fixed typos in nonlin chapter

Author: sveinlin

doc/.src/chapters/nonlin/nonlin_ode.do.txt

@@ -5,6 +5,8 @@ label{nonlin:timediscrete:logistic}
 
 === Algebraic equations ===
 
+idx{iterative methods}
+
 A linear, scalar, algebraic equation in $x$ has the form
 
 !bt
@@ -177,7 +179,7 @@ equation for the unknown value $u^{n+1}$ that we can easily solve:
 !bt
 \[ u^{n+1} = u^n + \Delta t\,u^n(1 - u^n)\tp\]
 !et
-The nonlinearity in the original equation poses in this case no difficulty
+In this case, the nonlinearity in the original equation poses no difficulty
 in the discrete algebraic equation.
 Any other explicit scheme in time will also give only linear
 algebraic equations
@@ -200,6 +202,8 @@ known in the next step, which is linear in the unknown $u^{n+1}$ .
 ===== Exact solution of nonlinear algebraic equations =====
 label{nonlin:timediscrete:logistic:roots}
 
+idx{`sympy`} idx{Taylor series} idx{`limit`}
+
 Switching to a Backward Euler scheme for
 (ref{nonlin:timediscrete:logistic:eq}),
 
@@ -259,7 +263,7 @@ In the present simple case, however, we can analyze the roots mathematically
 and provide an answer. The idea is to expand the roots
 in a series in $\Delta t$ and truncate after the linear term since
 the Backward Euler scheme will introduce an error proportional to
-$\Delta t$ anyway. Using `sympy` we find the following Taylor series
+$\Delta t$ anyway. Using `sympy`, we find the following Taylor series
 expansions of the roots:
 
 !bc pyshell
@@ -431,7 +435,7 @@ label{nonlin:timediscrete:logistic:BE:Picard:1it}
 !et
 which is a linear algebraic equation in the unknown $u^n$, making
 it easy to solve for $u^n$ without any need for
-any alternative notation.
+an alternative notation.
 
 We shall later refer to the strategy of taking one Picard step, or
 equivalently, linearizing terms with use of the solution at the
@@ -456,6 +460,8 @@ level as start for the Picard iteration.
 ===== Linearization by a geometric mean =====
 label{nonlin:timediscrete:logistic:geometric:mean}
 
+idx{geometric mean} idx{arithmetic mean}
+
 We consider now a Crank-Nicolson discretization of
 (ref{nonlin:timediscrete:logistic:eq}). This means that the
 time derivative is approximated by a centered
@@ -480,7 +486,7 @@ mean,
 \[ u^{n+\half}\approx \half(u^n + u^{n+1}),\]
 !et
 such that the scheme involves the unknown function only at the time levels
-where we actually compute it.
+where we actually intend to compute it.
 The same arithmetic mean applied to the nonlinear term gives
 
 !bt
@@ -549,6 +555,7 @@ u^\prime u$, i.e., a difference of size $\Delta t^2$).
 ===== Newton's method =====
 label{nonlin:timediscrete:logistic:Newton}
 
+idx{quadratic convergence}
 
 The Backward Euler scheme (ref{nonlin:timediscrete:logistic:eq:BE})
 for the logistic equation leads to a nonlinear algebraic equation
@@ -639,7 +646,7 @@ notation used in the literature.
 ===== Relaxation =====
 label{nonlin:timediscrete:logistic:relaxation}
 
-idx{relaxation (nonlinear equations)}
+idx{relaxation (nonlinear equations)} idx{relaxation parameter}
 
 One iteration in Newton's method or
 Picard iteration consists of solving a linear problem $\hat F(u)=0$.
@@ -670,6 +677,7 @@ label{nonlin:timediscrete:logistic:relaxation:Newton:formula}
 ===== Implementation and experiments =====
 label{nonlin:timediscrete:logistic:impl}
 
+idx{`logistic.py`}
 
 The program "`logistic.py`": "${src_nonlin}/logistic.py" contains
 implementations of all the methods described above.
@@ -828,13 +836,15 @@ where $f$ is a nonlinear function of $u$.
 
 === Explicit time discretization ===
 
-Explicit ODE methods like the Forward Euler scheme, Runge-Kutta methods,
+Explicit ODE methods like the Forward Euler scheme, Runge-Kutta methods and
 Adams-Bashforth methods all evaluate $f$ at time levels where
 $u$ is already computed, so nonlinearities in $f$ do not
 pose any difficulties.
 
