Errata for FINITE DIFFERENCE SCHEMES AND PARTIAL DIFFERENTIAL EQUATIONS by John C. Strikwerda

Date: November 9, 1994

In the list of corrections, the material that should replace the material in the text has been placed in boxes. Some of the corrections have been corrected in the second printing. I wish to thank all those who brought these mistakes to my attention.

This document is available by anonymous ftp as pub/misc/strik.errata.dvi from ftp.cs.wisc.edu.

Errata for Chapter 1

1. On page 4, in the fourth and fifth lines from the bottom, the initial conditions should have w^{1}(0, x) and w^{2}(0, x), instead of (x, 0).

2. In problem 5 on page 12, at the end of the last two lines, the expression 1 + x should be 1 + \frac{1}{3}x on both lines.

3. In the first paragraph on page 17, the sentence about the leapfrog scheme should be moved to the end of the paragraph.

4. In the middle of page 24, replace the formulas with

∥v^{n}∥_{h} ≤ (C_{T} ∑_{j=0}^{J} ∥v^{j}∥_{h}^{2})^{1/2},

which is equivalent to

∥v^{n}∥_{h} ≤ C_{T}^{∗} ∑_{j=0}^{J} ∥v^{j}∥_{h},  (1.5.3)

5. O. Hald has pointed out that the stability definition (Definition 1.5.1) on page 24 needs some modification to allow for conditionally stable schemes. I believe the following is adequate for the text and conforms to the usage of stability in the text.

Definition 1.5.1 A finite difference scheme P_{k,h}v_{m}^{n} = 0 for a first-order equation is stable in a stability region Λ if there is an integer J such that for any positive time T, there is a constant C_{T} such that

h ∑_{m=−∞}^{∞} |v_{m}^{n}|^{2} ≤ C_{T} h ∑_{j=0}^{J} ∑_{m=−∞}^{∞} |v_{m}^{j}|^{2}  (1.5.1)

for 0 ≤ nk ≤ T, with (k, h) ∈ Λ.

6. With the change in Definition 1.5.1, replace the last full paragraph on page 24 (the one discussing h_{0} and k_{0} ) with

A stability region is any bounded nonempty region of the first quadrant of R^{2} that has the origin as an accumulation point. That is, a stability region must contain a sequence (k_{ν}, h_{ν}) that converges to the origin as ν tends to infinity. A common example is a region of the form {(k, h) : 0 < k ≤ ch ≤ C} for some positive constants c and C.

Errata for Chapter 2

1. On page 34, in the third line of the first formula it should be e^{imhξ} rather than e^{−imhξ}.

2. On page 34, in the function defined at the bottom of the page, the top and bottom inequalities should be strict inequalities.

3. On page 34, replace formula (2.1.9) with:

∥v̂^{n}∥_{h} ≤ C_{T}^{∗} ∑_{j=0}^{J} ∥v̂^{j}∥_{h}, (2.1.9)

4. At the top of page 35, replace the formulas with

v̂(ξ) = \frac{h}{\sqrt{2π}} (\frac{1}{2}e^{iMhξ} + \frac{1}{2}e^{−iMhξ}) + \frac{h}{\sqrt{2π}} ∑_{m=−(M−1)}^{M−1} e^{−imhξ}

= \frac{h}{\sqrt{2π}} cos ξ + \frac{h}{\sqrt{2π}} \frac{sin(M − \frac{1}{2})hξ}{sin \frac{1}{2}hξ}

= \frac{h}{\sqrt{2π}} cos ξ + \frac{h}{\sqrt{2π}} \frac{sin Mhξ cos \frac{1}{2}hξ − cos Mhξ sin \frac{1}{2}hξ}{sin \frac{1}{2}hξ}

= \frac{h}{\sqrt{2π}} sin ξ cot \frac{1}{2}hξ.

5. On page 38, in exercise 2, the last Fourier transform should be \frac{1}{\sqrt{2π}} \frac{Γ(1 + α)}{(1 + iω)^{1+α}}

6. On page 39, in exercise 8, the Parseval’s relation should read

∑_{m=0}^{M−1} |v_{m}|^{2} = \frac{1}{M} ∑_{ℓ=0}^{M−1} |v̂_{ℓ}|^{2}.

