fixed typos in diffu chapter

Author: sveinlin

doc/.src/chapters/diffu/diffu_analysis.do.txt

@@ -2,6 +2,8 @@
 ======= Analysis of schemes for the diffusion equation =======
 label{diffu:pde1:analysis}
 
+idx{noise! sawtooth-like}
+
 The numerical experiments in Sections
 ref{diffu:pde1:FE:experiments} and ref{diffu:pde1:theta:experiments}
 reveal that there are some
@@ -25,6 +27,9 @@ Figures ref{diffu:pde1:FE:fig:F=0.5}-ref{diffu:pde1:CN:fig:F=10}.
 ===== Properties of the solution =====
 label{diffu:pde1:analysis:uex}
 
+idx{error function (erf)!} idx{step function}
+idx{error function (erf)! complementary}
+
 A particular characteristic of diffusive processes, governed
 by an equation like
 
@@ -102,11 +107,13 @@ label{diffu:analysis:pde1:p1:erf:uR}
 \tp
 \end{align}
 !et
-For small enough $t$, $u(0,t)\approx 1$ and $u(1,t)\approx 1$, but as
+For small enough $t$, $u(0,t)\approx 1$ and $u(1,t)\approx 0$, but as
 $t\rightarrow\infty$, $u(x,t)\rightarrow 1/2$ on $[0,1]$.
 
 === Solution for a Gaussian pulse ===
 
+idx{Dirac delta function}
+
 The standard diffusion equation $u_t = \dfc u_{xx}$ admits a
 Gaussian function as solution:
 
@@ -132,6 +139,8 @@ the pulse diffuse and flatten out.
 
 === Solution for a sine component ===
 
+idx{smoothing} idx{noise! removing} idx{signal processing}
+
 Also, (ref{diffu:pde1:eq}) admits a solution of the form
 
 !bt
@@ -150,8 +159,7 @@ inserting (ref{diffu:pde1:sol1}) in (ref{diffu:pde1:eq}) gives the constraint
 \end{equation*}
 !et
 
-
-A very important feature is that the initial shape $I(x)=Q\sin kx$
+A very important feature is that the initial shape $I(x)=Q\sin\left( kx\right)$
 undergoes a damping $\exp{(-\dfc k^2t)}$, meaning that
 rapid oscillations in space, corresponding to large $k$, are very much
 faster dampened than slow oscillations in space, corresponding to small
@@ -273,6 +281,8 @@ denotes $u$ at time $t_n$.
 
 === Stability ===
 
+idx{stability}
+
 The exact amplification factor is $\Aex=\exp{(-\dfc^2 k^2\Delta t)}$.
 We should therefore require $|A| < 1$ to have a decaying numerical
 solution as well. If
@@ -284,6 +294,8 @@ idx{amplification factor}
 
 === Accuracy ===
 
+idx{accuracy}
+
 To determine how accurately a finite difference scheme treats one
 wave component (ref{diffu:pde1:analysis:uni}), we see that the basic
 deviation from the exact solution is reflected in how well
@@ -297,6 +309,8 @@ make Taylor expansions of $A/\Aex$ to see the error more analytically.
 
 === Truncation error ===
 
+idx{diffusion equation! truncation error}
+
 As an alternative to examining the accuracy of the damping of a wave
 component, we can perform a general truncation error analysis as
 explained in ref[Appendix ref{ch:trunc}][ in
@@ -312,6 +326,8 @@ verifying codes based on empirical estimation of convergence rates.
 ===== Analysis of the Forward Euler scheme =====
 label{diffu:pde1:analysis:FE}
 
+idx{diffusion equation! numerical Fourier number}
+
 # 2DO: refer to vib and wave
 
 
@@ -402,6 +418,8 @@ The method hence becomes very expensive for fine spatial meshes.
 
 === Accuracy ===
 
+idx{Taylor series}
+
 Since $A$ is expressed in terms of $F$ and the parameter we now call
 $p=k\Delta x/2$, we should also express $\Aex$ by $F$ and $p$. The exponent
 in $\Aex$ is $-\dfc k^2\Delta t$, which equals $-F k^2\Delta x^2=-F4p^2$.
@@ -578,7 +596,7 @@ The exact numerical solution is hence
 
 !bt
 \begin{equation}
-u^n_q = \left(\frac{ 1 - 2F\sin^2p}{1 + 2F\sin^2p}\right)^ne^{ikp\Delta x}
+u^n_q = \left(\frac{ 1 - 2F\sin^2p}{1 + 2F\sin^2p}\right)^ne^{ikq\Delta x}
 \tp
 \end{equation}
 !et
@@ -605,21 +623,23 @@ cite{Langtangen_deqbook_trunc}]:
 ===== Analysis of the Leapfrog scheme =====
 label{diffu:pde1:analysis:leapfrog}
 
+idx{Leapfrog scheme}
+
 An attractive feature of the Forward Euler scheme is the explicit
 time stepping and no need for solving linear systems. However, the
 accuracy in time is only $\Oof{\Delta t}$. We can get an explicit
 *second-order* scheme in time by using the Leapfrog method:
 
