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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Temporal discretization</span></span>
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<p>In <a href="Applied_physics" title="Applied physics">applied physics</a> and <a href="Engineering" title="Engineering">engineering</a>, <b>temporal discretization</b> is a mathematical technique for solving <a href="Transient_(civil_engineering)" title="Transient (civil engineering)">transient</a> problems, such as <a href="Flow_problem" class="mw-redirect" title="Flow problem">flow problems</a>.
</p><p>Transient problems are often solved using <a href="Computer-aided_engineering" title="Computer-aided engineering">computer-aided engineering</a> (CAE) simulations, which require <a href="Discretization" title="Discretization">discretizing</a> the governing equations in both space and time. Temporal discretization involves the <a href="Integral" title="Integral">integration</a> of every term in various equations over a time step (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta t}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mi>t</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Delta t}</annotation>
</semantics>
</math></span><img src="./8c28867ecd34e2caed12cf38feadf6a81a7ee542.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.775ex; height:2.176ex;" alt="{\displaystyle \Delta t}" loading="lazy"></span>).
</p><p>The spatial domain can be discretized to produce a semi-discrete form:<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>
<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \varphi }{\partial t}}(x,t)=F(\varphi ).~}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>φ<!-- φ --></mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>t</mi>
</mrow>
</mfrac>
</mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>,</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>F</mi>
<mo stretchy="false">(</mo>
<mi>φ<!-- φ --></mi>
<mo stretchy="false">)</mo>
<mo>.</mo>
<mtext> </mtext>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \varphi }{\partial t}}(x,t)=F(\varphi ).~}</annotation>
</semantics>
</math></span></span>
</p><p>The first-order temporal discretization using <a href="Backward_differentiation_formula" title="Backward differentiation formula">backward differences</a> is <sup id="cite_ref-Selection_of_Spatial_and_Temporal_discretization_for_Wetland_modeling_2-0" class="reference"><a href="#cite_note-Selection_of_Spatial_and_Temporal_discretization_for_Wetland_modeling-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>
<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\varphi ^{n+1}-\varphi ^{n}}{\Delta t}}=F(\varphi ),}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<msup>
<mi>φ<!-- φ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<msup>
<mi>φ<!-- φ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mrow>
<mrow>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mi>t</mi>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<mi>F</mi>
<mo stretchy="false">(</mo>
<mi>φ<!-- φ --></mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {\varphi ^{n+1}-\varphi ^{n}}{\Delta t}}=F(\varphi ),}</annotation>
</semantics>
</math></span></span>
</p><p>And the second-order <a href="Discretization" title="Discretization">discretization</a> is
<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {3\varphi ^{n+1}-4\varphi ^{n}+\varphi ^{n-1}}{2\Delta t}}=F(\varphi ),}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mn>3</mn>
<msup>
<mi>φ<!-- φ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<mn>4</mn>
<msup>
<mi>φ<!-- φ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
<mo>+</mo>
<msup>
<mi>φ<!-- φ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mrow>
<mn>2</mn>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mi>t</mi>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<mi>F</mi>
<mo stretchy="false">(</mo>
<mi>φ<!-- φ --></mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {3\varphi ^{n+1}-4\varphi ^{n}+\varphi ^{n-1}}{2\Delta t}}=F(\varphi ),}</annotation>
</semantics>
</math></span></span>
where
</p>
<ul><li><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>φ<!-- φ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \varphi }</annotation>
</semantics>
</math></span><img src="./33ee699558d09cf9d653f6351f9fda0b2f4aaa3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.52ex; height:2.176ex;" alt="{\displaystyle \varphi }" loading="lazy"></span> is a <a href="Scalar_(physics)" title="Scalar (physics)">scalar</a></li>
<li><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n+1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n+1}</annotation>
</semantics>
