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Address review feedback: MathJax macros, figure improvements, grammar fixes
- Add MathJax macro definitions for HTML output in _quarto.yml - Regenerate stencil figures with legend explaining known/unknown values - Update figure captions to be more descriptive - Fix gendered pronoun in softeng2 appendix - Fix malformed figure captions in diffu_exer.qmd - Remove orphaned LaTeX label{} in diffu_app.qmd - Improve verbose phrasing in wave_app.qmd and nonlin_pde1D.qmd
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‎_quarto.yml‎

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@@ -41,6 +41,93 @@ format:
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number-depth: 3
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crossref:
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chapters: true
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include-in-header:
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text: |
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<script>
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MathJax = {
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tex: {
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macros: {
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half: "\\frac{1}{2}",
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halfi: "{1/2}",
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tp: "\\thinspace .",
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uex: "u_{\\mbox{\\footnotesize e}}",
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uexd: ["u_{\\mbox{\\footnotesize e}, #1}", 1],
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vex: "v_{\\mbox{\\footnotesize e}}",
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Vex: "V_{\\mbox{\\footnotesize e}}",
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vexd: ["v_{\\mbox{\\footnotesize e}, #1}", 1],
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Aex: "A_{\\mbox{\\footnotesize e}}",
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wex: "w_{\\mbox{\\footnotesize e}}",
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Ddt: ["\\frac{D #1}{dt}", 1],
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E: ["\\hbox{E}\\lbrack #1 \\rbrack", 1],
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Var: ["\\hbox{Var}\\lbrack #1 \\rbrack", 1],
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Std: ["\\hbox{Std}\\lbrack #1 \\rbrack", 1],
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xpoint: "\\boldsymbol{x}",
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normalvec: "\\boldsymbol{n}",
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Oof: ["\\mathcal{O}(#1)", 1],
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x: "\\boldsymbol{x}",
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X: "\\boldsymbol{X}",
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uu: "\\boldsymbol{u}",
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vv: "\\boldsymbol{v}",
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w: "\\boldsymbol{w}",
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acc: "\\boldsymbol{a}",
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rpos: "\\boldsymbol{r}",
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V: "\\boldsymbol{V}",
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e: "\\boldsymbol{e}",
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f: "\\boldsymbol{f}",
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F: "\\boldsymbol{F}",
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stress: "\\boldsymbol{\\sigma}",
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strain: "\\boldsymbol{\\varepsilon}",
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stressc: "{\\sigma}",
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strainc: "{\\varepsilon}",
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I: "\\boldsymbol{I}",
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T: "\\boldsymbol{T}",
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U: "\\boldsymbol{U}",
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q: "\\boldsymbol{q}",
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g: "\\boldsymbol{g}",
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dfc: "\\alpha",
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ii: "\\boldsymbol{i}",
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jj: "\\boldsymbol{j}",
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kk: "\\boldsymbol{k}",
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ir: "\\boldsymbol{i}_r",
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ith: "\\boldsymbol{i}_{\\theta}",
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iz: "\\boldsymbol{i}_z",
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Ix: "\\mathcal{I}_x",
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Iy: "\\mathcal{I}_y",
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Iz: "\\mathcal{I}_z",
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It: "\\mathcal{I}_t",
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If: "\\mathcal{I}_s",
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Ifd: "{I_d}",
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Ifb: "{I_b}",
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setb: ["#1^0", 1],
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sete: ["#1^{-1}", 1],
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setl: ["#1^-", 1],
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setr: ["#1^+", 1],
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seti: ["#1^i", 1],
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sequencei: ["\\left\\{ {#1}_i \\right\\}_{i\\in\\If}", 1],
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sequencej: ["\\left\\{ {#1}_j \\right\\}_{j\\in\\If}", 1],
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stepzero: "*",
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stephalf: "***",
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stepone: "**",
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basphi: "\\varphi",
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baspsi: "\\psi",
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refphi: "\\tilde\\basphi",
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psib: "\\boldsymbol{\\psi}",
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sinL: ["\\sin\\left((#1+1)\\pi\\frac{x}{L}\\right)", 1],
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xno: ["x_{#1}", 1],
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Xno: ["X_{(#1)}", 1],
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yno: ["y_{#1}", 1],
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Yno: ["Y_{(#1)}", 1],
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xdno: ["\\boldsymbol{x}_{#1}", 1],
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dX: "\\, \\mathrm{d}X",
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dx: "\\, \\mathrm{d}x",
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ds: "\\, \\mathrm{d}s",
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Real: "\\mathbb{R}",
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Integerp: "\\mathbb{N}",
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Integer: "\\mathbb{Z}"
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}
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}
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};
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</script>
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pdf:
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documentclass: scrbook
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classoption:

‎chapters/appendices/softeng2/softeng2.qmd‎

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@@ -778,8 +778,7 @@ user will not notice a change to properties.
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The only argument against direct attribute access in class `Mesh`
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is that the attributes are read-only so we could avoid offering
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a set function. Instead, we rely on the user that she does not
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assign new values to the attributes.
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a set function. Instead, we rely on the user not to assign new values to the attributes.
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:::
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## Class Function

‎chapters/diffu/diffu_app.qmd‎

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@@ -458,8 +458,7 @@ governed by the [Cable equation](http://en.wikipedia.org/wiki/Cable_equation):
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$$
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c_m \frac{\partial V}{\partial t} =
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\frac{1}{r_l}\frac{\partial^2 V}{\partial x^2} - \frac{1}{r_m}V
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label{}
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$$
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$$ {#eq-diffu-app-cable}
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where $V(x,t)$ is the voltage to be determined,
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$c_m$ is capacitance of the neuronal fiber, while
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$r_l$ and $r_m$ are measures of the resistance.

‎chapters/diffu/diffu_exer.qmd‎

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Running the `investigate` function, we get the following plots:
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![FIGURE: [fig-diffu/welding_gamma0_2, width=800 frac=1]](fig/welding_gamma0_025){width="100%"}
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![Temperature distribution for $\gamma = 0.025$: the heat source moves very slowly on the diffusion time scale.](fig/welding_gamma0_025){width="100%"}
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![FIGURE: [fig-diffu/welding_gamma5, width=800 frac=1]](fig/welding_gamma1){width="100%"}
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![Temperature distribution for $\gamma = 1$: the two scaling approaches give identical results.](fig/welding_gamma1){width="100%"}
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![For $\gamma\ll 1$ as in $\gamma = 0.025$, the heat source moves very
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slowly on the diffusion time scale and has hardly entered the medium,

‎chapters/diffu/diffu_fd3.qmd‎

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@@ -66,7 +66,7 @@ F_x = \frac{\dfc\Delta t}{\Delta x^2},\quad F_y = \frac{\dfc\Delta t}{\Delta y^2
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$$
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are the Fourier numbers in $x$ and $y$ direction, respectively.
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![3x2 2D mesh.](fig/mesh3x2){#fig-diffu-2D-fig-mesh3x2 width="500px"}
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![A 2D mesh with $N_x=3$ and $N_y=2$ cells, showing interior and boundary points.](fig/mesh3x2){#fig-diffu-2D-fig-mesh3x2 width="500px"}
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## Numbering of mesh points versus equations and unknowns {#sec-diffu-2D-numbering}
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@@ -262,7 +262,7 @@ p = (j-1)(N_x-1) + i,
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$$
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for $i=1,\ldots,N_x-1$, $j=1,\ldots,N_y-1$.
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![4x3 2D mesh.](fig/mesh4x3){#fig-diffu-2D-fig-mesh4x3 width="700px"}
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![A 2D mesh with $N_x=4$ and $N_y=3$ cells, illustrating the unknown numbering scheme.](fig/mesh4x3){#fig-diffu-2D-fig-mesh4x3 width="700px"}
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We can continue with illustrating a bit larger mesh, $N_x=4$ and $N_y=3$,
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see Figure @fig-diffu-2D-fig-mesh4x3. The corresponding coefficient matrix

‎chapters/nonlin/nonlin_pde1D.qmd‎

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Other entries are as in the $2\times 2$ system.
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### Newton's method
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The Jacobian must be derived in order to use Newton's method. Here it means
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Using Newton's method requires deriving the Jacobian. Here it means
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that we need to differentiate $F(u)=A(u)u - b(u)$ with respect to
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the unknown parameters
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$u_0,u_1,\ldots,u_m$ ($m=N_x$ or $m=N_x-1$, depending on whether the