7. On page 40, in exercise 9, the formulas for v̂ should read

v̂_{ℓ} = v̂_{ℓ}^{c} − iv̂_{ℓ}^{s} for l = 0, …, \frac{M}{2}
v̂_{ℓ} = v̂_{ℓ_{0}}^{c} + iv̂_{ℓ_{0}}^{s} for l = \frac{M}{2} + 1, …, M − 1, with ℓ_{0} = M − ℓ

8. On page 42, in the summation just above the middle of the page there should not be a ˆ on the v.

9. On page 42, Theorem 2.2.1 should be replaced with

Theorem 2.2.1 A one-step finite difference scheme (with constant coefficients) is stable in a stability region Λ if and only if there is a constant K (independent of θ, k, and h) such that

|g(θ, k, h)| ≤ 1 + Kk  (2.2.6)

with (k, h) ∈ Λ. If g(θ, k, h) is independent of h and k, the stability condition (2.2.6) can be replaced with

|g(θ)| ≤ 1.  (2.2.7)

10. On page 46, the second sentence of Example 2.2.2 should read ‘ Replacing v_{m}^{n} by g^{n}e^{imθ} ....’ 11. In the middle of page 46, the reference to Corollary 2.2.1 should be to Corollary 2.2.2.

12. The last line on page 48 should read:

g(θ) = cos θ + 4akh^{−3}i sin θ sin^{2} \frac{1}{2}θ,

13. On page 49, the first lines should read:

and it is easily shown (see Exercise 2.2.3) that the scheme is stable only if

4|a|kh^{−3}

is bounded.

14. On page 49, the first exercise in EXERCISES 2.2 should read:

1. Show that the backward-time central-space scheme (1.6.1) is consistent with equation (1.1.1) and is unconditionally stable.

15. On page 50, on line 9 from the bottom, the word ‘discontinuity’ should replace the word ‘discontinuities.’ 16. There are several corrections to the first two exercises in EXERCISES 2.3. The correct reading should

be:

EXERCISES 2.3

1. Use the unstable forward-time forward-space scheme (1.3.1) for u_{t} + u_{x} = 0 with the initial data

u_{0}(x) = \{ \begin{matrix}1 − |x| & if |x| ≤ 1, \\ 0 & otherwise, \end{matrix}

on the interval [−1, 3] for 0 ≤ t ≤ 1. Use a grid spacing of 0.1 and λ equal to 0.8. Demonstrate that the instability grows by approximately |g(π)| per time step. Comment on the appearance of the graph of v_{m}^{n} as a function of m. Use the boundary condition u(t, −1) = 0, at the left and use v_{M}^{n+1} = v_{M−1}^{n+1} at the right boundary.

2. Use the unstable forward-time central-space scheme (1.3.3) for u_{t} + u_{x} = 0 with the following two sets of initial data on the interval [−1, 3] for 0 ≤ t ≤ 4 :

(a) u_{0}(x) = \{ \begin{matrix}1 − |x| & if |x| ≤ 1, \\ 0 & otherwise. \end{matrix}

(b) u_{0}(x) = sin x

Use a grid spacing of 0.1 and λ equal to 0.8. Demonstrate that the instability is evident sooner with the less smooth initial data (a) than it is for the smooth data (b). Show that the growth in the instability for each case is approximately |g(π/2)|. For (a) use the boundary condition u(t, −1) = 0, and for (b) use the boundary condition u(t, −1) = − sin(1 + t). Use v_{M}^{n+1} = v_{M−1}^{n+1} at the right boundary.