</math></span><img src="./2a135e65a42f2d73cccbfc4569523996ca0036f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n+1}" loading="lazy"></span> is the value at the next time, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t+\Delta t}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>t</mi>
<mo>+</mo>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mi>t</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle t+\Delta t}</annotation>
</semantics>
</math></span><img src="./e896baa706e07d8100439dc56c5b27771e354e4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.456ex; height:2.343ex;" alt="{\displaystyle t+\Delta t}" loading="lazy"></span></li>
<li><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span> is the value at the current time, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>t</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle t}</annotation>
</semantics>
</math></span><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"></span></li>
<li><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n-1}</annotation>
</semantics>
</math></span><img src="./fbd0b0f32b28f51962943ee9ede4fb34198a2521.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.398ex; height:2.343ex;" alt="{\displaystyle n-1}" loading="lazy"></span> is the value at the previous time, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t-\Delta t}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>t</mi>
<mo>−<!-- − --></mo>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mi>t</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle t-\Delta t}</annotation>
</semantics>
</math></span><img src="./0bd38c11b3d50579d1613733fcb9df63e599fecf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.456ex; height:2.343ex;" alt="{\displaystyle t-\Delta t}" loading="lazy"></span></li></ul>
<p>The function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(\varphi )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>F</mi>
<mo stretchy="false">(</mo>
<mi>φ<!-- φ --></mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F(\varphi )}</annotation>
</semantics>
</math></span><img src="./881491b47250b1ec9630e89bdf106c4497136cc1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.07ex; height:2.843ex;" alt="{\displaystyle F(\varphi )}" loading="lazy"></span> is evaluated using implicit- and explicit-time integration.<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>
<meta property="mw:PageProp/toc">
<div class="mw-heading mw-heading2"><h2 id="Description">Description</h2></div>
<p>Temporal discretization is done by <a href="Integral" title="Integral">integrating</a> the general discretized equation over time. First, values at a given control volume <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>P</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle P}</annotation>
</semantics>
</math></span><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"></span> at time interval <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>t</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle t}</annotation>
</semantics>
</math></span><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"></span> are assumed, and then value at time interval <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t+\Delta t}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>t</mi>
<mo>+</mo>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mi>t</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle t+\Delta t}</annotation>
</semantics>
</math></span><img src="./e896baa706e07d8100439dc56c5b27771e354e4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.456ex; height:2.343ex;" alt="{\displaystyle t+\Delta t}" loading="lazy"></span> is found. This method states that the time integral of a given variable is a weighted average between current and future values. The <a href="Integral" title="Integral">integral</a> form of the equation can be written as:
<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\varphi ^{n+1}-\varphi ^{n}}{\Delta t}}=f\cdot F(\varphi ^{n+1})+(1-f)\cdot F(\varphi ^{n}),}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<msup>
<mi>φ<!-- φ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<msup>
<mi>φ<!-- φ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mrow>
<mrow>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mi>t</mi>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<mi>f</mi>
<mo>⋅<!-- ⋅ --></mo>
<mi>F</mi>
<mo stretchy="false">(</mo>
<msup>
<mi>φ<!-- φ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mi>f</mi>
<mo stretchy="false">)</mo>
<mo>⋅<!-- ⋅ --></mo>
<mi>F</mi>
<mo stretchy="false">(</mo>
<msup>
<mi>φ<!-- φ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {\varphi ^{n+1}-\varphi ^{n}}{\Delta t}}=f\cdot F(\varphi ^{n+1})+(1-f)\cdot F(\varphi ^{n}),}</annotation>
</semantics>
</math></span></span>
where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> is a weight between 0 and 1.
