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Errata for Strikwerda, 2nd edition, first printing

This is a collection of comments and corrections to "Finite Difference Schemes and Partial Differential Equations." Note that some of the corrections are typeset in \( \LaTeX \).

Chapter #1 — Hyperbolic PDEs

Page 2, Figure 1.1:   [From Scott R. Fulton's errata page]

The ordinate label should be "\(u\)" (not "\(t\)").

Page 7, Exercise 1.1.1:   [From Scott R. Fulton's errata page]

The solution \( u(t,x) \) is not differentiable along the lines \(x=0\) and \( (x-at)=0 \), so in fact it is a generalized solution (Peter D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, SIAM, 1973).

Page 8, Exercise 1.1.8:   [From Scott R. Fulton's errata page]

Following the notation in the rest of this section, the initial conditions should be \( u(0,x)=u_0(x) \) and \( v(0,x)=v_0(x) \), i.e., the \( t \) and \( x \) are reversed here.

Page 12, Equation (1.2.7):   [From Scott R. Fulton's errata page]

The second part should read \( w_2(t,1) = -w_1(t,1) \) [the minus sign was left out--compare this with the middle of page 13, where the boundary condition is correct).

Page 34, Second paragraph of proof of theorem 1.6.1:   [From Scott R. Fulton's errata page]

Both references to \( u(1,x) \) should be \( u(1,0) \) instead.

Chapter #2 — Analysis of Finite Difference Schemes

Page 38, First equation on the page (example 2.1.1):

The left hand side is \( \tilde{u}(x) \) — the argument should be \( \omega \), and the tilde should be the hat in order to be consistemt with eqn (2.1.1). The correct expression should be \( \hat{u}(\omega) = ... \)

Page 43, Second equation on the page:

The numerator, \( \partial x \), of the integral on the left-hand-side is missing the power r; the correct integrand should be \( \left| \frac{ \partial^{r} u(x) } { \partial x^{r} } \right|^{2} \).

Chapter #3 — Order of Accuracy of Finite Difference Schemes

Page 66, First sentence:

It is a little odd to say "...is consistent with the one-way wave-equation according to Definition 3.1.2." Definition 3.1.2 defines the order of accuracy r, and a better wording may be "...is consistent with the one-way wave-equation, and the order of accuracy according to Defintion 3.1.2 is one (r=1)."

Page 67, Corollary 3.1.2:

For clarity it may be better to change "... order r ..." to "... order \( \rho \) " and "(\( h^{r} \))" to "(\( h^{\rho} \))", so that there is no confusion between the symbol, and the order...

Page 69, Theorem 3.1.3:

For clarity it may be better to change "... order r ..." to "... order \( \rho \) ..." and "(\( h^{r} \))" to "(\( h^{\rho} \))", so that there is no confusion between the symbol, and the order... This change should carry through to the proof and discussion on page 70, in the first three equations, and also to the sentence and un-numbered equation following equation (3.1.24) on page 73.

Page 88, exercise 3.4.1c:   [From Scott R. Fulton's errata page]

The boundary condition (3.4.1b) must be translated to apply at the left endpoint.

Chapter #4 — Stability for Multistep Schemes

Page 103, The two equations following (4.2.1):   [From Scott R. Fulton's errata page]

Both of these need additional multiplicative factors on the right-hand side to obtain polynomials. The first must be multiplied by \( g^{\sigma-1} \), where \( \sigma \) is the number of steps in the scheme as defined following (4.2.2), and the second must be multiplied by \( (e^{sk})^{\sigma-1} \). This assumes that the difference scheme (4.2.1) is written as normally done in this text, i.e., going from time level \( t_{n-\sigma+1} \) to time level \( t_{n+1} \).

Page 105, Theorem 4.2.2:   [From Scott R. Fulton's errata page]

The arguments of \( \Phi \) should be \( (g,\theta,k,h) \).

Chapter #5 — Dissipation and Dispersion

Page 126, Mid-page:

The reference to figure 1.3.6 is wrong. Is should be figure 1.6 (in section 1.3).

Chapter #6 — Parabolic Partial Differential Equations

Page 155, exercise 6.3.6:   [From Scott R. Fulton's errata page]

In the fourth equation there is a factor of "b" missing: replace "k" with "kb".

Chapter #7 — Systems of PDEs in Higher Dimensions

Page 170–171:

The reference to [72] is wrong, should be to Yanenko [71].

The reference to [25] is wrong, should be to Gottlieb [24].

The reference to [58] is wrong, should be to Strang [57].

The reference to [39] is wrong, should be to LeVeque and Oliger [38].

The reference to [41] is wrong, should be to MacCormack [40].

Page 175:

The reference to [50] is wrong, should be to Peaceman and Rachford [49].

Page 180:

The reference to [44] is wrong, should be to Mitchell and Fairweather [43].

Page 183, exercise 7.3.3:

The boundary conditions should be given at the boundary! So, the second sentence should read "Specify the exact solution along the sides with \( y = -1 \) and \( x = -1 \)."

Chapter #8 — Second-Order Equations

Page 199, exercise 8.2.6:

\( u_1(x) \) should be "\( c \pi \sin(\pi x) \)," the multiplicative factor of \( \pi \) is missing.

Chapter #12 — Elliptic Partial Differential Equations and Difference Schemes

Page 317:

The reference to [61] is wrong, should be to Taylor [60].

Page 318 and 319:

The reference to [23] is wrong, should be to Garabedian [22].

Page 329:

The reference to [54] is wrong, should be to Rosser [53].