1. On page 59, Theorem 3.1.3 should be corrected to be:

Theorem 3.1.3 An explicit one-step scheme for hyperbolic equations that has the form

v_{m}^{n+1} = ∑_{ℓ=−∞}^{∞} α_{ℓ} v_{m+ℓ}^{n}  (3.1.16)

for homogeneous problems can be at most first-order accurate if all the coefficients α_{ℓ} are nonnegative, except for the trivial schemes for the one-way wave equation with |aλ| = ℓ, where ℓ is an integer, given by

v_{m}^{n+1} = v_{m−ℓ}^{n}  (3.1.17)

2. On page 59, there should also be a reference with Theorem 3.1.3 indicating that it was originally proved independently by S.K. Godunov and P.D. Lax, and extended by A. Harten, J.M. Hyman, and P.D. Lax. The references are:

Godunov, S.K. 1959. Finite difference methods for numerical computation of discontinuous solutions of equations of fluid dynamics. Mat. Sb. 47:27-295. (In Russian.) Lax, P.D. 1961. On the stability of difference approximations to solutions of hyperbolic with variable coefficients. Communications on Pure and Applied Mathematics, 14:497-520. Harten, A., J.M. Hyman, and P.D. Lax. 1976. On finite-difference approximations and entropy conditions for shocks, Communications on Pure and Applied Mathematics, 29:297-322.

3. On page 61, the proof of Theorem 3.1.3 should be modified by replacing (3.1.21) and the following text

with:

This means there must be a constant c such that

\sqrt{α_{ℓ}} ℓ = c\sqrt{α_{ℓ}} for all ℓ,  (3.1.21)

and this implies that at most one α_{ℓ} is nonzero. It is then easy to check that the only way these relations can be satisfied is if |aλ| is an integer, and the resulting schemes are the trivial schemes (3.1.17). This proves the theorem.

4. On page 65, equation (3.1.23) should read:

\begin{aligned}\frac{1}{2k} & [(v_{m}^{n+1} + v_{m+1}^{n+1}) − (v_{m}^{n} + v_{m+1}^{n})] \\ & + \frac{a}{2h} [(v_{m+1}^{n+1} − v_{m}^{n+1}) + (v_{m+1}^{n} − v_{m}^{n})] \\ = & \frac{1}{4} (f_{m+1}^{n+1} + f_{m}^{n+1} + f_{m+1}^{n} + f_{m}^{n}) \end{aligned} (3.1.23)

5. On page 67, in exercise 10, the symbol should be

\frac{e^{sk} − cos hξ}{k} − 4ai\frac{sin^{2} \frac{1}{2}hξ sin hξ}{h^{3}}

6. On page 71 in the sixth line of the first equation, the ( after the = should be removed.

7. On page 71 in the last line there should be an additional ) before the +O(k^{2}).

8. On page 72 in the fourth line the reference should be to Exercise (3.2.8). (See the revised numbering for the exercises.)

9. Remove Exercise 3.2.8 on page 74 and correct the following two exercises as:

8. Show that the scheme derived in Example 3.2.2,

\begin{aligned}\frac{v_{m}^{n+1} − v_{m}^{n}}{k} & + aδ_{0} (\frac{v_{m}^{n+1} + 2v_{m}^{n}}{3} ) − \frac{k}{6} a^{2}δ^{2} (\frac{v_{m}^{n+1} + 2v_{m}^{n}}{3} ) \\ & = \frac{f_{m}^{n+1} + f_{m}^{n}}{2} − \frac{ak}{6} δ_{0} (\frac{f_{m}^{n+1} + 2f_{m}^{n}}{3} ) , \end{aligned}

is stable for |aλ| ≤ 3.

9. Use the relationship ∂ = ln(1 + hδ_{+})/h to derive the second-order accurate one-sided approximation

\frac{du}{dx}(x_{0}) ≈ \frac{−3u(x_{0}) + 4u(x_{1}) − u(x_{2})}{2h}.

10. On page 77 in the fourth line below the figure, the exact solution is sin 2π(x − t).

11. On page 81 in Exercise 1, it should be p_{1} = 0.

12. On page 81 the third part of Exercise 3 should read:

3. The solution w_{i} is then obtained as w_{i} = x_{i} −ry_{i} where r = \frac{x_{0} − x_{m}}{y_{0} − y_{m}} .

Errata for Chapter 4

1. The last formula on page 84 should read:

B(ξ) = [\frac{1}{2}a^{2}λ^{2} sin^{2} hξ + O(hξ)^{4}] v̂^{0}(ξ).

2. The middle formula on page 87 should read:

\frac{d}{dθ} (4 − cos θ) sin θ = 1 + 4 cos θ − 2 cos^{2} θ = 0.