</p>
<ul><li><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=0.0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>=</mo>
<mn>0.0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f=0.0}</annotation>
</semantics>
</math></span><img src="./029a22a48f0ffc54e720b2cdc100c243b7c2a1a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.349ex; height:2.509ex;" alt="{\displaystyle f=0.0}" loading="lazy"></span> yields the fully <a href="Explicit_and_implicit_methods" title="Explicit and implicit methods">explicit scheme</a>.</li>
<li><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=1.0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>=</mo>
<mn>1.0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f=1.0}</annotation>
</semantics>
</math></span><img src="./11fda8800d0e7d2da3c232620fda6074d406c36d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.349ex; height:2.509ex;" alt="{\displaystyle f=1.0}" loading="lazy"></span> yields the fully <a href="Explicit_and_implicit_methods" title="Explicit and implicit methods">implicit scheme</a>.</li>
<li><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=0.5}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>=</mo>
<mn>0.5</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f=0.5}</annotation>
</semantics>
</math></span><img src="./3003a00bce9edc5da0be6a7c424b26cd0658502d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.349ex; height:2.509ex;" alt="{\displaystyle f=0.5}" loading="lazy"></span> yields the <a href="Crank%E2%80%93Nicolson_method" title="Crank–Nicolson method">Crank-Nicolson scheme</a>.</li></ul>
<p>This integration holds for any control volume and any discretized variable. The following equation is obtained when applied to the governing equation, including full discretized <a href="Diffusion" title="Diffusion">diffusion</a>, <a href="Convection" title="Convection">convection</a>, and <a href="Current_source" title="Current source">source</a> terms.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>
<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{t}^{t+\Delta t}F(\varphi )\,dt=[f\cdot F_{\varphi }^{t+\Delta t}+(1-f)\cdot F_{\varphi }^{t}]\,\Delta t}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msubsup>
<mo>∫<!-- ∫ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
<mo>+</mo>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mi>t</mi>
</mrow>
</msubsup>
<mi>F</mi>
<mo stretchy="false">(</mo>
<mi>φ<!-- φ --></mi>
<mo stretchy="false">)</mo>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>t</mi>
<mo>=</mo>
<mo stretchy="false">[</mo>
<mi>f</mi>
<mo>⋅<!-- ⋅ --></mo>
<msubsup>
<mi>F</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>φ<!-- φ --></mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
<mo>+</mo>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mi>t</mi>
</mrow>
</msubsup>
<mo>+</mo>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mi>f</mi>
<mo stretchy="false">)</mo>
<mo>⋅<!-- ⋅ --></mo>
<msubsup>
<mi>F</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>φ<!-- φ --></mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>t</mi>
</mrow>
</msubsup>
<mo stretchy="false">]</mo>
<mspace width="thinmathspace"></mspace>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mi>t</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \int _{t}^{t+\Delta t}F(\varphi )\,dt=[f\cdot F_{\varphi }^{t+\Delta t}+(1-f)\cdot F_{\varphi }^{t}]\,\Delta t}</annotation>
</semantics>
</math></span></span>
</p>
<div class="mw-heading mw-heading2"><h2 id="Methods_for_evaluating_function_F(φ)">Methods for evaluating function <i>F</i>(<i>φ</i>)</h2></div>
<p>After discretizing the time derivative, function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(\varphi )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>F</mi>
<mo stretchy="false">(</mo>
<mi>φ<!-- φ --></mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F(\varphi )}</annotation>
</semantics>
</math></span><img src="./881491b47250b1ec9630e89bdf106c4497136cc1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.07ex; height:2.843ex;" alt="{\displaystyle F(\varphi )}" loading="lazy"></span> remains to be evaluated. The function is now evaluated using implicit and explicit-time integration.<sup id="cite_ref-Notes_on_Spatial_and_Temporal_Discretization_5-0" class="reference"><a href="#cite_note-Notes_on_Spatial_and_Temporal_Discretization-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading3"><h3 id="Implicit-time_integration">Implicit-time integration</h3></div>
<p>This methods evaluates the function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(\varphi )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>F</mi>
<mo stretchy="false">(</mo>
<mi>φ<!-- φ --></mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F(\varphi )}</annotation>
</semantics>
</math></span><img src="./881491b47250b1ec9630e89bdf106c4497136cc1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.07ex; height:2.843ex;" alt="{\displaystyle F(\varphi )}" loading="lazy"></span> at a future time.
