Randy LeVeque -- Finite Difference Methods for ODEs and PDEs

Finite Difference Methods for Ordinary and Partial Differential Equations
Steady State and Time Dependent Problems

Randall J. LeVeque

Errata for books published after mid-2010

p. 23, equation (2.28), $\epsilon$ in both denominators should be $\epsilon^2$.
p. 29, line after (2.52), $\beta = 0$ should be $\beta = 3$.
p. 31, last line, and p. 32, first equation of (2.57): $\sigma$ should be replaced by $-\sigma$ since the one-sided finite difference approximates $-u'(x_0)$.
p. 32, first equation. The elements of the first row should come from the equation at the bottom of p. 31, after negating: $-3h/2,~ 2h,$ and $-h/2$. Also the second and third rows have an extra 0 (blank) at the start. (The $-2$ elements should all be on the diagonal.) The system should be: $$ \frac{1}{h^2} \left[ \begin{array}{ccccccccccccccc} -3h/2 & 2h & -h/2\\ 1&-2&1\\ &1&-2&1\\ &&&\ddots\\ &&&&1&-2&1\\ &&&&&0&h^2 \end{array} \right] ~ \left[ \begin{array}{ccccccccccccccc} U_0 \\ U_1 \\ U_2 \\ \vdots \\ U_m \\ U_{m+1} \end{array} \right] = \left[ \begin{array}{ccccccccccccccc} \sigma \\ f(x_1) \\ f(x_2) \\ \vdots \\ f(x_m) \\ \beta \end{array} \right] $$
p. 33, equation (2.6), the last part of the equation should be $= u'(0) + \int_0^1 f(x) \, dx$.
p. 130, the last line before section 5.8: "\texttt{typeode45}" should be "type \textt{ode45}"
p. 143, the displayed equation between (6.19) and (6.20): The factor $\frac 5 2$ should be $\frac 1 2$ in the dominant term of the truncation error.
p. 158, line -5, "diagonal matrix of eigenvectors" should be "... eigenvalues".
p. 172, line -10, the fraction $\frac{1 - \frac 1 2 k\lambda}{1 + \frac 1 2 k\lambda}$ should be $\frac{1 + \frac 1 2 k\lambda}{1 - \frac 1 2 k\lambda}$.
p. 172, line -6, $(1 + k\lambda)^{-1} \approx - 10^{-6}$ should be $(1 - k\lambda)^{-1} \approx 10^{-5}$.
p. 175, equation (8.5): the entry for $r=3: \alpha = 88^\circ$ should read $r=3: \alpha \approx 86^\circ$ and for $r \geq 4$ the $\alpha = \ldots$ equalities should be $\alpha \approx \ldots$. For $r=6$, $\alpha \approx 17^\circ$ would be on the safer side. See scholarpedia article for more precise values.
p. 222, (10.48): the first $=$ should be $+$.
p. 321, (E.36) LHS: $\bar{t}$ should be $\bar{x}$
p. 321, the first sentence below (E.36): $\beta\rightarrow 0$ should be $\beta\rightarrow +\infty$
p. 321, the first line below (E.36): $\bar{t}$ should be $\bar{x}$

Errata for books published before mid-2010

p. 8, second line of (1.12), $\frac{1}{12}$ should be $-\frac{1}{3}$.
p. 13, line 4: "(ODEs)" should be "(PDEs)"
p. 23, equation (2.28), $\epsilon$ in both denominators should be $\epsilon^2$.
p. 26, line -4: $u_{m+1}=\beta$ should be captial U.
p. 28, displayed equation in middle of page defining the inf-norm of $B$: the max should be over $i$, not $j$.
p. 32, equation (2.57), the first row of the matrix is missing. To the top of the matrix add a row
[3h/2 -2h h/2 ]
p. 35, line -3: "chain rule" should be "product rule".
p. 63, in the lines before (3.14), it should refer to the parameters p and q (not p and k) in two places.
p. 95: the description of the PCG algorithm is not correct. In the first unnumbered displayed equation, $w_k$ should be defined to be $C^T \tilde w_k$ not $C^{-1} \tilde w_k$.
In the algorithm at the bottom of p. 95, the definition of $\alpha_{k-1}$ should have $(z_{k-1}^T r_{k-1})$ in the numerator and the definition of $\beta_{k-1}$ should have $(z_{k-1}^T r_{k-1})$ in the denominator.
The last sentence of this page should then read "... and then use the $z$ vector in place of $r$ in several places in the algorithm."
p. 121, un-numbered displayed equation between (5.24) and (5.25): $\frac{1}{12}$ should be $-\frac{1}{3}$.
p. 130, last sentence of Section 5.7, there should be a space in "type ode45"
p. 213, equation (10.29): there is a factor of 1/2 missing in front of the $\nu$.
P. 213, equation (10.31): Second line, $+$ sign should be $-$ sign. the $\nu$.
P. 213, equation (10.32): First line, $4$ should be $4\nu^2$.
P. 213, equation (10.33): final term should be $e^{i\xi(j-1)h}$ (remove the minus sign).
p. 223, equation (10.52): $\nu$ should be $\nu^2$ in last denominator.