 === Backward Euler discretization ===
 
+idx{`ODE_Picard_tricks.py`}
+
 Approximating $u^{\prime}$ by a backward difference leads to a Backward Euler
 scheme, which can be written as
 
@@ -863,8 +873,8 @@ until a stopping criterion is fulfilled.
 !bnotice Explicit vs implicit treatment of nonlinear terms
 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 known $u$, as in the manual linearization
-like $(u^{-})^2u$, and then the treatment of $f$ is ``more implicit''
+$f$ 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
 of $u^{\prime}=f(u,t)$, where evaluating $f$ for known $u$ values gives
 explicit schemes, while treating $f$ or parts of $f$ implicitly,
@@ -894,7 +904,7 @@ implicitly.
 
 A trick to treat $f$ implicitly in Picard iteration is to
 evaluate it as $f(u^{-},t)u/u^{-}$. For a polynomial $f$, $f(u,t)=u^m$,
-this corresponds to $(u^{-})^{m}u/u^{-1}=(u^{-})^{m-1}u$. Sometimes this more implicit
+this corresponds to $(u^{-})^{m}u/u^{-}=(u^{-})^{m-1}u$. Sometimes this more implicit
 treatment has no effect, as with $f(u,t)=\exp(-u)$ and $f(u,t)=\ln (1+u)$,
 but with $f(u,t)=\sin(2(u+1))$, the $f(u^{-},t)u/u^{-}$ trick
 leads to 7, 9, and 11 iterations during the first three steps, while
@@ -904,7 +914,7 @@ $f(u^{-},t)$ demands 17, 21, and 20 iterations.
 
 Newton's method applied to a Backward Euler discretization of
 $u^{\prime}=f(u,t)$
-requires the computation of the derivative
+requires computation of the derivative
 
 !bt
 \[ F^{\prime}(u) = 1 - \Delta t\frac{\partial f}{\partial u}(u,t_n)\tp\]
@@ -966,6 +976,8 @@ while Newton's method can apply the general formula
 ===== Systems of ODEs =====
 label{nonlin:ode:generic:sys:pendulum}
 
+idx{system of algebraic equations} idx{coupled system}
+
 We may write a system of ODEs
 
 !bt
@@ -1094,7 +1106,7 @@ compactly as
 where $u$ is a vector of unknowns $u=(u_0,\ldots,u_N)$, and
 $F$ is a vector function: $F=(F_0,\ldots,F_N)$.
 The system at the end of Section ref{nonlin:ode:generic:sys:pendulum} fits
-this notation with $N=2$, $F_0(u)$ given by the left-hand side of
+this notation with $N=1$, $F_0(u)$ given by the left-hand side of
 (ref{nonlin:ode:generic:sys:pendulum:u0}), while $F_1(u)$ is
 the left-hand side of (ref{nonlin:ode:generic:sys:pendulum:u1}).
 
@@ -1118,6 +1130,8 @@ of these methods, can be found in Kelley cite{Kelley_1995}.
 ===== Picard iteration =====
 label{nonlin:systems:alg:Picard}
 
+idx{linear system}
+
 We cannot apply Picard iteration to nonlinear equations unless there is
 some special structure. For the commonly arising case
 $A(u)u=b(u)$ we can linearize the
@@ -1153,6 +1167,8 @@ more details.
 ===== Newton's method =====
 label{nonlin:systems:alg:Newton}
 
+idx{Jacobian}
+
 The natural starting point for Newton's method is the general
 nonlinear vector equation $F(u)=0$.
 As for a scalar equation, the idea is to approximate $F$
@@ -1316,7 +1332,7 @@ k>k_{\max}\tp
 ===== Example: A nonlinear ODE model from epidemiology =====
 label{nonlin:systems:alg:SI}
 
-The simplest model spreading of a disease, such as a flu, takes
+A very simple model for the spreading of a disease, such as a flu, takes
 the form of a $2\times 2$ ODE system
 
 !bt
@@ -1345,7 +1361,7 @@ algebraic equations in the unknowns $S^{n+1}$ and $I^{n+1}$:
 \frac{\nu}{2}(I^n + I^{n+1})\tp
 \end{align}
 !et
-Introducing $S$ for $S^{n+1}$, $S^{(1)}$ for $S^n$, $I$ for $I^{n+1}$,
+Introducing $S$ for $S^{n+1}$, $S^{(1)}$ for $S^n$, $I$ for $I^{n+1}$ and
 $I^{(1)}$ for $I^n$, we can rewrite the system as
 