3. In the discussion at the bottom of page 90, change g_{0}(θ_{0}) to g_{1}(θ_{0}) and formula (4.2.9) and the line

before it should read: If |g_{1}(θ_{0})| is less than 1, then this quantity is bounded by a multiple of

(|g_{1}(θ_{0})| log |g_{1}(θ_{0})|^{−1})^{−1} |B_{1}|  (4.2.9)

4. On page 91, in exercise 4.2.1, ’Vandermonde’ is misspelled.

5. In the third line on page 96, the expression |7 + 2iβ| should be replaced by |7 − 4iβ|.

6. In exercise 11 on page 99, the second displayed equation should read:

f_{j+1} = \frac{f_{j}(z)f_{j}^{∗}(−1) − f_{j}^{∗}(z)f_{j}(−1)}{z + 1}.

1. On page 102, the last sentence before the exercises should read:

For a dissipative scheme the amplification

polynomial is a Schur polynomial for θ not equal to zero.

2. On page 109, the second displayed formula is missing an h. It should read:

\begin{aligned}γ̃(hω) & = \frac{(ω + ξ_{0})α(hω + hξ_{0}) − ξ_{0}α(hξ_{0})}{ω} \\ & = \frac{(φ + θ_{0})α(φ + θ_{0}) − θ_{0}α(θ_{0})}{φ} \end{aligned}

3. On page 109, the fifth line should refer to the Taylor series in α rather than γ̃.

4. On page 109, the middle formula has a α in the integral that should be a γ. Also, the formula should be numbered (5.3.10), and the next formula should be numbered (5.3.11). The new formula (5.3.10) should read:

ṽ(t, x) = e^{iξ_{0}[x−α(hξ_{0})t]} \frac{1}{\sqrt{2π}} ∫_{−∞}^{∞}e^{iω[x−γ(hξ_{0})t]}e^{ihtr(ω)}p̂(ω) dω (5.3.10)

5. In the line above formula (5.3.10) there is an extraneous ‘of’.

Errata for Chapter 6

1. On page 116, replace the first exercise with:

1. Determine the behavior of the quantity C_{t,ℓ,m} in (6.1.4) as a function of t. In particular, show that C_{t,ℓ,m} is unbounded as t approaches 0. Also determine the behavior of the bounds on the L^{2} norm of the derivatives of u as t approaches 0.

2. On page 121, the order of accuracy of the Du Fort–Frankel scheme is given by O (h^{2}) + O (k^{2}) + O (k^{2}h^{−2}) .

3. On page 127, the second equation in (6.3.14) should have a b after the k.

Errata for Chapter 7

1. The matrix near the bottom of page 137 should read:

G(hξ) = ( \begin{matrix}a_{0} & a_{1} & … & a_{K−1} & a_{K} \\ I & 0 & … & 0 & 0 \\ 0 & I & … & 0 & 0 \\ ⋮ & ⋮ & ... & ⋮ & ⋮ \\ 0 & 0 & … & I & 0 \end{matrix} ) .

2. On page 139, in equation (7.2.5) the parentheses should be removed.

3. On page 147, in the last line of the second paragraph, the word ‘index’ should be inserted before the word ‘corresponding.’

4. On page 147, in the third line from the bottom, the expression ṽ_{L,m} should read ṽ_{ℓ,M}.

5. On page 149, the expressions given in the paragraph before equation (7.3.18) are incorrect. The second sentence should read:

We can eliminate the terms with δ_{x}^{2}v^{n+1/2} from

(7.3.17) by multiplying the first equation by b_{1}μ_{1} + 1/6 and the second by b_{1}μ_{1} − 1/6, and then eliminating the terms containing δ_{x}^{2}v^{n+1/2}.

6. On page 153, in exercise 14, in the last line, the word ‘only’ should be omitted.

Errata for Chapter 8

1. On page 158, Definition 8.2.1 must be modified as was Definition 1.5.1.

Definition 8.2.1 A finite difference scheme P_{k,h}v_{m}^{n} = 0 for an equation that is second-order in t is stable in a stability region Λ if there is an integer J such that for any positive time T, there is a constant C_{T} such that

h ∑_{m=−∞}^{∞} |v_{m}^{n}|^{2} ≤ (1 + n^{2}) C_{T} h ∑_{j=0}^{J} ∑_{m=−∞}^{∞} |v_{m}^{j}|^{2}  (8.2.1)

for 0 ≤ nk ≤ T with (k, h) ∈ Λ.

