charset="UTF-8"/><meta name="viewport" content="width=device-width, initial-scale=1.0"/><title>Accuracy · OrnsteinZernike.jl Documentation</title><meta name="title" content="Accuracy · OrnsteinZernike.jl Documentation"/><meta property="og:title" content="Accuracy · OrnsteinZernike.jl Documentation"/><meta property="twitter:title" content="Accuracy · OrnsteinZernike.jl Documentation"/><meta name="description" content="Documentation for OrnsteinZernike.jl Documentation."/><meta property="og:description" content="Documentation for OrnsteinZernike.jl Documentation."/><meta property="twitter:description" content="Documentation for OrnsteinZernike.jl Documentation."/><script data-outdated-warner src="assets/warner.js"></script><link href="https://cdnjs.cloudflare.com/ajax/libs/lato-font/3.0.0/css/lato-font.min.css" rel="stylesheet" type="text/css"/><link href="https://cdnjs.cloudflare.com/ajax/libs/juliamono/0.050/juliamono.min.css" rel="stylesheet" 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href="#"></a><nav class="breadcrumb"><ul class="is-hidden-mobile"><li><a class="is-disabled">Tutorials</a></li><li class="is-active"><a href="Accuracy.html">Accuracy</a></li></ul><ul class="is-hidden-tablet"><li class="is-active"><a href="Accuracy.html">Accuracy</a></li></ul></nav><div class="docs-right"><a class="docs-navbar-link" href="https://github.com/IlianPihlajamaa/OrnsteinZernike.jl" title="View the repository on GitHub"><span class="docs-icon fa-brands"></span><span class="docs-label is-hidden-touch">GitHub</span></a><a class="docs-navbar-link" href="https://github.com/IlianPihlajamaa/OrnsteinZernike.jl/blob/main/docs/src/Accuracy.md" title="Edit source on GitHub"><span class="docs-icon fa-solid"></span></a><a class="docs-settings-button docs-navbar-link fa-solid fa-gear" id="documenter-settings-button" href="#" title="Settings"></a><a class="docs-article-toggle-button fa-solid fa-chevron-up" id="documenter-article-toggle-button" href="javascript:;" title="Collapse all docstrings"></a></div></header><article class="content" id="documenter-page"><h1 id="Accuracy"><a class="docs-heading-anchor" href="#Accuracy">Accuracy</a><a id="Accuracy-1"></a><a class="docs-heading-anchor-permalink" href="#Accuracy" title="Permalink"></a></h1><p>The discrete Fourier tranforms used by the <code>FourierIteration</code>, and <code>NgIteration</code> solvers to represent their continuous counterparts using first order accuracy in <span>$n$</span> dimensions and second order accuracy in three dimensions (with the trapezoidal rule). To obtain the latter accuracy, it is important that any discontinuities of the interaction potential lie on an exact multiple of <code>dr</code>. To test this, we can compute the pressure of a hard-sphere system, and compare to the exact value. Below, we compute the relative error for different values of the number of gridpoints <code>M</code>, and plot the result on a log-log-scale</p><pre><code class="language-julia hljs">using OrnsteinZernike,  Plots
+
+# Make sure the discontinuity is a multiple of dr
+Rmax = 10.0
+M_array = 10 * round.(Int,  10 .^ (range(1,4,length=20)))
+p = zeros(length(M_array))
+ρ = 0.3
+kBT = 1.0
+dims = 3
+pot = HardSpheres(1.0)
+system = SimpleLiquid(dims, ρ, kBT, pot)
+
+for (i,M) in enumerate(M_array)
+    dr = Rmax/M
+    method = NgIteration(M=M, dr=dr, verbose=false)
+    sol = solve(system, PercusYevick(), method)
+    pressure = compute_virial_pressure(sol, system)
+    p[i] = pressure/ρ/kBT-1.0
+    println(&quot;The pressure = &quot;, round(pressure, digits=8), &quot; with M = $(M) gridpoints.&quot;)
+end</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">The pressure = 0.58024311 with M = 100 gridpoints.
+The pressure = 0.5831676 with M = 140 gridpoints.
+The pressure = 0.58483445 with M = 210 gridpoints.
+The pressure = 0.58550373 with M = 300 gridpoints.
+The pressure = 0.5858285 with M = 430 gridpoints.
+The pressure = 0.58598643 with M = 620 gridpoints.
+The pressure = 0.58606101 with M = 890 gridpoints.
+The pressure = 0.58609654 with M = 1270 gridpoints.
+The pressure = 0.58611421 with M = 1830 gridpoints.
+The pressure = 0.58612271 with M = 2640 gridpoints.
+The pressure = 0.58612675 with M = 3790 gridpoints.
+The pressure = 0.58612871 with M = 5460 gridpoints.
+The pressure = 0.58612966 with M = 7850 gridpoints.
+The pressure = 0.58613011 with M = 11290 gridpoints.
+The pressure = 0.58613033 with M = 16240 gridpoints.
+The pressure = 0.58613044 with M = 23360 gridpoints.
+The pressure = 0.58613049 with M = 33600 gridpoints.
+The pressure = 0.58613051 with M = 48330 gridpoints.
+The pressure = 0.58613053 with M = 69520 gridpoints.
+The pressure = 0.58613053 with M = 100000 gridpoints.</code></pre><p>We can see that the method has well-behaved second order convergence, and that with <span>$M=10^4$</span>, we get almost 6 digits of relative accuracy.</p><pre><code class="language-julia hljs">η = ρ/6*π
+p_exact = (1+2η+3η^2)/(1-η)^2-1.0
+scatter(M_array, abs.(p.-p_exact)./p_exact)
+plot!(ylabel=&quot;relative error&quot;, xlabel=&quot;M&quot;, xscale=:log, yscale=:log)</code></pre><img src="Accuracy-995f857c.svg" alt="Example block output"/><p>In principle, these results can be extrapolated to improve the accuracy further.</p></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="OtherDimensions.html">« Other dimensions</a><a class="docs-footer-nextpage" href="ThermodynamicConsistency.html">Thermodynamic Consistency »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.7.0 on <span class="colophon-date" title="Friday 4 October 2024 09:00">Friday 4 October 2024</span>. Using Julia version 1.10.5.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
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