 !bt

doc/.src/chapters/nonlin/nonlin_pde1D.do.txt

@@ -459,7 +459,7 @@ nonlinear problem and minimize the spatial discretization details.
 The nonlinearity in the differential equation
 (ref{nonlin:alglevel:1D:pde}) poses no more difficulty than a variable
 coefficient, as in the term $(\dfc(x)u^{\prime})^{\prime}$.  We can
-therefore use a standard finite difference approach to discretizing
+therefore use a standard finite difference approach when discretizing
 the Laplace term with a variable coefficient:
 
 !bt
@@ -612,7 +612,7 @@ case with a small mesh, say just two cells ($N_x=2$). We use $u^{-}_i$
 for the $i$-th component in $u^{-}$.
 
 The starting point is the basic expressions for the
-nonlinear equations at mesh point $i=0$ and $i=1$ are
+nonlinear equations at mesh point $i=0$ and $i=1$:
 
 !bt
 \begin{align}
@@ -701,7 +701,7 @@ where
 !bt
 \begin{align}
 B_{0,0} &=\frac{1}{2\Delta x^2}(\dfc(u^-_{-1}) + 2\dfc(u^-_{0}) + \dfc(u^-_{1}))
-+ a\\
++ a,\\
 B_{0,1} &=
 -\frac{1}{2\Delta x^2}(\dfc(u^-_{-1}) + 2\dfc(u^-_{0})
 + \dfc(u^-_{1})),\\

doc/.src/chapters/nonlin/nonlin_pde_gen.do.txt

@@ -255,7 +255,7 @@ which in 2D can be written
 \]
 !et
 We do not dive into the details of handling boundary conditions now.
-Dirichlet and Neumann conditions are handled as in a
+Dirichlet and Neumann conditions are handled as in
 corresponding linear, variable-coefficient diffusion problems.
 
 Writing the scheme out, putting the unknown values on the
@@ -334,7 +334,7 @@ contribution to the Jacobian since $F_{i,j}$ contains $u_{i\pm 1,j}$,
 $u_{i,j\pm 1}$, and $u_{i,j}$. This means that $J_{i,j,r,s}$ has
 nonzero contributions only if $r=i\pm 1$, $s=j\pm 1$, as well as $r=i$
 and $s=j$.  The corresponding terms in $J_{i,j,r,s}$ are
-$J_{i,j,i-1,j}$, $J_{i,j,i+1,j}$, $J_{i,j,i,j-1}$, $J_{i,j,i,j+1}$,
+$J_{i,j,i-1,j}$, $J_{i,j,i+1,j}$, $J_{i,j,i,j-1}$, $J_{i,j,i,j+1}$
 and $J_{i,j,i,j}$.  Therefore, the left-hand side of the Newton
 system, $\sum_r\sum_s J_{i,j,r,s}\delta u_{r,s}$ collapses to
 
@@ -380,7 +380,7 @@ of an index.
 
 ===== Continuation methods =====
 
-idx{continuation method}
+idx{continuation! method} idx{continuation! parameter}
 
 Picard iteration or Newton's method may diverge when solving PDEs with
 severe nonlinearities. Relaxation with $\omega <1$

doc/.src/chapters/nonlin/nonlin_split.do.txt

@@ -2,6 +2,8 @@
 ======= Operator splitting methods =======
 label{nonlin:splitting}
 
+idx{ADI methods} idx{split-step methods} idx{dimensional splitting}
+
 Operator splitting is a natural and old idea. When a PDE or system of PDEs
 contains different terms expressing different physics, it is natural to
 use different numerical methods for different physical processes. This can
@@ -119,6 +121,8 @@ with global solution set as $u(t_{n+1}) = u^{\stepone}(t_{n+1})$.
 ===== Example: Logistic growth =====
 label{nonlin:splitting:logistic}
 
+idx{logistic growth} idx{`split_logistic.py`}
+
 Let us split the (scaled) logistic equation
 
 !bt
@@ -284,7 +288,7 @@ The algorithm goes like this:
  o Solve the diffusion problem for one time step as usual.
  o Solve the reaction ODEs at each mesh point in $[t_n,t_n+\Delta t]$,
    using the diffusion solution in 1. as initial condition.
-   The solution of the ODEs constitute the solution of the original problem
+   The solution of the ODEs constitutes the solution of the original problem
    at the end of each time step.
 
