From b7ed2a835817d660237b924e3f6d5886f01cadd0 Mon Sep 17 00:00:00 2001 From: danieljfarrell Date: Fri, 19 Apr 2024 15:17:51 +0100 Subject: [PATCH] Fix another symbol --- _sources/solving_linear_equations.rst.txt | 4 ++-- doctrees/environment.pickle | Bin 255455 -> 255783 bytes doctrees/solving_linear_equations.doctree | Bin 9383 -> 9389 bytes solving_linear_equations.html | 4 ++-- source/solving_linear_equations.rst | 4 ++-- 5 files changed, 6 insertions(+), 6 deletions(-) diff --git a/_sources/solving_linear_equations.rst.txt b/_sources/solving_linear_equations.rst.txt index a5d7160..6bdfd02 100644 --- a/_sources/solving_linear_equations.rst.txt +++ b/_sources/solving_linear_equations.rst.txt @@ -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, diff --git a/doctrees/environment.pickle b/doctrees/environment.pickle index 5607ad5a06593e40dafa7d5620b1b2e1b27d9f87..016caf63cab441a89ba4cc6887b24cb7bb52e078 100644 GIT binary patch literal 255783 zcmeFa37i~9eJ^eud+I(c+t|j9jOE?+&aPIr!Imu>*|Ll!OLlxA!rG(WncnSb&86uX z?XJZV5<((5O*lM+BP7HGa|Cida=g4h2}c4UAtaASLQImE1QL<}c}X4w$i?IT{r#${ zyQ^k;H9I?_6$X9m+3xDA_t^`Pfr->LZZU`^I8)`PwSl~Q@CG*=JS-dV~zC_ebe5xgcr z`}J!5fqKyES2FY!cVNgJbl17--3{(WcN5A>wMxd3l`^H`bZ({|Y>-V+_q}(D2Vwhr ze5YFD9yoJVr|9QOMZf-NJ;K9Wb*z9kd-h!24*K(Uai(U^pmD1mGTY;qAV-vVXv19vA-Ax5O)l$};ue-y+ z0AHo23stmjUA~mD^G+Jb;~w^jdmr7p6u7EYoV4m)dODYPigp3r+;O^6DpeQ8POE1N zr-7)2dSJ~s)haI2c0OO4O;6Vze|$b&t=L6BZ&w+MXM-zZ_i1!_rsDX1I+M43-|_2b 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Z8s%}-dD@1xnp|8{pvu59n_D?GqycFE8ngfa diff --git a/solving_linear_equations.html b/solving_linear_equations.html index e7fed8d..87cdfe3 100644 --- a/solving_linear_equations.html +++ b/solving_linear_equations.html @@ -48,10 +48,10 @@

Here we will rearrange the \(\theta\)-method form of the advection-diffusion equation,

-\[\frac{w^{n+1} - w^{n}}{\tau} & = \theta F(w^{n+1}) + (1-\theta)F(w^{n})\]
+\[\frac{w^{n+1} - w^{n}}{\tau} = \theta F(w^{n+1}) + (1-\theta)F(w^{n})\]

into the form of a linear system \(A\cdot x = d\). Firstly lets move all terms involving future time points the l.h.s,

-\[w^{n+1} - \theta \tau F(w^{n+1}) = w^{n} + (1-\theta) F(w^{n+1})\]
+\[w^{n+1} - \theta \tau F(w^{n+1}) = w^{n} + (1-\theta) \tau F(w^{n+1})\]

Replacing the \(F\) terms with the full matrix expressions and factoring yields,

\[\underbrace{(I - \theta\tau M^{n+1})}_{A}\underbrace{w^{n+1}}_{x} = \underbrace{(I + (1-\theta)\tau M^{n})w^{n}}_{d}\]
diff --git a/source/solving_linear_equations.rst b/source/solving_linear_equations.rst index a5d7160..6bdfd02 100644 --- a/source/solving_linear_equations.rst +++ b/source/solving_linear_equations.rst @@ -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,

Solving linear equationsΒΆ