fixed typos in wave Chapter

Author: sveinlin

doc/.src/chapters/wave/wave1D_fd1.do.txt

@@ -39,6 +39,7 @@ in the forthcoming text by finite difference methods.
 label{wave:string}
 idx{waves!on a string}
 idx{wave equation!1D}
+idx{wave! velocity}
 
 We begin our study of wave equations by simulating one-dimensional
 waves on a string, say on a guitar or violin.
@@ -107,6 +108,8 @@ idx{mesh!finite differences}
 ===== Discretizing the domain =====
 label{wave:string:mesh}
 
+idx{mesh! uniform}
+
 The temporal domain $[0,T]$ is represented by a finite number of mesh points
 
 !bt
@@ -156,6 +159,9 @@ develop algebraic equations for computing the mesh function.
 ===== Fulfilling the equation at the mesh points =====
 label{wave:string:samplingPDE}
 
+idx{PDE! algebraic version}
+idx{finite difference scheme}
+
 In the finite difference method, we relax
 the condition that (ref{wave:pde1}) holds at all points in
 the space-time domain $(0,L)\times (0,T]$ to the requirement that the PDE is
@@ -222,14 +228,18 @@ label{wave:pde1:step3a}
 
 === Interpretation of the equation as a stencil ===
 
+idx{finite difference scheme}
+idx{difference equations}
+idx{sampling (a PDE)}
+
 A characteristic feature of (ref{wave:pde1:step3b}) is that it
 involves $u$ values from neighboring points only: $u_i^{n+1}$,
 $u^n_{i\pm 1}$, $u^n_i$, and $u^{n-1}_i$.  The circles in Figure
 ref{wave:pde1:fig:mesh} illustrate such neighboring mesh points that
 contribute to an algebraic equation. In this particular case, we have
 sampled the PDE at the point $(2,2)$ and constructed
-(ref{wave:pde1:step3b}), which then involves a coupling of $u_nm1^1$,
-$u_n^2$, $u_nm1^2$, $u_3^2$, and $u_nm1^3$.  The term *stencil* is often
+(ref{wave:pde1:step3b}), which then involves a coupling of $u_1^2$,
+$u_2^3$, $u_2^2$, $u_2^1$, and $u_3^2$.  The term *stencil* is often
 used about the algebraic equation at a mesh point, and the geometry of
 a typical stencil is illustrated in Figure
 ref{wave:pde1:fig:mesh}. One also often refers to the algebraic
@@ -269,6 +279,8 @@ The other initial condition can be computed by
 ===== Formulating a recursive algorithm =====
 label{wave:string:alg}
 
+idx{wave equation! 1D, discrete}
+
 We assume that $u^n_i$ and $u^{n-1}_i$ are available for
 $i=0,\ldots,N_x$.  The only unknown quantity in
 (ref{wave:pde1:step3b}) is therefore $u^{n+1}_i$, which we now can
@@ -333,18 +345,18 @@ label{wave:pde1:step4:1}
 \end{equation}
 !et
 Figure ref{wave:pde1:fig:stencil:u1} illustrates how (ref{wave:pde1:step4:1})
-connects four instead of five points: $u^1_2$, $u_n^0$, $u_nm1^0$, and $u_3^0$.
+connects four instead of five points: $u^1_2$, $u_1^0$, $u_2^0$, and $u_3^0$.
 
 FIGURE: [mov-wave/D_stencil_gpl/stencil_n0_interior, width=500] Modified stencil for the first time step. label{wave:pde1:fig:stencil:u1}
 
 We can now summarize the computational algorithm:
 
  o Compute $u^0_i=I(x_i)$ for $i=0,\ldots,N_x$
- o Compute $u^1_i$ by (ref{wave:pde1:step4:1}) and set $u_i^1=0$
-   for the boundary points $i=0$ and $i=N_x$, for $n=1,2,\ldots,N-1$,
+ o Compute $u^1_i$ by (ref{wave:pde1:step4:1}) for $i=1,2,\ldots,N_x-1$ and set $u_i^1=0$
+   for the boundary points given by $i=0$ and $i=N_x$,
  o For each time level $n=1,2,\ldots,N_t-1$
    o apply (ref{wave:pde1:step4}) to find $u^{n+1}_i$ for $i=1,\ldots,N_x-1$
-   o set $u^{n+1}_i=0$ for the boundary points $i=0$, $i=N_x$.
+   o set $u^{n+1}_i=0$ for the boundary points having $i=0$, $i=N_x$.
 