 !bt
-\[ [D_{2t} u = \dfc D_xDx u + f]^n_i\tp\]
+\[ [D_{2t} u = \dfc D_xDx u + f]^n_q\tp\]
 !et
 Written out,
 
 !bt
-\[ u^{n+1} = u^{n-1} + \frac{2\dfc\Delta t}{\Delta x^2}
-(u^{n}_{i+1} - 2u^n_i + u^n_{i-1}) + f(x_i,t_n)\tp\]
+\[ u_q^{n+1} = u_q^{n-1} + \frac{2\dfc\Delta t}{\Delta x^2}
+(u^{n}_{q+1} - 2u^n_q + u^n_{q-1}) + f(x_q,t_n)\tp\]
 !et
-We need some formula for the first step, $u^1_i$, but for that we can use
+We need some formula for the first step, $u^1_q$, but for that we can use
 a Forward Euler step.
 
 Unfortunately, the Leapfrog scheme is always unstable for the
@@ -690,7 +710,8 @@ respectively, and we see how short waves pollute the overall solution.
 ===== Analysis of the 2D diffusion equation =====
 label{diffu:2D:analysis}
 
-We first consider the 2D diffusion equation
+Diffusion in several dimensions is treated later, but it is appropriate to
+include the analysis here. We first consider the 2D diffusion equation
 
 !bt
 \[ u_{t} = \dfc(u_{xx} + u_{yy}),\]
@@ -711,7 +732,7 @@ and the schemes have discrete versions of this Fourier component:
 For the Forward Euler discretization,
 
 !bt
-\[ [D_t^+u = \dfc(D_xD_x u + D_yD_y u)]_{i,j}^n,\]
+\[ [D_t^+u = \dfc(D_xD_x u + D_yD_y u)]_{q,r}^n,\]
 !et
 we get
 
@@ -752,7 +773,7 @@ The complete numerical solution for a wave component is
 !bt
 \begin{equation}
 u^{n}_{q,r} = A(1 - 4F_x\sin^2 p_x - 4F_y\sin^2 p_y)^n
-e^{i(k_xp\Delta x + k_yq\Delta y)}\tp
+e^{i(k_xq\Delta x + k_yr\Delta y)}\tp
 label{diffu:2D:analysis:FE:numexact}
 \end{equation}
 !et
@@ -781,7 +802,7 @@ where $d$ is the number of space dimensions: $d=1,2,3$.
 The Backward Euler method,
 
 !bt
-\[ [D_t^-u = \dfc(D_xD_x u + D_yD_y u)]_{i,j}^n,\]
+\[ [D_t^-u = \dfc(D_xD_x u + D_yD_y u)]_{q,r}^n,\]
 !et
 results in
 
@@ -809,9 +830,9 @@ label{diffu:2D:analysis:BN:numexact}
 With a Crank-Nicolson discretization,
 
 !bt
-\[ [D_tu]^{n+\half}_{i,j} =
-\half [\dfc(D_xD_x u + D_yD_y u)]_{i,j}^{n+1} +
-\half [\dfc(D_xD_x u + D_yD_y u)]_{i,j}^n,\]
+\[ [D_tu]^{n+\half}_{q,r} =
+\half [\dfc(D_xD_x u + D_yD_y u)]_{q,r}^{n+1} +
+\half [\dfc(D_xD_x u + D_yD_y u)]_{q,r}^n,\]
 !et
 we have, after some algebra,
 
@@ -853,8 +874,8 @@ label{diffu:2D:analysis:CN:numexact}
 
 The behavior of the solution generated by Forward Euler discretization in time (and centered
 differences in space) is summarized at the end of
-Section ref{diffu:pde1:FE:experiments}. Can we from the analysis
-above explain the behavior?
+Section ref{diffu:pde1:FE:experiments}. Can we, from the analysis
+above, explain the behavior?
 
 We may start by looking at Figure ref{diffu:pde1:FE:fig:F=0.51}
 where $F=0.51$. The figure shows that the solution is unstable and

doc/.src/chapters/diffu/diffu_app.do.txt

@@ -8,6 +8,8 @@ and heat conduction (include Robin/cooling).]
 ===== Diffusion of a substance =====
 label{diffu:app:substance}
 
+idx{Fick's law}
+
 The first process to be considered is a substance that gets
 transported through a fluid at rest by pure diffusion. We consider an
 arbitrary volume $V$ of this fluid, containing the substance with
@@ -124,6 +126,9 @@ label{diffu:app:substance:PDE}
 ===== Heat conduction =====
 label{diffu:app:heat}
 
+idx{internal energy} idx{radioactive rock} idx{equation of state}
+idx{heat capacity} idx{Fourier's law}
+
 Heat conduction is a well-known diffusion process. The governing PDE
 is in this case based on the first law of thermodynamics: the increase
 in energy of a system is equal to the work done on the system, plus
@@ -252,7 +257,7 @@ c_v\partial T/\partial t$ in the PDE gives
 !et
 We still have four unknown scalar fields ($T$ and $\q$). To close the
 system, we need a relation between the heat flux $\q$ and the temperature $T$
-called Fourier's law:
+called *Fourier's law*:
 
 !bt
 \[ \q = -k\nabla T,\]
@@ -265,7 +270,10 @@ $k$ is constant. The value of $k$ reflects how easy heat is
 conducted through the medium, and $k$ is named the *coefficient of
 heat conduction*.
 