2. On page 159, in lines 9 and 10 it should read that J is at least 1.

3. On page 161, in equation (8.2.4) there should be a δ^{2} before the v_{m}^{n} on the right-hand side.

4. On page 163, the scheme in exercise 8.2.2 should be:

\begin{matrix}\frac{v_{m}^{n+2} − 2v_{m}^{n} + v_{m}^{n−2}}{4k^{2}} − a^{2} (1 + \frac{h^{2}}{12} δ_{x}^{2})^{−1} δ_{x}^{2}v_{m}^{n} \\ − \frac{a^{2}}{48} (1 + \frac{h^{2}}{12} δ_{x}^{2})^{−1} δ_{x}^{2} (v_{m}^{n+2} + 12v_{m}^{n+1} − 26v_{m}^{n} + 12v_{m}^{n−1} + v_{m}^{n−2}) = 0 \end{matrix}

5. On page 168, the scheme in exercise 8.4.4 should be:

\begin{aligned}(1 − \frac{k^{2}}{4} a_{11}δ_{x}^{2}) ṽ_{ℓ,m}^{n+1/2} & = (1 + \frac{k^{2}}{4} a_{22}δ_{y}^{2}) v_{ℓ,m}^{n} + \frac{k^{2}}{2} a_{12}δ_{0x}δ_{0y}v_{ℓ,m}^{n} \\ (1 − \frac{k^{2}}{4} a_{22}δ_{y}^{2}) ṽ_{ℓ,m}^{n+1} & = (1 + \frac{k^{2}}{4} a_{11}δ_{x}^{2}) ṽ_{ℓ,m}^{n+1/2} + \frac{k^{2}}{2} a_{12}δ_{0x}δ_{0y}v_{ℓ,m}^{n} \\ v_{ℓ,m}^{n+1} & = 2ṽ_{ℓ,m}^{n+1} − v_{ℓ,m}^{n−1}. \end{aligned}

1. On page 173, in the last formula on the page the subscript on B_{−} should be removed.

2. On page 174, the formula in the middle of the page should read:

\begin{aligned}|û(t, ω)| & ≤ (|A(ω)| + |B(ω)|) e^{q̄t} \\ & ≤ [|û(0, ω)| \frac{|q_{+}(ω)| + |q_{−}(ω)|}{|q_{+}(ω) − q_{−}(ω)|} + \frac{2|û_{t}(0, ω)|}{|q_{+}(ω) − q_{−}(ω)|}] e^{q̄t}. \end{aligned}

3. At the bottom of page 174, the functions A′ and B′ should be denoted with different symbols. (They are not the derivatives of the functions A and B used earlier.) Replace the lines near the bottom of page 174 with:

For |ω| less than 2c_{0} we write the equation for û(t, ω) as

û(t, ω) = C(ω) [\frac{e^{q_{+}(ω)t} + e^{q_{−}(ω)t}}{2} ] + D(ω) [\frac{e^{q_{+}(ω)t} − e^{q_{−}(ω)t}}{q_{+}(ω) − q_{−}(ω)} ] .

We have that

C(ω) = û(0, ω)

and

D(ω) = −û(0, ω) (\frac{q_{+}(ω) + q_{−}(ω)}{2}) + û_{t}(0, ω).

