Resources for Project Finite Differences in 1D

Notes and errata

• The equation displayed at the bottom of the page 252 is missing a factor of 1/2 in its last term. The corrected equation is: \[ u(x,t_k) = u(x_j, t_k) + \frac{\partial u}{\partial x}\Big|_{(x_j, t_k)} (x - x_j) + \frac{1}{2} \frac{\partial^2 u}{\partial x^2}\Big|_{(x_j, t_k)} (x - x_j)^2 + \cdots. \] The next two equations (page 253) are also missing the 1/2 factors. The corrected equations are: \begin{align*} u(x_{j-1},t_k) &= u(x_j, t_k) + \frac{\partial u}{\partial x}\Big|_{(x_j, t_k)} (x_{j-1} - x_j) + \frac{1}{2} \frac{\partial^2 u}{\partial x^2}\Big|_{(x_j, t_k)} (x_{j-1} - x_j)^2 + \cdots, \\ u(x_{j+1},t_k) &= u(x_j, t_k) + \frac{\partial u}{\partial x}\Big|_{(x_j, t_k)} (x_{j+1} - x_j) + \frac{1}{2} \frac{\partial^2 u}{\partial x^2}\Big|_{(x_j, t_k)} (x_{j+1} + x_j)^2 + \cdots. \end{align*}
• Page 255, equation (20.9) should be \[ \lambda_j = 1 - 2r \big( 1 - \cos \frac{j\pi}{n+1} \big), \quad j = 1,2,\ldots,n. \]
• Page 256, equation (20.11):  $u_{j+1}^{j+1}$ should be $u_{j+1}^{k+1}$.
• Page 258, the first displayed equation should be \[ \lambda_j = \frac{1}{ 1 + 2r \big( 1 - \cos \frac{j\pi}{n+1} \big)}, \quad j = 1,2,\ldots,n. \]
• Page 264, equation (20.20): The $\frac{1}{2}$ in the exponent should be $\frac{1}{4}$, that is, the correct equation is \[ u_\text{ex}(x,t) = e^{-\frac{1}{4}\pi^2 t} \cos\frac{1}{2}\pi x \]
• Page 265, second line from the bottom: $x=0$ should be $x=1$.
• On page 267, line 4, we have:
  max error at time 0.2 is 0.00999372
  That should be
  max error at time 0.2 is 0.00799498
  
• In the graphs shown on pages 268 and 269, the orientations of the graphs in the rightmost column are inconsistent with those of the first three columns. For consistency, they should have been rotated about their vertical axes by 90 degrees clockwise (as viewed from above.)
• Page 275, Listing 20.5, line 1: It would be a good style to add const qualifiers to the parameters a and c, as in:

  static void trisolve(int n, const double *a, double *d,
                     const double *c, double *b, double *x)

  These assure the compiler that the vectors a and c will not be changed in this function.

New version!

The implemenation of this chapter's programs can be significantly streamlined. For that reason, I have written an alternative presetnation which I recommend that you follow instead of the one in the printed book. Should there be a second edition of the book some day, this version will replace the old.

Replacement for Chapter 20: finite-differences.pdf

Auxiliary files for Chapter 20: plot3d.h plot3d.c

Programming Projects in C