Fix DocOnce artifacts causing PDF formatting issues

- Replace DocOnce-style quotes (``text'') with standard quotes
- Remove URL incorrectly used as section heading (diffu_fd2.qmd)
- Remove leftover Mako conditionals (vib_undamped.qmd)
- Remove DocOnce author comments appearing as headings
- Fix truncation error appendix reference

Author: ggorman

chapters/advec/advec.qmd

@@ -764,7 +764,7 @@ numerical solutions.
 ## Leapfrog in time, centered differences in space
 ### Method {#sec-advec-1D-leapfrog}
 
-Another explicit scheme is to do a ``leapfrog'' jump over $2\Delta t$ in
+Another explicit scheme is to do a "leapfrog" jump over $2\Delta t$ in
 time and combine it with central differences in space:
 $$
 [D_{2t} u + vD_{2x} u = 0]_i^n,
@@ -1458,7 +1458,7 @@ signals to the left and the right. The value $u(0)=0$ is transported
 through the domain if $\epsilon$ is small, and $u\approx 0$ except in
 the vicinity of $x=1$, where $u(1)=1$ and the diffusion transports
 some information about $u(1)=1$ to the left. For large $\epsilon$,
-diffusion dominates and the $u$ takes on the ``average'' value, i.e.,
+diffusion dominates and the $u$ takes on the "average" value, i.e.,
 $u$ gets a linear variation from 0 to 1 throughout the domain.
 
 It turns out that we can find an exact solution to the differential

chapters/appendices/softeng2/softeng2.qmd

@@ -2174,11 +2174,11 @@ Simulate for $C=1, 0.9, 0.75$ and make an animation comparing the
 three curves (use the `animate_archives.py` program to combine the
 curves and make animations on the screen and video files).  Perform
 the investigations for different types of initial profiles: a Gaussian
-pulse, a ``cosine hat'' pulse, half a``cosine hat'' pulse, and a plug
+pulse, a "cosine hat" pulse, half a"cosine hat" pulse, and a plug
 pulse.
 
 !bsol
-We make a little Python script for running one ``pulse'' simulation:
+We make a little Python script for running one "pulse" simulation:
 
 ```python
 ```

chapters/diffu/diffu_analysis.qmd

@@ -790,7 +790,7 @@ the $u_x=0$ condition from the previous subexercise that is
 qualitatively wrong for large $t$.)
 Develop a diffusion problem for the error in the solution using
 (@eq-diffu-pde1-Gaussian-xL-cooling) as boundary condition.
-Assume one can take $u_S=0$ ``outside the domain'' since
+Assume one can take $u_S=0$ "outside the domain" since
 $\uex\rightarrow 0$ as $x\rightarrow\infty$.
 Find a function $q=q(t)$ such that the exact solution
 obeys the condition (@eq-diffu-pde1-Gaussian-xL-cooling).

chapters/diffu/diffu_exer.qmd

@@ -60,7 +60,7 @@ energy of the system. A mathematical tradition has introduced the
 notion *energy* in this context.
 
 The estimate (@eq-diffu-exer-estimates-p1-result) says that the
-``size of $u$'' never exceeds that of the initial condition,
+"size of $u$" never exceeds that of the initial condition,
 or more precisely, it says that the area under the $u$ curve decreases
 with time.
 
@@ -260,7 +260,7 @@ The surface temperature is set as
 $$
 T(0,t) = T_m + A\sin(\omega t)\tp
 $$
-With only one ``active'' spatial coordinate we get the initial-boundary
+With only one "active" spatial coordinate we get the initial-boundary
 value problem
 
 \begin{alignat*}{2}
@@ -329,7 +329,7 @@ Inserting $u(\bar x, \bar t)=e^{-\bar x}\sin (\bar t - \bar x)$ in the
 PDE shows that this is a solution. It also obeys
 the boundary condition $\bar u(0,\bar t)=sin(\bar t)$. As
 $\bar t\rightarrow\infty$, the initial condition has no longer impact
-on the solution and is ``forgotten'' and of no interest.
+on the solution and is "forgotten" and of no interest.
 The boundary condition at $\bar x=\bar L$ is never compatible with the
 given solution unless $\bar u$ is damped to zero, which happens
 mathematically as $\bar L\rightarrow\infty$. For a numerical solution,