4. Exercise 4 on page 177 should read:

4. Show that the following two equations are well-posed.

a.  u_{tt} + 2au_{txxx} + bu_{xxxx} = 0 for all real a and b, a ≠ 0
b.  u_{tt} + 2au_{txx} + b^{2}u_{xxxx} = 0 if a ≤ 0

5. On page 187, in the last line on the page the summation should go to n − 1.

6. On page 192, on the second to the last line it should be ‘hermitian’ rather than ‘symmetric.’

7. On page 193, the sequence of equalities and inequalities should read:

∥A^{n}∥ = ∥T^{−1}T A^{n}T^{−1}T ∥ ≤ ∥T^{−1}∥ ∥T ∥ ∥T A^{n}T^{−1}∥

≤ 2∥T^{−1}∥ ∥T ∥ sup_{|x|=1} |(T A^{n}T^{−1}x, x)|

= 2∥T^{−1}∥ ∥T ∥ sup_{|x|=1} |(Ω A^{n}T^{−1}x, T^{−1}x)|

≤ 2∥T^{−1}∥ ∥T ∥ sup_{|x|=1} |(Ω T^{−1}x, T^{−1}x)|

= 2∥T^{−1}∥ ∥T ∥ ≤ 2C_{ω}.

It should also mention that ∥T ∥ and ∥T^{−1}∥ are bounded by C_{ω}^{1/2}.

1. On page 202, in the formula for the factorization of e^{q(ξ)t_{n}} −g(hξ)^{n} in the middle of the page the power of z in the summation should be n − j − 1 rather than n − j.

2. The first equation on page 205 should read:

∥Eu − T u∥_{h}^{2} ≤ C(σ)^{2}h^{2σ}∫_{−π/h}^{π/h} ^{∑′} |û(ξ + 2πℓh^{−1})|^{2} |ξ + 2πℓh^{−1}|^{2σ} dξ

3. In the last equation on page 205 in the second line, the expression Sŵ^{n} should be Ŝw^{n}.

4. On page 206, the equation in the middle of the page should read

p(s, ξ) = q_{0}(ξ)s + q_{1}(ξ).

5. On page 207, the proof of Theorem 10.1.1 is incorrect as given. It requires the use of the Taylor series with the integral form of the remainder. Starting with the eighth line:

We begin with the

Taylor series with remainder for the exponential function for complex variables in the form

e^{z} = ∑_{j=0}^{J−1} \frac{z^{j}}{j!} + \frac{z^{J}}{(J − 1)!} ∫_{0}^{1}(1 − t)^{J−1}e^{tz} dt.

From this we obtain

e^{kq(ξ)} = ∑_{j=0}^{J−1} \frac{k^{j}q(ξ)^{j}}{j!} + \frac{k^{J}q(ξ)^{J}}{(J − 1)!} ∫_{0}^{1}(1 − t)^{J−1}e^{tkq(ξ)} dt.

This requires that formula (10.1.17) and the following lines be:

\begin{matrix}\frac{e^{kq(ξ)} − g(hξ, k, h)}{k} = ∑_{j=0}^{J−1} \frac{1}{j!} k^{j−1}q(ξ)^{j} − ∑_{i=0}^{I−1} \frac{k^{−1}h^{i}ξ^{i}}{i!} g^{(i)}(0) \\ + \frac{k^{J−1}q(ξ)^{J}}{(J − 1)!} ∫_{0}^{1}(1 − t)^{J−1}e^{tkq(ξ)} dt − \frac{k^{−1}h^{I}ξ^{I}}{I!} g^{(I)}(θ′) = O(h^{r}). \end{matrix}

(10.1.17)

Now, recalling that k = Λ(h), we can choose J and I large enough so that the two sums combine to be O(h^{r}). Similarly, k^{J−1}, which is Λ(h)^{J−1}, and k^{−1}h^{I} which is Λ(h)^{−1}h^{I}, can be made to be O(h^{r}). The value of g^{(I)}(θ′) is then bounded by some constant, and the integral is bounded independently of ξ because the equation is well-posed. Finally, q(ξ) is a rational function of ξ, and so the growth of (10.1.17) in ξ is at most polynomial. Thus (10.1.17) is bounded by h^{r} times a constant multiple of (1 + |ξ|)^{ρ} for some nonnegative integer ρ.

6. On page 208, in exercise 1, the scheme is accurate of order [1, 1].

7. On page 208, in exercise 4, in the first equation t should be t_{n}.

8. On page 210, the references to equations (5.3.9) and (5.3.10) in the tenth line should be to (5.3.10) and (5.3.11), respectively. (This numbering corresponds to the corrections made in Chapter 5.)