-We have now one scalar PDE for the unknown temperature field $T(\x,t)$:
+idx{heat conduction! coefficient of} idx{mass balance}
+idx{incompressible fluid}
+
+We now have one scalar PDE for the unknown temperature field $T(\x,t)$:
 
 !bt
 \begin{equation}
@@ -277,6 +285,8 @@ label{diffu:app:heat:PDE}
 ===== Porous media flow =====
 label{diffu:app:porous}
 
+idx{Darcy's law} idx{dynamic viscosity}
+
 The requirement of mass balance for
 flow of a single, incompressible fluid through a deformable (elastic) porous
 medium leads to the equation
@@ -292,7 +302,7 @@ coefficient of the medium (related to the compressibility of the
 fluid and the material in the medium), and $\alpha$ is another coefficient.
 In many circumstances, the last term with $\u$ can be neglected,
 an assumption that decouples the equation above from a model for
-the deformation of the medium. The famous Darcy's law relates
+the deformation of the medium. The famous *Darcy's law* relates
 $\q$ to $p$:
 
 !bt
@@ -309,11 +319,13 @@ S\frac{\partial p}{\partial t} = \mu^{-1}\nabla(K\nabla p) + \frac{\varrho g}{\m
 label{diffu:app:porous:PDE}
 \end{equation}
 !et
-Boundary conditions consist of specifying $p$ or $\q\cdot\normalvec$ at
-(normal velocity) each point of the boundary.
+Boundary conditions consist of specifying $p$ or $\q\cdot\normalvec$ (i.e., normal velocity) at
+ each point of the boundary.
 
 ===== Potential fluid flow =====
 
+idx{viscous effects}
+
 Let $\v$ be the velocity of a fluid. The condition $\nabla\times\v =0$
 is relevant for many flows, especially in geophysics when viscous effects
 are negligible. From vector calculus it is known
@@ -341,6 +353,8 @@ $\phi$ at a point.
 
 ===== Streamlines for 2D fluid flow =====
 
+idx{stationary fluid flow} idx{stream function}
+
 The streamlines in a two-dimensional
 stationary fluid flow are lines tangential to the flow.
 The "stream function": "https://en.wikipedia.org/wiki/Stream_function"
@@ -413,6 +427,8 @@ equation in that case (anisotropic conduction?). Can you figure it out?]
 ===== Development of flow between two flat plates =====
 label{diffu:app:Couette}
 
+idx{Navier-Stokes equations}
+
 Diffusion equations may also arise as simplified versions of other
 mathematical models, especially in fluid flow. Consider a fluid
 flowing between two flat, parallel plates. The velocity is
@@ -460,7 +476,7 @@ a very standard one-dimensional diffusion equation:
 !bt
 \[
 \varrho\frac{\partial u}{\partial t} =
-\mu\frac{\partial^2 u}{\partial z^2} + \beta(t) + \varrho\gamma g,\quad
+\mu\frac{\partial^2 u}{\partial x^2} + \beta(t) + \varrho\gamma g,\quad
 x\in [0,L],\ t\in (0,T]\tp
 \]
 !et
@@ -490,6 +506,8 @@ the adjusted boundary condition
 ===== Flow in a straight tube =====
 label{diffu:app:pipeflow}
 
+idx{cylindrical coordinates}
+
 Now we consider viscous fluid flow in a straight tube with radius $R$
 and rigid walls.
 The governing equations are the Navier-Stokes equations, but as
@@ -515,6 +533,8 @@ the tube. The associated boundary condition is $u(R,t)=0$.
 
 ===== Tribology: thin film fluid flow =====
 
+idx{friction}
+
 Thin fluid films are extremely important inside machinery to reduce friction
 between gliding surfaces. The mathematical model for the fluid motion takes
 the form of a diffusion problem and is quickly derived here.
@@ -554,7 +574,7 @@ The idea is to use this expression locally
 also when the surfaces are not flat,
 but slowly varying, and if $U$, $V$, or $p$ varies in time, provided the
 time variation is sufficiently slow. This is a common quasi-static
-approximation much used in mathematical modeling.
+approximation, much used in mathematical modeling.
 
 Inserting the expression for $\q$ via $p$, $U$, and $V$ in the
 equation $\nabla\q = 0$ gives a diffusion PDE for $p$:
@@ -572,6 +592,8 @@ The boundary conditions must involve $p$ or $\q$ at the boundary.
 
 ===== Propagation of electrical signals in the brain =====
 
+idx{neuronal fibers} idx{cable equation}
+
 # http://icwww.epfl.ch/~gerstner/SPNM/node17.html
 # http://www.uio.no/studier/emner/matnat/ifi/INF5610/h09/Lecture04.pdf
 # http://people.mbi.ohio-state.edu/schwemmer.2/Publications/Schwemmer_Dissertation_Final.pdf