 We may use a much smaller time step when solving the reaction part, adapted
@@ -296,6 +300,9 @@ splitting methods.
 ===== Example: Reaction-Diffusion with linear reaction term =====
 label{nonlin:splitting:RD_linearR}
 
+idx{`split_diffu_react.py`} idx{verification! convergence rates}
+idx{Adams-Bashforth} idx{Odespy}
+
 The methods above may be explored in detail through a specific
 computational example in which we compute the convergence rates
 associated with four different solution approaches for the

doc/pub/book/pdf/fdm-book-4print.pdf

@@ -7,7 +7,7 @@
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/hplgit/doconce/" />
 <meta name="description" content="Solving nonlinear ODE and PDE problems">
-<meta name="keywords" content="linearization explicit time integration,linearization,Picard iteration,successive substitutions,fixed-point iteration,linearization Picard iteration,linearization successive substitutions,linearization fixed-point iteration,stopping criteria (nonlinear problems),single Picard iteration technique,relaxation (nonlinear equations),stopping criteria (nonlinear problems),continuation method,operator splitting,splitting ODEs,fractional step methods,Strang splitting,continuation method">
+<meta name="keywords" content="iterative methods,linearization explicit time integration,Taylor series,linearization,Picard iteration,successive substitutions,fixed-point iteration,linearization Picard iteration,linearization successive substitutions,linearization fixed-point iteration,stopping criteria (nonlinear problems),single Picard iteration technique,geometric mean,arithmetic mean,quadratic convergence,relaxation (nonlinear equations),relaxation parameter,system of algebraic equations,coupled system,linear system,Jacobian,stopping criteria (nonlinear problems),continuation  method,continuation  parameter,ADI methods,split-step methods,dimensional splitting,operator splitting,splitting ODEs,fractional step methods,Strang splitting,logistic growth,verification  convergence rates,Adams-Bashforth,Odespy,continuation method">
 
 <title>Solving nonlinear ODE and PDE problems</title>
 
@@ -419,7 +419,7 @@
 <center>[3] <b>Department of Process, Energy and Environmental Technology, University College of Southeast Norway</b></center>
 <br>
 <p>
-<center><h4>Sep 29, 2016</h4></center> <!-- date -->
+<center><h4>Dec 21, 2016</h4></center> <!-- date -->
 <br>
 <p>
 Note: Preliminary version (expect typos).

doc/pub/book/pdf/fdm-book-4screen-sol.pdf

@@ -7,7 +7,7 @@
 <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
 <meta name="generator" content="DocOnce: https://github.com/hplgit/doconce/" />
 <meta name="description" content="Solving nonlinear ODE and PDE problems">
-<meta name="keywords" content="linearization explicit time integration,linearization,Picard iteration,successive substitutions,fixed-point iteration,linearization Picard iteration,linearization successive substitutions,linearization fixed-point iteration,stopping criteria (nonlinear problems),single Picard iteration technique,relaxation (nonlinear equations),stopping criteria (nonlinear problems),continuation method,operator splitting,splitting ODEs,fractional step methods,Strang splitting,continuation method">
+<meta name="keywords" content="iterative methods,linearization explicit time integration,Taylor series,linearization,Picard iteration,successive substitutions,fixed-point iteration,linearization Picard iteration,linearization successive substitutions,linearization fixed-point iteration,stopping criteria (nonlinear problems),single Picard iteration technique,geometric mean,arithmetic mean,quadratic convergence,relaxation (nonlinear equations),relaxation parameter,system of algebraic equations,coupled system,linear system,Jacobian,stopping criteria (nonlinear problems),continuation  method,continuation  parameter,ADI methods,split-step methods,dimensional splitting,operator splitting,splitting ODEs,fractional step methods,Strang splitting,logistic growth,verification  convergence rates,Adams-Bashforth,Odespy,continuation method">
 