9. On page 215, the first line of text should read: If we choose α equal to r/ρ, then both terms have h to the power 2σr/ρ; this

10. On page 217, in line 11, the reference should be to Theorem 10.1.3.

11. On page 217, the third case for defining the function γ should read:

−\frac{1}{2}e^{−|x|} if x < 0.

12. The Fourier transform of γ given three lines later should be:

γ̂(ω) = \frac{1}{\sqrt{2π}} \frac{−iω}{1 + ω^{2}}

13. The second to the last equation on page 217 should read:

\begin{aligned}Êγ(ξ) − T̂ γ(ξ) & = \begin{matrix}∞ \\ ∑ & ′ \\ ℓ=−∞ \end{matrix} γ̂(ξ + 2πℓh^{−1}) \\ & = \frac{−i}{\sqrt{2π}} ∑_{ℓ=1}^{∞} \frac{ξ + 2πℓh^{−1}}{1 + (ξ + 2πℓh^{−1})^{2}} + \frac{ξ − 2πℓh^{−1}}{1 + (ξ − 2πℓh^{−1})^{2}} \\ & = \frac{−i}{\sqrt{2π}} ∑_{ℓ=1}^{∞} \frac{2ξ [1 + ξ^{2} − (2πℓh^{−1})^{2}]}{[1 + (ξ + 2πℓh^{−1})^{2}] [1 + (ξ − 2πℓh^{−1})^{2}]}, \end{aligned}

14. On page 220, before equation (10.4.3), the reference should be to equation (10.1.5) rather than (10.4.5).

15. On page 223, in the ninth line, the reference should be to equation (10.1.3) rather than (10.3.10).

16. On page 223, the definition of ϕ_{h} should read:

ϕ_{h}(ξ) = \{ \begin{matrix}|e^{q(ξ)t_{n}} − g(hξ)^{n}|^{2} |û_{0}(ξ)|^{2} & if |ξ| ≤ \frac{π}{h}, \\ |e^{q(ξ)t_{n}} û_{0}(ξ)|^{2} & if |ξ| > \frac{π}{h} . \end{matrix}

17. On page 224, in formula (10.5.5), the parameters h_{M} and k_{M} in the amplification factor should be interchanged.

On page 245, the boundary condition in exercise 11.2.1 should be

v_{0}^{n+1}(1 + aλ) − aλv_{1}^{n+1} − v_{0}^{n} = β^{n+1}

Errata for Chapter 12

1. On page 270, in the sixth line, the word ‘condition’ should be inserted after the phrase ‘Neumann boundary.’

2. On page 277, in exercise 4, the reference should be to Exercise 12.3.2.

3. On page 279, in exercise 1. a, the condition on u should be u = r^{α} on θ = \frac{3}{2}π, 0 < r < 1.

4. On page 280, in the sixth line, the reference should be to (12.5.1) rather than (12.3.1).

5. On page 284, equation (12.5.7) should have a power of 2 on the norm for f.

6. On page 285, the inequality on the fourth line has Ω_{1} and Ω interchanged. It should read:

\begin{aligned}∥u − v∥_{h,s,Ω_{1}}^{2} & ≤ C_{s−2} (∥L_{h}u − R_{h}Lu∥_{h,s−2,Ω}^{2} + ∥u − v∥_{h,0,Ω}^{2}) \\ & ≤ C_{s−2}h^{2r}∥u∥_{s}^{2} \end{aligned}

7. On page 285, the thirteenth line, should read:

estimates (12.5.8) and (12.5.9) with r equal to 4, then

8. On page 290, in exercise 3, the coefficients are in error in the last expression in the numerator. It should read:

\begin{matrix}\frac{∂}{∂x} & [\frac{a_{11} (\frac{∂a_{12}}{∂x} + \frac{∂a_{22}}{∂y} ) − a_{12} (\frac{∂a_{11}}{∂x} + \frac{∂a_{12}}{∂y} )}{a_{11}a_{22} − a_{12}^{2}} ] \\ + & \frac{∂}{∂y} [\frac{a_{12} (\frac{∂a_{11}}{∂x} + \frac{∂a_{12}}{∂y} ) − a_{22} (\frac{∂a_{12}}{∂x} + \frac{∂a_{22}}{∂y} )}{a_{11}a_{22} − a_{12}^{2}} ] = 0. \end{matrix}

1. On page 299, in Definition 13.2.2, A_{1} and A_{2} must be square matrices.

2. On page 300, in the first line of equation (13.2.15) the last x_{i} should be x_{j}. Also, there is a | missing in the last expression in this inequality.

