H-Matrix approximation for the operator exponential with applications

104 I. P. Gavrilyuk et al.
5 On the choice of computational parameters and numerics
In this section we discuss how the parameters of the parabola influence our
method.
Let Γ 0 = {z = ξ ± iη = aη 2 + γ 0 ± iη} be the parabola containing the
spectrum of L whose parameters a, γ 0 are determined by the coefficients of
L. Given a, we choose the integration path Γ b(k) = {z = ξ 1 ± iη 1 = a k η2 1 +
b ± iη 1 : η 1 ∈ (−∞, ∞)} with b(k) =γ 0 − k−1
4a
. In this case
(
the integrand
)
can be extended analytically into the strip D d with d = 1 − √ 1 k
k 2a and
the estimate (2.18) holds with constants given by (2.19).
First, let us estimate the constant M 1 in (2.19). Since the absolute value
of an analytic function attains its maximum on the boundary, we have
{
}
(5.1) M 1 = max
where
(5.2) f ± (η) =
It is easy to see that
sup
η∈(−∞,∞)
f − (η),
sup
η∈(−∞,∞)
f + (η)
|2 a k
(η ± id) − i|
∣∣
a
1+√
k (η ± id)2 + b − i(η ± id) ∣ .
(5.3) f± 2 (2 a k
≤
η)2 +(2 a k d ∓ 1)2
1+ ∣ a
k (η2 ± 2ηdi − d 2 )+b + d ∓ iη ∣ .
Further we have for the function f +
f+(η) 2 ≤
4 a2 η 2 +(2 a k 2 k d − 1)2
1+ √ ( a k (η2 − d 2 )+b + d) 2 +(2 a k d − 1)2 η 2
(5.4) ≤
4 a2 η 2 +(2 a k 2 k d − 1)2
1+ a k (η2 − d 2 )+b + d = 4 a2 η 2 + 1 k 2 k
a
k η2 +1+γ 0
= 4a +4γ 0 − 1
k(ξ +1+γ 0 ) 2 ,
where ξ = aη 2 /k, ξ ∈ (0, ∞). The latter function increases monotonically
and max f + = f + (∞) = 4a/k provided that 4a +4γ 0 > 1, while it
decreases monotonically and max f + = f + (0) =
1
k(1+γ 0 )
provided that
0 < 4a +4γ 0 < 1. Similarly one can see that max f − = f − (∞) =4a/k
provided that 4a(1 − γ 0 )/k > (2 − 1/k) 2 , while max f − = f − (0) =
(2 − 1/ √ k) 2 /(1 + γ 0 ) if 4a(1 − γ 0 )/k < (2 − 1/k) 2 . Thus, we have
{
}
4a
(5.5) M 1 ≤ max
k , 1
k(1 + γ 0 ) , (2 − 1/√ k) 2
.
1+γ 0
,