 The algorithm essentially consists of moving a finite difference
 stencil through all the mesh points, which can be seen as an animation
@@ -369,10 +381,6 @@ memory at as few time levels as possible.
 In a Python implementation of this algorithm, we use the array
 elements `u[i]` to store $u^{n+1}_i$, `u_n[i]` to store $u^n_i$, and
 `u_nm1[i]` to store $u^{n-1}_i$.
-Our naming convention is to use `u` for the
-unknown new spatial field to be computed and have all previous time
-levels in a list `u_n` that we index as `u_n`, `u_nm1`, `u_n[-2]`
-and so on. For the wave equation, `u_n` has just length 2.
 
 The following Python snippet realizes the steps in the computational
 algorithm.
@@ -413,6 +421,8 @@ for n in range(1, Nt):
 
 ======= Verification =======
 
+idx{source term}
+
 Before implementing the algorithm, it is convenient to add a source
 term to the PDE (ref{wave:pde1}), since that gives us more freedom in
 finding test problems for verification. Physically, a source term acts
@@ -484,6 +494,8 @@ label{wave:pde2:step3c}
 ===== Using an analytical solution of physical significance =====
 label{wave:pde2:fd:standing:waves}
 
+idx{verification! convergence rates}
+
 Many wave problems feature sinusoidal oscillations in time
 and space. For example, the original PDE problem
 (ref{wave:pde1})-(ref{wave:pde1:bc:L}) allows an exact solution
@@ -539,6 +551,8 @@ label{wave:pde2:fd:MMS}
 
 === Specifying the solution and computing corresponding data ===
 
+idx{manufactured solution}
+
 One problem with the exact solution (ref{wave:pde2:test:ue}) is
 that it requires a simplification ($V=0, f=0$) of the implemented problem
 (ref{wave:pde2})-(ref{wave:pde2:bc:L}). An advantage of using
@@ -569,6 +583,9 @@ u_t(x,0) &= V(x) = x(L-x)\tp
 
 === Defining a single discretization parameter ===
 
+idx{verification! convergence rates}
+idx{discretization parameter}
+
 To verify the code, we compute the convergence rates in a series of
 simulations, letting each simulation use a finer mesh than the
 previous one. Such empirical estimation of convergence rates relies on
@@ -609,8 +626,8 @@ C_t h^r + C_x\left(\frac{c}{C}\right)^r h^r
 We choose an initial discretization parameter $h_0$ and run
 experiments with decreasing $h$: $h_i=2^{-i}h_0$, $i=1,2,\ldots,m$.
 Halving $h$ in each experiment is not necessary, but it is a common
-choice.  For each experiment we must record $E$ and $h$.  A standard
-choice of error measure is the $\ell^2$ or $\ell^\infty$ norm of the
+choice.  For each experiment we must record $E$ and $h$.  Standard
+choices of error measure are the $\ell^2$ and $\ell^\infty$ norms of the
 error mesh function $e^n_i$:
 
 !bt
@@ -619,7 +636,7 @@ E &= ||e^n_i||_{\ell^2} = \left( \Delta t\Delta x
 \sum_{n=0}^{N_t}\sum_{i=0}^{N_x}
 (e^n_i)^2\right)^{\half},\quad e^n_i = \uex(x_i,t_n)-u^n_i,
 label{wave:pde2:fd:MMS:E:l2}\\
-E &= ||e^n_i||_{\ell^\infty} = \max_{i,n} |e^i_n|\tp
+E &= ||e^n_i||_{\ell^\infty} = \max_{i,n} |e^n_i|\tp
 label{wave:pde2:fd:MMS:E:linf}
 \end{align}
 !et
@@ -641,12 +658,12 @@ E &= ||e^n_i||_{\ell^2} = \left( \Delta x\sum_{i=0}^{N_x}
 E &= ||e^n_i||_{\ell^\infty} = \max_{0\leq i\leq N_x} |e^{n}_i|\tp
 \end{align}
 !et
-The important issue is that our error measure $E$ must be one number
-that represents the error in the simulation.
+The important point is that the error measure ($E$) for the simulation is represented by a single number.
 