‎chapters/wave/fig/stencil_generator.py‎

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#!/usr/bin/env python
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"""
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Generate stencil figures for the wave equation chapter.
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This script generates three stencil diagrams:
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- stencil_n_interior.png: Standard 5-point stencil at interior point
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- stencil_n0_interior.png: Modified 4-point stencil for first time step
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- stencil_n_left.png: Modified stencil at left boundary (Neumann condition)
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Each figure includes a legend explaining:
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- Filled blue circles: Known values (computed at previous time steps)
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- Empty black circle: Unknown value (to be computed)
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"""
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import matplotlib.patches as mpatches
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import matplotlib.pyplot as plt
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def create_stencil_figure(
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known_points,
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unknown_point,
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title,
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filename,
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xlim=(0, 5),
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ylim=(0, 5),
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):
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"""
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Create a stencil figure with legend.
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Parameters
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----------
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known_points : list of tuples
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List of (i, n) coordinates for known values
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unknown_point : tuple
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(i, n) coordinate for the unknown value
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title : str
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Figure title
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filename : str
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Output filename
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xlim, ylim : tuples
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Axis limits
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"""
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fig, ax = plt.subplots(figsize=(8, 6))
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# Plot grid
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ax.set_xlim(xlim)
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ax.set_ylim(ylim)
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ax.set_xticks(range(xlim[0], xlim[1] + 1))
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ax.set_yticks(range(ylim[0], ylim[1] + 1))
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ax.grid(True, linestyle='--', alpha=0.5)
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ax.set_aspect('equal')
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# Plot known points (filled blue circles)
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for i, n in known_points:
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circle = plt.Circle(
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(i, n), 0.15, fill=True, color='blue', linewidth=2
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)
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ax.add_patch(circle)
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# Plot unknown point (empty black circle)
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i, n = unknown_point
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circle = plt.Circle(
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(i, n), 0.15, fill=False, color='black', linewidth=2
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)
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ax.add_patch(circle)
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# Labels
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ax.set_xlabel('index i', fontsize=12)
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ax.set_ylabel('index n', fontsize=12)
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ax.set_title(title, fontsize=14)
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# Create legend
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known_patch = mpatches.Patch(
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facecolor='blue', edgecolor='blue',
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label='Known (from previous time steps)'
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)
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unknown_patch = mpatches.Patch(
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facecolor='white', edgecolor='black',
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label='Unknown (to be computed)'
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)
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ax.legend(
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handles=[known_patch, unknown_patch],
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loc='upper right',
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fontsize=10,
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framealpha=0.9
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)
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plt.tight_layout()
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plt.savefig(filename, dpi=150, bbox_inches='tight')
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plt.close()
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print(f"Generated: {filename}")
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def main():
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# Figure 1: Standard interior stencil (5 points)
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# Computing u[2]^3 from u[1]^2, u[2]^2, u[3]^2, u[2]^1
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create_stencil_figure(
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known_points=[(1, 2), (2, 2), (3, 2), (2, 1)],
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unknown_point=(2, 3),
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title='Stencil at interior point',
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filename='stencil_n_interior.png'
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)
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# Figure 2: First time step stencil (4 points, no n-1 level)
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# Computing u[2]^1 from u[1]^0, u[2]^0, u[3]^0
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create_stencil_figure(
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known_points=[(1, 0), (2, 0), (3, 0)],
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unknown_point=(2, 1),
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title='Stencil at interior point (first time step)',
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filename='stencil_n0_interior.png'
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)
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# Figure 3: Left boundary stencil (Neumann condition)
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# Computing u[0]^3 from u[0]^2, u[1]^2, u[0]^1
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create_stencil_figure(
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known_points=[(0, 2), (1, 2), (0, 1)],
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unknown_point=(0, 3),
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title='Stencil at boundary point (Neumann condition)',
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filename='stencil_n_left.png'
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)
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print("\nAll stencil figures generated successfully!")
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print("\nLegend explanation:")
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print(" - Filled blue circles: Known values (computed at previous time steps)")
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print(" - Empty black circle: Unknown value (to be computed)")
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if __name__ == '__main__':
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main()

‎chapters/wave/fig/stencil_n0_interior.png‎

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‎chapters/wave/fig/stencil_n_interior.png‎

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‎chapters/wave/fig/stencil_n_left.png‎

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