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<title>Numerical diffusion</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Numerical diffusion</span></span>

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<p><b>Numerical diffusion</b> is a difficulty with <a href="Computer_simulation" title="Computer simulation">computer simulations</a> of continua (such as <a href="Fluid" title="Fluid">fluids</a>) wherein the simulated medium exhibits a higher <a href="Eddy_diffusion" title="Eddy diffusion">diffusivity</a> than the true medium. This phenomenon can be particularly egregious when the system should not be diffusive at all, for example an ideal fluid acquiring some spurious viscosity in a numerical model.

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<div class="mw-heading mw-heading2"><h2 id="Explanation">Explanation</h2></div>

<p>In <a href="Eulerian_method" class="mw-redirect" title="Eulerian method">Eulerian simulations</a>, time and space are divided into a discrete grid and the continuous <a href="Differential_equation" title="Differential equation">differential equations</a> of motion (such as the <a href="Navier%E2%80%93Stokes_equation" class="mw-redirect" title="Navier–Stokes equation">Navier–Stokes equation</a>) are <a href="Discretization" title="Discretization">discretized</a> into <a href="Finite-difference_equation" class="mw-redirect" title="Finite-difference equation">finite-difference equations</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The discrete equations are in general more <a href="Diffusion" title="Diffusion">diffusive</a> than the original differential equations, so that the simulated system behaves differently than the intended physical system.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The amount and character of the difference depends on the system being simulated and the type of discretization that is used. Most fluid dynamics or <a href="Magnetohydrodynamics" title="Magnetohydrodynamics">magnetohydrodynamic</a> simulations seek to reduce numerical diffusion to the minimum possible, to achieve high fidelity — but under certain circumstances diffusion is added deliberately into the system to avoid <a href="Mathematical_singularity" class="mw-redirect" title="Mathematical singularity">singularities</a>. For example, <a href="Shock_wave" title="Shock wave">shock waves</a> in fluids and <a href="Current_sheet" title="Current sheet">current sheets</a> in <a href="Plasma_(physics)" title="Plasma (physics)">plasmas</a> are infinitely thin in some approximations; this can cause difficulty for numerical codes. A simple way to avoid the difficulty is to add diffusion that smooths out the shock or current sheet. Higher order numerical methods (including spectral methods) tend to have less numerical diffusion than low order methods.

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<div class="mw-heading mw-heading2"><h2 id="Example">Example</h2></div>

<p>As an example of numerical diffusion, consider an Eulerian simulation using an explicit time-advance of a drop of green dye diffusing through water. If the water is flowing diagonally through the simulation grid, then it is impossible to move the dye in the exact direction of the flow: at each time step the simulation can at best transfer some dye in each of the vertical and horizontal directions. After a few time steps, the dye will have spread out through the grid due to this sideways transfer. This numerical effect takes the form of an extra high diffusion rate.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

</p><p>When numerical diffusion applies to the components of the <a href="Momentum" title="Momentum">momentum</a> vector, it is called numerical viscosity; when it applies to a magnetic field, it is called <a href="Numerical_resistivity" title="Numerical resistivity">numerical resistivity</a>.

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<p> Consider a <a href="Phase_field_models" class="mw-redirect" title="Phase field models">Phasefield-problem</a> with a high pressure loaded air bubble (blue) within a phase of water. Since there are no chemical or thermodynamical reactions during expansion of air in water there is no possibility to come up with another (i.e. non red or blue) phase during the simulation. These inaccuracies between single phases are based on numerical diffusion and can be decreased by <a href="Polygon_mesh" title="Polygon mesh">mesh</a> refining.

</p><div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>

<ul><li><a href="False_diffusion" title="False diffusion">False diffusion</a></li>

<li><a href="Numerical_dispersion" title="Numerical dispersion">Numerical dispersion</a></li>

<li><a href="Numerical_error" title="Numerical error">Numerical error</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.mathematik.tu-dortmund.de/~kuzmin/cfdintro/lecture10.pdf">Analysis of numerical dissipation and dispersion </a>, Dortmund.</span>

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</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20110401023456/http://people.maths.ox.ac.uk/trefethen/5all.pdf">"Dissipation, Dispersion and Group Velocity"</a> <span class="cs1-format">(PDF)</span>. Archived from <a rel="nofollow" class="external text" href="http://people.maths.ox.ac.uk/trefethen/5all.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2011-04-01.</cite></span>

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<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFPierre_Schoeffler" class="citation web cs1">Pierre Schoeffler. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220120232527/http://www.smhi.se/polopoly_fs/1.158070!/RMK_34.pdf">"Dissipation, dispersion and stability of numerical schemes for advection and diffusion"</a> <span class="cs1-format">(PDF)</span>. Archived from <a rel="nofollow" class="external text" href="http://www.smhi.se/polopoly_fs/1.158070!/RMK_34.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2022-01-20.</cite></span>

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