 === Computing rates ===
 
-Let $E_i$ be the error measure in experiment (mesh) number $i$ and
+Let $E_i$ be the error measure in experiment (mesh) number $i$ 
+(not to be confused with the spatial index $i$) and
 let $h_i$ be the corresponding discretization parameter ($h$).
 With the error model $E_i = Dh_i^r$, we can
 estimate $r$ by comparing two consecutive
@@ -714,8 +731,7 @@ and $V(x)={\half}x(L-x)$.
 
 To realize that the chosen $\uex$ is also an exact
 solution of the discrete equations,
-we first remind ourselves that $t_n=n\Delta t$ before we
-establish that
+we first remind ourselves that $t_n=n\Delta t$ so that
 
 !bt
 \begin{align}

doc/.src/chapters/wave/wave1D_fd2.do.txt

@@ -55,7 +55,6 @@ idx{homogeneous Dirichlet conditions}
 idx{boundary conditions!Neumann}
 idx{boundary conditions!Dirichlet}
 
-
 When a wave hits a boundary and is to be reflected back, one applies
 the condition
 
@@ -103,7 +102,7 @@ $x=0$ and $t=t_n$ by the difference
 
 !bt
 \begin{equation}
-[D_{2x} u]^n_0 = \frac{u_{-1}^n - u_n^n}{2\Delta x} = 0
+[D_{2x} u]^n_0 = \frac{u_{-1}^n - u_1^n}{2\Delta x} = 0
 \tp
 label{wave:pde1:Neumann:0:cd}
 \end{equation}
@@ -112,7 +111,6 @@ The problem is that $u_{-1}^n$ is not a $u$ value that is being
 computed since the point is outside the mesh. However, if we combine
 (ref{wave:pde1:Neumann:0:cd}) with the scheme
 # (ref{wave:pde1:step4})
-for $i=0$,
 
 !bt
 \begin{equation}
@@ -121,8 +119,8 @@ u^{n+1}_i = -u^{n-1}_i + 2u^n_i + C^2
 label{wave:pde1:Neumann:0:scheme}
 \end{equation}
 !et
-we can eliminate the fictitious value $u_{-1}^n$. We see that
-$u_{-1}^n=u_n^n$ from (ref{wave:pde1:Neumann:0:cd}), which
+for $i=0$, we can eliminate the fictitious value $u_{-1}^n$. We see that
+$u_{-1}^n=u_1^n$ from (ref{wave:pde1:Neumann:0:cd}), which
 can be used in (ref{wave:pde1:Neumann:0:scheme}) to
 arrive at a modified scheme for the boundary point $u_0^{n+1}$:
 
@@ -359,7 +357,7 @@ A lot of test examples are also included in the program:
    typical initial shape of a guitar string.
  * A sinusoidal variation of $u$ at $x=0$ and either $u=0$ or
    $u_x=0$ at $x=L$.
- * An exact analytical solution $u(x,t)=\cos(m\pi t/L)\sin({\half}m\pi x/L)$, which can be used for convergence rate tests.
+ * An analytical solution $u(x,t)=\cos(m\pi t/L)\sin({\half}m\pi x/L)$, which can be used for convergence rate tests.
 !enotice
 
 [hpl: Should include some experiments here or make exercises. Qualitative
@@ -402,6 +400,8 @@ label{wave:pde1:Neumann:ghost}
 
 === Idea ===
 
+idx{ghost! cells} idx{ghost! points} idx{ghost! values}
+
 Instead of modifying the scheme at the boundary, we can introduce
 extra points outside the domain such that the fictitious values
 $u_{-1}^n$ and $u_{N_x+1}^n$ are defined in the mesh.  Adding the
@@ -483,7 +483,7 @@ for i in Ix:
            0.5*dt2*f(x[i-Ix[0]], t[0])
 !ec
 