</p>
<div class="mw-heading mw-heading4"><h4 id="Formulation">Formulation</h4></div>
<p>The evaluation using implicit-time integration is given as:
<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\varphi ^{n+1}-\varphi ^{n}}{\Delta t}}=F(\varphi ^{n+1}),}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<msup>
<mi>φ<!-- φ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<msup>
<mi>φ<!-- φ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mrow>
<mrow>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mi>t</mi>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<mi>F</mi>
<mo stretchy="false">(</mo>
<msup>
<mi>φ<!-- φ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {\varphi ^{n+1}-\varphi ^{n}}{\Delta t}}=F(\varphi ^{n+1}),}</annotation>
</semantics>
</math></span></span>
</p><p>This is called implicit integration as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi ^{n+1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>φ<!-- φ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \varphi ^{n+1}}</annotation>
</semantics>
</math></span><img src="./bfa5a613574bf485da214e2d64cb24a143f69cf6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.839ex; height:3.176ex;" alt="{\displaystyle \varphi ^{n+1}}" loading="lazy"></span> in a given cell is related to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi ^{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>φ<!-- φ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \varphi ^{n}}</annotation>
</semantics>
</math></span><img src="./2bfefb926a7e3f934344edc2d382566ad9c37c30.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.739ex; height:2.843ex;" alt="{\displaystyle \varphi ^{n}}" loading="lazy"></span> in neighboring cells through <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(\varphi ^{n+1})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>F</mi>
<mo stretchy="false">(</mo>
<msup>
<mi>φ<!-- φ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F(\varphi ^{n+1})}</annotation>
</semantics>
</math></span><img src="./cc576e7f235c8c4b8065d2251d00cce716c37106.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.389ex; height:3.176ex;" alt="{\displaystyle F(\varphi ^{n+1})}" loading="lazy"></span>:
<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi ^{n+1}=\varphi ^{n}+\Delta tF(\varphi ^{n+1}),}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>φ<!-- φ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
<mo>=</mo>
<msup>
<mi>φ<!-- φ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
<mo>+</mo>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mi>t</mi>
<mi>F</mi>
<mo stretchy="false">(</mo>
<msup>
<mi>φ<!-- φ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \varphi ^{n+1}=\varphi ^{n}+\Delta tF(\varphi ^{n+1}),}</annotation>
</semantics>
</math></span></span>
</p><p>In case of implicit method, the setup is unconditionally stable and can handle large time step (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta t}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mi>t</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Delta t}</annotation>
</semantics>
</math></span><img src="./8c28867ecd34e2caed12cf38feadf6a81a7ee542.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.775ex; height:2.176ex;" alt="{\displaystyle \Delta t}" loading="lazy"></span>). But, stability doesn't mean accuracy. Therefore, large <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta t}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mi>t</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Delta t}</annotation>
</semantics>
</math></span><img src="./8c28867ecd34e2caed12cf38feadf6a81a7ee542.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.775ex; height:2.176ex;" alt="{\displaystyle \Delta t}" loading="lazy"></span> affects accuracy and defines time resolution. But, behavior may involve physical timescale that needs to be resolved.
</p>
<div class="mw-heading mw-heading3"><h3 id="Explicit-time_integration">Explicit-time integration</h3></div>
<p>This methods evaluates the function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(\varphi )}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>F</mi>
<mo stretchy="false">(</mo>
<mi>φ<!-- φ --></mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F(\varphi )}</annotation>
</semantics>
</math></span><img src="./881491b47250b1ec9630e89bdf106c4497136cc1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.07ex; height:2.843ex;" alt="{\displaystyle F(\varphi )}" loading="lazy"></span> at a current time.