 <title>Solving nonlinear ODE and PDE problems</title>
 
@@ -576,7 +576,7 @@ <h2 id="nonlin:timediscrete:logistic:FE">Linearization by explicit time discreti
 
 $$ u^{n+1} = u^n + \Delta t\,u^n(1 - u^n)\tp$$
 
-The nonlinearity in the original equation poses in this case no difficulty
+In this case, the nonlinearity in the original equation poses no difficulty
 in the discrete algebraic equation.
 Any other explicit scheme in time will also give only linear
 algebraic equations
@@ -661,7 +661,7 @@ <h2 id="nonlin:timediscrete:logistic:roots">Exact solution of nonlinear algebrai
 and provide an answer. The idea is to expand the roots
 in a series in \( \Delta t \) and truncate after the linear term since
 the Backward Euler scheme will introduce an error proportional to
-\( \Delta t \) anyway. Using <code>sympy</code> we find the following Taylor series
+\( \Delta t \) anyway. Using <code>sympy</code>, we find the following Taylor series
 expansions of the roots:
 
 <p>
@@ -824,7 +824,7 @@ <h3 id="___sec10">A single Picard iteration </h3>
 
 which is a linear algebraic equation in the unknown \( u^n \), making
 it easy to solve for \( u^n \) without any need for
-any alternative notation.
+an alternative notation.
 
 <p>
 We shall later refer to the strategy of taking one Picard step, or
@@ -877,7 +877,7 @@ <h2 id="nonlin:timediscrete:logistic:geometric:mean">Linearization by a geometri
 $$ u^{n+\half}\approx \half(u^n + u^{n+1}),$$
 
 such that the scheme involves the unknown function only at the time levels
-where we actually compute it.
+where we actually intend to compute it.
 The same arithmetic mean applied to the nonlinear term gives
 
 $$ (u^{n+\half})^2\approx \frac{1}{4}(u^n + u^{n+1})^2,$$
@@ -1228,7 +1228,7 @@ <h2 id="nonlin:ode:generic">Generalization to a general nonlinear ODE</h2>
 <h3 id="___sec16">Explicit time discretization </h3>
 
 <p>
-Explicit ODE methods like the Forward Euler scheme, Runge-Kutta methods,
+Explicit ODE methods like the Forward Euler scheme, Runge-Kutta methods and
 Adams-Bashforth methods all evaluate \( f \) at time levels where
 \( u \) is already computed, so nonlinearities in \( f \) do not
 pose any difficulties.
@@ -1261,8 +1261,8 @@ <h3 id="___sec17">Backward Euler discretization </h3>
 <div class="alert alert-block alert-success alert-text-normal"><b>Explicit vs implicit treatment of nonlinear terms.</b>
 Evaluating \( f \) for a known \( u^{-} \) is referred to as <em>explicit</em> treatment of
 \( f \), while if \( f(u,t) \) has some structure, say \( f(u,t) = u^3 \), parts of
-\( f \) can involve the known \( u \), as in the manual linearization
-like \( (u^{-})^2u \), and then the treatment of \( f \) is &quot;more implicit&quot;
+\( f \) can involve the unknown \( u \), as in the manual linearization
+\( (u^{-})^2u \), and then the treatment of \( f \) is &quot;more implicit&quot;
 and &quot;less explicit&quot;. 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,
@@ -1296,7 +1296,7 @@ <h3 id="___sec17">Backward Euler discretization </h3>
 <p>
 A trick to treat \( f \) implicitly in Picard iteration is to
 evaluate it as \( f(u^{-},t)u/u^{-} \). For a polynomial \( f \), \( f(u,t)=u^m \),
-this corresponds to \( (u^{-})^{m}u/u^{-1}=(u^{-})^{m-1}u \). Sometimes this more implicit
+this corresponds to \( (u^{-})^{m}u/u^{-}=(u^{-})^{m-1}u \). Sometimes this more implicit
 treatment has no effect, as with \( f(u,t)=\exp(-u) \) and \( f(u,t)=\ln (1+u) \),
 but with \( f(u,t)=\sin(2(u+1)) \), the \( f(u^{-},t)u/u^{-} \) trick
 leads to 7, 9, and 11 iterations during the first three steps, while
@@ -1308,7 +1308,7 @@ <h3 id="___sec17">Backward Euler discretization </h3>
 <p>
 Newton's method applied to a Backward Euler discretization of
 \( u^{\prime}=f(u,t) \)
-requires the computation of the derivative
+requires computation of the derivative
 
 $$ F^{\prime}(u) = 1 - \Delta t\frac{\partial f}{\partial u}(u,t_n)\tp$$