3. On page 305, in the fourth line from the bottom, the word ‘plan’ should be ‘plane.’

4. On page 310, Theorem 13.4.1 is incorrect. Replace the bottom of page 310 and the top of page 311 with:

Theorem 13.4.1 If A is symmetric and positive definite, then the iterative method (13.1.8) based on the splitting (13.1.7) is convergent if

Re B > \frac{1}{2}A  (13.4.1)

or, equivalently, that B^{T} + C is symmetric and positive definite, i.e.,

B^{T} + C > 0  (13.4.2)

Proof We first establish that the two conditions in the conclusion are equivalent. The matrix Re B is \frac{1}{2}(B + B^{T}), and thus (13.4.1) is equivalent to

B^{T} + B − A > 0.  (13.4.3)

The defining relation of the splitting (13.1.7) shows that this is equivalent to (13.4.2) and also shows that B^{T} + C is symmetric.

We now begin the proof. We measure the error in the norm induced by A, i.e., ∥x∥_{A} = (x, Ax)^{1/2}. In this norm we have the relation

∥e^{k+1}∥_{A} = ∥B^{−1}Ce^{k}∥_{A} ≤ ∥B^{−1}C∥_{A}∥e^{k}∥_{A}

(see Appendix A). If the norm of B^{−1}C is less than 1, then the error will decrease at each iteration and the method will converge. We have the norm of B^{−1}C given by

∥B^{−1}C∥_{A}^{2} = sup_{x≠0} \frac{(B^{−1}Cx, AB^{−1}Cx)}{(x, Ax)} = sup_{x≠0} \frac{(x, C^{T} B^{−T} AB^{−1}Cx)}{(x, Ax)}.

Thus the condition ∥B^{−1}C∥_{A} < 1 is equivalent to C^{T} B^{−T} AB^{−1}C < A, and we consider now the matrix C^{T} B^{−T} AB^{−1}C.

5. On page 316, formulas (13.5.10) should read:

\begin{aligned}R_{J+1}^{k+1} & = R_{J}^{k+1} + (r_{ℓ,m}^{k+1} − \overline{r}_{J}^{k+1})^{2} \frac{J}{J + 1} \\ V_{J+1}^{k+1} & = V_{J}^{k+1} + (v_{ℓ,m}^{k+1} − \overline{v}_{J}^{k+1})^{2} \frac{J}{J + 1} \end{aligned} (13.5.10)

6. The formula at the bottom of page 316 should read:

u_{xx} + u_{yy} = −5 sin (x + 2y)

1. On page 323, there should be a citation of the work of P. Concus, G. Golub, D. O’Leary on which the material is based. The reference is:

Concus, P., G.H. Golub, and D.P. O’Leary. 1976. A generalized conjugate gradient method for the numerical solution of elliptic partial differential equations, in Sparse Matrix Computations, Eds, J.R. Bunch and D.J. Rose. 309-322. Academic Press, New York

2. On page 324, in the last line it should be −F (0) rather than +F (0).

3. On page 325, in formula (14.1.10c) the r_{k} should be r^{k}.

4. On page 325, in Theorem 14.1.1, the last expression should be x^{0}, not x_{0}.

5. On page 326, in EXERCISES 14.1, exercise 2, λ_{1} and λ_{2} are the greatest, not least eigenvalues and λ_{3} is the least eigenvalue.

6. On page 327, in EXERCISES 14.1, exercise 7, A must also be symmetric.

7. On page 336, in the first line of the proof, the references should be to (14.2.9c) and (14.2.9d), respectively.

8. On page 338, equation (14.4.9) should read:

T_{k}(μ) = \{ \begin{matrix}cos(k cos^{−1} μ) & if |μ| ≤ 1 \\ [sign(μ)]^{k} cosh(k cosh^{−1} |μ|) & if |μ| ≥ 1. \end{matrix}  (14.4.9)