chapters/diffu/diffu_fd1.qmd

@@ -1170,7 +1170,7 @@ which written out reads
 $$
 \frac{u^{n}_i-u^{n-1}**i}{\Delta t} = \dfc\frac{u^{n}**{i+1} - 2u^n_i + u^n_{i-1}}{\Delta x^2} + f_i^n\tp
 $$ {#eq-diffu-pde1-step3bBE}
-Now we assume $u^{n-1}_i$ is already computed, but that all quantities at the ``new''
+Now we assume $u^{n-1}_i$ is already computed, but that all quantities at the "new"
 time level $n$ are unknown. This time it is not possible to solve
 with respect to $u_i^{n}$ because this value couples to its neighbors
 in space, $u^n_{i-1}$ and $u^n_{i+1}$, which are also unknown.

chapters/diffu/diffu_fd2.qmd

@@ -270,8 +270,7 @@ shows the result.
 
 ![Solution of the stationary diffusion equation corresponding to a *smoothed* piecewise constant diffusion coefficient.](fig/flow_in_layers_case1_eps){#fig-diffu-fd2-pde-st-sol-pc-fig2 width="400px"}
 
-## Axi-symmetric diffusion
-## http://www.ewp.rpi.edu/hartford/~ernesto/S2004/CHT/Notes/s06.pdf {#sec-diffu-fd2-radial}
+## Axi-symmetric diffusion {#sec-diffu-fd2-radial}
 
 Suppose we have a diffusion process taking place in a straight tube
 with radius $R$. We assume axi-symmetry such that $u$ is just a
@@ -453,7 +452,7 @@ $$
 $$
 For $\alpha$ constant we immediately realize that we can reuse a
 solver in Cartesian coordinates to compute $v$. With variable $\alpha$,
-a ``reaction'' term $v/r$ needs to be added to the Cartesian solver.
+a "reaction" term $v/r$ needs to be added to the Cartesian solver.
 The boundary condition $\partial u/\partial r=0$ at $r=0$, implied
 by symmetry, forces $v(0,t)=0$, because
 $$

chapters/diffu/diffu_fd3.qmd

@@ -141,7 +141,7 @@ to $p$:
 (2,3)\rightarrow 11\tp
 \end{align*}
 That is, we number the points along the $x$ axis, starting with $y=0$,
-and then progress one ``horizontal'' mesh line at a time.
+and then progress one "horizontal" mesh line at a time.
 In Figure @fig-diffu-2D-fig-mesh3x2 you can see that the $(i,j)$ and the
 corresponding single index ($p$) are listed for each mesh point.
 