-It remains to update the solution at ghost points, i.e, `u[0]`
+It remains to update the solution at ghost points, i.e., `u[0]`
 and `u[-1]` (or `u[Nx+2]`). For a boundary condition $u_x=0$,
 the ghost value must equal the value at the associated inner mesh
 point. Computer code makes this statement precise:
@@ -557,6 +557,8 @@ or (as always) `u[Ix[0]:Ix[-1]+1]`.
 ======= Generalization: variable wave velocity =======
 label{wave:pde2:var:c}
 
+idx{wave! variable velocity} idx{wave! reflected} idx{wave! transmitted}
+
 Our next generalization of the 1D wave equation (ref{wave:pde1}) or
 (ref{wave:pde2}) is to allow for a variable wave velocity $c$:
 $c=c(x)$, usually motivated by wave motion in a domain composed of
@@ -704,7 +706,7 @@ terms $qu_{xx} + q_xu_x$. The typical discretization is
 
 !bt
 \begin{equation}
-D_x q D_x u + D_{2x}q D_{2x} u]_i^n,
+[D_x q D_x u + D_{2x}q D_{2x} u]_i^n,
 label{wave:pde2:var:c:chainrule_scheme}
 \end{equation}
 !et
@@ -717,7 +719,7 @@ $[D_x(q D_x u)]^n_i$ with harmonic averaging of $q$ yields
 a better solution than arithmetic averaging or
 (ref{wave:pde2:var:c:chainrule_scheme}).
 In the literature, the discretization $[D_x(q D_x u)]^n_i$ totally
-dominates and very few mention the possibility of
+dominates and very few mention the alternative in
 (ref{wave:pde2:var:c:chainrule_scheme}).
 !ewarning
 
@@ -730,6 +732,8 @@ dominates and very few mention the possibility of
 ===== Computing the coefficient between mesh points =====
 label{wave:pde2:var:c:means}
 
+idx{interpolation}
+
 If $q$ is a known function of $x$, we can easily evaluate
 $q_{i+\half}$ simply as $q(x_{i+\half})$ with $x_{i+\half} = x_i +
 \half\Delta x$.  However, in many cases $c$, and hence $q$, is only
@@ -810,10 +814,12 @@ label{wave:pde2:var:c:scheme:impl}
 ===== How a variable coefficient affects the stability =====
 label{wave:pde2:var:c:stability}
 
-The stability criterion derived in Section ref{wave:pde1:stability}
+idx{time step! spatially varying}
+
+The stability criterion derived later (Section ref{wave:pde1:stability})
 reads $\Delta t\leq \Delta x/c$. If $c=c(x)$, the criterion will depend
 on the spatial location. We must therefore choose a $\Delta t$ that
-is small enough such that no mesh cell has $\Delta x/c(x) >\Delta t$.
+is small enough such that no mesh cell has $\Delta t > \Delta x/c(x)$.
 That is, we must use the largest $c$ value in the criterion:
 
 !bt
@@ -971,6 +977,8 @@ u[1:-1] = - u_nm1[1:-1] + 2*u_n[1:-1] + \
 
 ===== A more general PDE model with variable coefficients =====
 
+idx{coefficients! variable}
+
 Sometimes a wave PDE has a variable coefficient in front of
 the time-derivative term:
 
@@ -1009,6 +1017,8 @@ also without any difficulty:
 
 ===== Generalization: damping =====
 
+idx{wave! damping}
+
 Waves die out by two mechanisms. In 2D and 3D the energy of the wave
 spreads out in space, and energy conservation then requires
 the amplitude to decrease. This effect is not present in 1D.
@@ -1078,8 +1088,8 @@ label{wave:pde2:software:bcL}
 \end{align}
 !et
 
-The only new feature here is the time-dependent Dirichlet conditions.
-These are trivial to implement:
+The only new feature here is the time-dependent Dirichlet conditions, but
+they are trivial to implement:
 
 !bc pycod
 i = Ix[0]  # x=0
@@ -1169,6 +1179,9 @@ cite{Langtangen_deqbook_softeng2}].
 
 ===== Pulse propagation in two media =====
 
+idx{pulse propagation} idx{plug} idx{Gaussian function} idx{cosine hat}
+idx{discontinuous medium} idx{discretization parameters}
+
 The function `pulse` in `wave1D_dn_vc.py` demonstrates wave motion in
 heterogeneous media where $c$ varies. One can specify an interval
 where the wave velocity is decreased by a factor `slowness_factor`