Elements of Numerical Methods for PDEs - Computational Methods for Quantitative Finance - page 19

(N − 1 ) × (N − 1 )

∈ R

Lemma 2.3.5 Let X be a tridiagonal matrix given by

⎛

⎞
⎠

αβ
γα . . .
. . .

⎝

X =

. . . β
γα

Then , X v () = μ v () , =

1 ,...,N − 1, with the eigensystem

2 β β − 1 γ cos N − 1 π , = (β − 1 γ) j/ 2 sin (N − 1 jπ) N − 1

j

()

μ = α +

1 .

=

For G resulting from the finite differences discretization, we obtain

Corollary 2.3.6 The eigensystem of the matrix G in ( 2.9 ) is given by

sin 2 π
2 (N +

, sin jπ

N +

N
j

4
h 2

()

μ = = , = 1 ,...,N. (2.19)

1 ) 1

= 1

Using Corollary 2.3.6 , we find that

θ)h − 2 k sin 2 (π/( 2 (N 4 ( 1 − + 1 ))) − 1

λ ( A θ ) = , = 1 ,...,N.

4 θh − 2 k sin 2 (π/( 2 (N 1 + + 1 )))

2 θ)h − 2 k

≤
θ< 2 , we obtain the so-called CFL-condition (after the seminal paper of Courant,
Friedrichs and Lewy [43]),

A θ 2 = max | λ | A θ 2 ≤

− ≤

Since ,wehave 1, if 2 ( 1 1. For 0

k
h 2 ≤

1

2 θ) .

(2.20)

2 ( 1 −

Therefore, we obtain directly from Theorem 2.3.4 :

Lemma 2.3.7 (Stability of the θ -scheme)

1

(i) If

2 ≤

θ ≤ 1, the scheme ( 2.10 ) is stable for all k and h .

θ< 2 , the scheme ( 2.10 ) is stable if and only if the CFL-condition ( 2.20 ) (ii) If 0 ≤

holds .

We can now combine the results obtained so far using Lemma 2.3.7 , Proposi-
tion 2.3.3 and Proposition 2.3.2 to obtain a convergence result for the θ -scheme. We
measure the error using the quantity sup m h 2 ε m

2 which is a discrete version of

the L ∞ (J ; L 2 (G)) -norm.