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Fix another symbol
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danieljfarrell committed Apr 19, 2024
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4 changes: 2 additions & 2 deletions _sources/solving_linear_equations.rst.txt
Original file line number Diff line number Diff line change
Expand Up @@ -9,12 +9,12 @@ Time stepping the advection-diffusion equation
Here we will rearrange the :math:`\theta`-method form of the advection-diffusion equation,

.. math::
\frac{w^{n+1} - w^{n}}{\tau} & = \theta F(w^{n+1}) + (1-\theta)F(w^{n})
\frac{w^{n+1} - w^{n}}{\tau} = \theta F(w^{n+1}) + (1-\theta)F(w^{n})
into the form of a linear system :math:`A\cdot x = d`. Firstly lets move all terms involving future time points the l.h.s,

.. math::
w^{n+1} - \theta \tau F(w^{n+1}) = w^{n} + (1-\theta) F(w^{n+1})
w^{n+1} - \theta \tau F(w^{n+1}) = w^{n} + (1-\theta) \tau F(w^{n+1})
Replacing the :math:`F` terms with the full matrix expressions and factoring yields,

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4 changes: 2 additions & 2 deletions solving_linear_equations.html
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Expand Up @@ -48,10 +48,10 @@ <h1>Solving linear equations<a class="headerlink" href="#solving-linear-equation
<h2>Time stepping the advection-diffusion equation<a class="headerlink" href="#time-stepping-the-advection-diffusion-equation" title="Link to this heading"></a></h2>
<p>Here we will rearrange the <span class="math notranslate nohighlight">\(\theta\)</span>-method form of the advection-diffusion equation,</p>
<div class="math notranslate nohighlight">
\[\frac{w^{n+1} - w^{n}}{\tau} &amp; = \theta F(w^{n+1}) + (1-\theta)F(w^{n})\]</div>
\[\frac{w^{n+1} - w^{n}}{\tau} = \theta F(w^{n+1}) + (1-\theta)F(w^{n})\]</div>
<p>into the form of a linear system <span class="math notranslate nohighlight">\(A\cdot x = d\)</span>. Firstly lets move all terms involving future time points the l.h.s,</p>
<div class="math notranslate nohighlight">
\[w^{n+1} - \theta \tau F(w^{n+1}) = w^{n} + (1-\theta) F(w^{n+1})\]</div>
\[w^{n+1} - \theta \tau F(w^{n+1}) = w^{n} + (1-\theta) \tau F(w^{n+1})\]</div>
<p>Replacing the <span class="math notranslate nohighlight">\(F\)</span> terms with the full matrix expressions and factoring yields,</p>
<div class="math notranslate nohighlight">
\[\underbrace{(I - \theta\tau M^{n+1})}_{A}\underbrace{w^{n+1}}_{x} = \underbrace{(I + (1-\theta)\tau M^{n})w^{n}}_{d}\]</div>
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4 changes: 2 additions & 2 deletions source/solving_linear_equations.rst
Original file line number Diff line number Diff line change
Expand Up @@ -9,12 +9,12 @@ Time stepping the advection-diffusion equation
Here we will rearrange the :math:`\theta`-method form of the advection-diffusion equation,

.. math::
\frac{w^{n+1} - w^{n}}{\tau} & = \theta F(w^{n+1}) + (1-\theta)F(w^{n})
\frac{w^{n+1} - w^{n}}{\tau} = \theta F(w^{n+1}) + (1-\theta)F(w^{n})
into the form of a linear system :math:`A\cdot x = d`. Firstly lets move all terms involving future time points the l.h.s,

.. math::
w^{n+1} - \theta \tau F(w^{n+1}) = w^{n} + (1-\theta) F(w^{n+1})
w^{n+1} - \theta \tau F(w^{n+1}) = w^{n} + (1-\theta) \tau F(w^{n+1})
Replacing the :math:`F` terms with the full matrix expressions and factoring yields,

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