@@ -912,7 +912,7 @@ for n in It[0:-1]:
 The most tricky part of this code snippet is the loading of values from
 the one-dimensional array `c`
 into the two-dimensional array `u`. With our numbering of unknowns
-from left to right along ``horizontal'' mesh lines, the correct
+from left to right along "horizontal" mesh lines, the correct
 reordering of the one-dimensional array `c` as a two-dimensional array
 requires first a reshaping to an `(Ny+1,Nx+1)` two-dimensional
 array and then taking the transpose. The result is an `(Nx+1,Ny+1)`
@@ -1597,7 +1597,7 @@ i = slice(1, None, 2);  j = slice(2, None, 2)
 That is, we need four sets of slices. The simplest way of implementing
 the algorithm is to make a function with variables for the slices
 representing $i$, $i-1$, $i+1$, $j$, $j-1$, and $j+1$, here called
-`ic` (``i center''), `im1` (``i minus 1'', `ip1` (``i plus 1''), `jc`, `jm1`,
+`ic` ("i center"), `im1` ("i minus 1", `ip1` ("i plus 1"), `jc`, `jm1`,
 and `jp1`, respectively.
 
 ```python

chapters/diffu/diffu_rw.qmd

@@ -13,9 +13,9 @@ point in a small time interval and small spatial volume.
 
 Random walk is a principally different kind of modeling procedure
 compared to the reasoning behind partial differential equations.  The
-idea in random walk is to have a large number of ``particles'' that
+idea in random walk is to have a large number of "particles" that
 undergo random movements. Averaging can then be used afterwards to
-compute macroscopic quantities like concentration. The``particles''
+compute macroscopic quantities like concentration. The"particles"
 and their random movement represent a very simplified microscopic
 behavior of molecules, much simpler and computationally much more
 efficient than direct [molecular simulation](https://en.wikipedia.org/wiki/Molecular_dynamics), yet the random
@@ -98,7 +98,7 @@ $$
 \left(\frac{1}{W}\sum_{j=0}^{W-1} \bar x_{j,k}\right)^2\tp
 $$
 That is, we take the statistics for a given $K$ across the ensemble
-of random walks (``vertically'' in
+of random walks ("vertically" in
 Figure @fig-diffu-randomwalk-1D-fig-ensemble). The key quantities
 to record are $\sum_i \bar x_{i,k}$ and $\sum_i \bar x_{i,k}^2$.
 

chapters/nonlin/nonlin_exer.qmd

@@ -183,7 +183,7 @@ numerical answers should be identical.
 Implement unit tests that check the asymptotic limit of the solutions:
 $u\rightarrow M$ as $t\rightarrow\infty$.
 
-:::{.callout-tip title="You need to experiment to find what ``infinite time'' is"}
+:::{.callout-tip title="You need to experiment to find what "infinite time" is"}
 (increases substantially with $p$) and what the
 appropriate tolerance is for testing the asymptotic limit.
 :::

chapters/nonlin/nonlin_ode.qmd

@@ -368,7 +368,7 @@ approach in science and technology, but with some limitations if
 $\Delta t$ is not sufficiently small (as will be illustrated later).
 
 :::{.callout-note title="Equation (@eq-nonlin-timediscrete-logistic-BE-Picard-1it) does not"}
-correspond to a ``pure'' finite difference method where the equation
+correspond to a "pure" finite difference method where the equation
 is sampled at a point and derivatives replaced by differences (because
 the $u^{n-1}$ term on the right-hand side must then be $u^n$). The
 best interpretation of the scheme
@@ -535,7 +535,7 @@ notation used in the literature.
 One iteration in Newton's method or
 Picard iteration consists of solving a linear problem $\hat F(u)=0$.
 Sometimes convergence problems arise because the new solution $u$
-of $\hat F(u)=0$ is ``too far away'' from the previously computed
+of $\hat F(u)=0$ is "too far away" from the previously computed
 solution $u^{-}$. A remedy is to introduce a relaxation, meaning that
 we first solve $\hat F(u^*)=0$ for a suggested value $u^*$ and
 then we take $u$ as a weighted mean of what we had, $u^{-}$, and
@@ -772,16 +772,16 @@ until a stopping criterion is fulfilled.
 Evaluating $f$ for a known $u^{-}$ is referred to as *explicit* treatment of
 $f$, while if $f(u,t)$ has some structure, say $f(u,t) = u^3$, parts of
 $f$ can involve the unknown $u$, as in the manual linearization
-$(u^{-})^2u$, and then the treatment of $f$ is ``more implicit''
-and ``less explicit''. This terminology is inspired by time discretization
+$(u^{-})^2u$, and then the treatment of $f$ is "more implicit"
+and "less explicit". This terminology is inspired by time discretization
 of $u^{\prime}=f(u,t)$, where evaluating $f$ for known $u$ values gives
 explicit schemes, while treating $f$ or parts of $f$ implicitly,
 makes $f$ contribute to the unknown terms in the equation at the new
 time level.
 
 Explicit treatment of $f$ usually means stricter conditions on
 $\Delta t$ to achieve stability of time discretization schemes. The same
-applies to iteration techniques for nonlinear algebraic equations: the ``less''
+applies to iteration techniques for nonlinear algebraic equations: the "less"
 we linearize $f$ (i.e., the more we keep of $u$ in the original formula),
 the faster the convergence may be.
 
@@ -1002,7 +1002,7 @@ Given $A(u)u=b(u)$ and an initial guess $u^{-}$, iterate until convergence:
  1. $u^{-}\ \leftarrow\ u$
 :::
 
-``Until convergence'' means that the iteration is stopped when the
+"Until convergence" means that the iteration is stopped when the
 change in the unknown, $||u - u^{-}||$, or the residual $||A(u)u-b||$,
 is sufficiently small, see Section @sec-nonlin-systems-alg-terminate for
 more details.