</p>
<div class="mw-heading mw-heading4"><h4 id="Formulation_2">Formulation</h4></div>
<p>The evaluation using explicit-time integration is given as:
<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\varphi ^{n+1}-\varphi ^{n}}{\Delta t}}=F(\varphi ^{n}),}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<msup>
<mi>φ<!-- φ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<msup>
<mi>φ<!-- φ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mrow>
<mrow>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mi>t</mi>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<mi>F</mi>
<mo stretchy="false">(</mo>
<msup>
<mi>φ<!-- φ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {\varphi ^{n+1}-\varphi ^{n}}{\Delta t}}=F(\varphi ^{n}),}</annotation>
</semantics>
</math></span></span>
</p><p>And is referred as explicit integration since <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi ^{n+1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>φ<!-- φ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \varphi ^{n+1}}</annotation>
</semantics>
</math></span><img src="./bfa5a613574bf485da214e2d64cb24a143f69cf6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.839ex; height:3.176ex;" alt="{\displaystyle \varphi ^{n+1}}" loading="lazy"></span> can be expressed explicitly in the existing solution values, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi ^{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>φ<!-- φ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \varphi ^{n}}</annotation>
</semantics>
</math></span><img src="./2bfefb926a7e3f934344edc2d382566ad9c37c30.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.739ex; height:2.843ex;" alt="{\displaystyle \varphi ^{n}}" loading="lazy"></span>:
<span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi ^{n+1}=\varphi ^{n}+\Delta t\,F(\varphi ^{n}),}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>φ<!-- φ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
<mo>=</mo>
<msup>
<mi>φ<!-- φ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
<mo>+</mo>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mi>t</mi>
<mspace width="thinmathspace"></mspace>
<mi>F</mi>
<mo stretchy="false">(</mo>
<msup>
<mi>φ<!-- φ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
<mo stretchy="false">)</mo>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \varphi ^{n+1}=\varphi ^{n}+\Delta t\,F(\varphi ^{n}),}</annotation>
</semantics>
</math></span></span>
</p><p>Here, the time step (<span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta t}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mi>t</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Delta t}</annotation>
</semantics>
</math></span><img src="./8c28867ecd34e2caed12cf38feadf6a81a7ee542.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.775ex; height:2.176ex;" alt="{\displaystyle \Delta t}" loading="lazy"></span>) is restricted by the stability limit of the solver (i.e., time step is limited by the <a href="Courant%E2%80%93Friedrichs%E2%80%93Lewy_condition" title="Courant–Friedrichs–Lewy condition">Courant–Friedrichs–Lewy condition</a>). To be accurate with respect to time the same time step should be used in all the domain, and to be stable the time step must be the minimum of all the local time steps in the domain. This method is also referred to as "global time stepping".
</p>
<div class="mw-heading mw-heading4"><h4 id="Examples">Examples</h4></div>
<p>Many schemes use explicit-time integration. Some of these are as follows:
</p>
<ul><li><a href="Lax%E2%80%93Wendroff_method" title="Lax–Wendroff method">Lax–Wendroff method</a></li>
<li><a href="Runge%E2%80%93Kutta_method" class="mw-redirect" title="Runge–Kutta method">Runge–Kutta method</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="Courant%E2%80%93Friedrichs%E2%80%93Lewy_condition" title="Courant–Friedrichs–Lewy condition">Courant–Friedrichs–Lewy condition</a>.</li>
<li><a href="Von_Neumann_stability_analysis" title="Von Neumann stability analysis">Von Neumann stability analysis</a>.</li>
<li><a href="Finite_element_method" title="Finite element method">Finite element method</a></li>
<li><a href="Explicit_and_implicit_methods" title="Explicit and implicit methods">Explicit and implicit methods</a></li>
<li><a href="Chi-Wang_Shu" title="Chi-Wang Shu">Chi-Wang Shu</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20160305032706/http://aerojet.engr.ucdavis.edu/fluenthelp/html/ug/node993.htm">"Spatial and Temporal Discretization"</a>. Archived from <a rel="nofollow" class="external text" href="http://aerojet.engr.ucdavis.edu/fluenthelp/html/ug/node993.htm">the original</a> on 5 March 2016.</cite></span>
</li>
<li id="cite_note-Selection_of_Spatial_and_Temporal_discretization_for_Wetland_modeling-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Selection_of_Spatial_and_Temporal_discretization_for_Wetland_modeling_2-0">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://gwmftp.jacobs.com/Peer_Review/resolution.pdf">Selection of Spatial and Temporal discretization</a></span>
</li>
<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.cfd-online.com/Wiki/Discretization_of_the_transient_term">"Discretisation of transient term"</a>.</cite></span>
</li>
<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.xmswiki.com/xms/GMS:Temporal_Discretization">"Examples of Temporal Discretization"</a>.</cite></span>
</li>
<li id="cite_note-Notes_on_Spatial_and_Temporal_Discretization-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-Notes_on_Spatial_and_Temporal_Discretization_5-0">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.pc-progress.com/Documents/Notes_on_Spatial_and_Temporal_Discretization.pdf">Jirka Simunek</a></span>
</li>
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