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MATHEMATICS OF COMPUTATION Volume 68, Number 227, Pages 1067–1078 S 0025-5718(99)01059-5 Article electronically published on February 5, 1999
ERROR ESTIMATION OF HERMITE SPECTRAL METHOD FOR NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS GUO BEN-YU
Abstract. Hermite approximation is investigated. Some inverse inequalities, imbedding inequalities and approximation results are obtained. A Hermite spectral scheme is constructed for Burgers equation. The stability and convergence of the proposed scheme are proved strictly. The techniques used in this paper are also applicable to other nonlinear problems in unbounded domains.
1. Introduction A number of physical problems are set in unbounded domains. Some conditions at infinity are given by certain asymptotic behaviors for solutions. When we use the finite difference method or the finite element method to solve such problems numerically, we often restrict calculations to some bounded domains, and impose certain conditions on artificial boundaries. They cause numerical errors usually. If we use spectral methods associated with some orthogonal systems in unbounded domains, then the above troubles could be avoided. While the spectral methods provide numerical solutions with high accuracies. Maday, Pernaud-Thomas and Vandeven [1], Coulaud, Funaro and Kavian [2], and Funaro [3] used the Laguerre spectral method for several linear partial differential equations. Iranzo and Falqu`es [4] provided some Laguerre pseudospectral schemes and Laguerre tau schemes. Mavriplis [5] and Black [6] developed the Laguerre spectral element method. Also, Funaro and Kavian [7], and Weideman [8] considered the Hermite spectral method and the Hermite pseudospectral method. In particular, Funaro and Kavian [7] proved the convergence of a spectral scheme using the Hermite functions for some linear problems. But so far, there is no paper concerning error estimates of the Hermite spectral method using Hermite polynomials. Another spectral method for partial differential equations in unbounded domains is based on the rational basis functions, see Christov [9], Boyd [10], Iranzo and Falqu`es [4], and Weideman [8]. The purpose of this paper is to study spectral approximation using Hermite polynomials and their applications to nonlinear problems. Some inverse inequalities, imbedding inequalities and approximation results are given, which play important roles in analysis of the Hermite spectral method. We use the Burgers equation as an example showing how to construct Hermite spectral schemes for nonlinear problems. The generalized stability and the convergence of the proposed scheme are proved Received by the editor October 16, 1997 and, in revised form, January 2, 1998. 1991 Mathematics Subject Classification. Primary 65N30, 76D99. Key words and phrases. Hermite approximation, Burgers equation, error estimations. c
1999 American Mathematical Society
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GUO BEN-YU
strictly. The main idea and techniques used in this paper are also applicable to various nonlinear problems arising in fluid dynamics, quantum mechanics and other fields. 2. Hermite approximation 2
Let Λ = {x| − ∞ < x < ∞} and ω(x) = e−x . For 1 ≤ p ≤ ∞, set Lpω (Λ) = {v|v is measurable and k v kLpω (Λ) < ∞}, where k v kLpω (Λ) =
Z 1 ( |v(x)|p ω(x)dx) p ,
1 ≤ p < ∞,
Λ
ess sup |v(x)|,
p = ∞.
x∈Λ
In particular, L2ω (Λ) is a Hilbert space with the inner product Z (u, v)L2ω (Λ) = u(x)v(x)w(x)dx. Λ
Further, let ∂x v =
∂v ∂x ,
and for any non-negative integer m,
Hωm (Λ) = {v|∂xk v ∈ L2ω (Λ), 0 ≤ k ≤ m}. The semi-norm and the norm of Hωm (Λ) are given by |v|
m (Λ) Hω
=k
∂xm v
k
L2ω (Λ)
, kvk
m (Λ) Hω
m X 1 =( |v|2Hωk (Λ) ) 2 . k=0
For any real r ≥ 0, we define the space Hωr (Λ) with the norm k v kHωr (Λ) by the space interpolation as in Adams [11]. For simplicity, we denote the inner product (u, v)L2ω (Λ) , the semi-norm |v|Hωr (Λ) , the norms k v kHωr (Λ) and k v kLpω (Λ) , by (u, v)ω , |v|r,ω , k v kr,ω and k v kLpω , respectively. In particular, k v kω =k v k0,ω . Besides, let c denote a generic positive constant in this paper. The Hermite polynomial of degree l is defined by 2
2
Hl (x) = (−1)l ex ∂xl (e−x ). It is the l-th eigenfunction of a singular Liouville problem (2.1)
2
2
∂x (e−x ∂x v(x)) + λe−x v(x) = 0, x ∈ Λ.
The corresponding eigenvalue λl = 2l. Clearly H0 (x) = 1 and H1 (x) = 2x. The Hermite polynomials satisfy the recurrence relations (2.2)
Hl+1 (x) − 2xHl (x) + 2lHl−1 (x) = 0, l ≥ 1,
and ∂x Hl (x) = 2lHl−1 (x), l ≥ 1.
(2.3)
The set of Hermite polynomials is an orthogonal system with the weight function ω(x) on the whole line Λ, namely, Z √ (2.4) Hl (x)Hm (x)ω(x)dx = 2l l! πδl,m . Λ
By (2.3), the set of ∂x Hl (x) is also an orthogonal system with the same weight, i,e., Z √ (2.5) ∂x Hl (x)∂x Hm (x)ω(x)dx = 2l+1 ll! πδl,m . Λ
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ERROR ESTIMATION OF THE HERMITE SPECTRAL METHOD
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For any function v ∈ L2ω (Λ), v(x) =
∞ X
vˆl Hl (x),
l=0
where vˆl is the Hermite coefficient, vˆl (x) =
1 √ 2l l! π
Z Λ
v(x)Hl (x)ω(x)dx, l ≥ 0.
We now consider the Hermite approximation. Let N be any positive integer and PN be the set of polynomials of degree at most N . In numerical analysis of the Hermite spectral method, we need some inverse inequalities. The first is due to Nessel and Wilmes [12], stated in the following lemma. Lemma 2.1. For any φ ∈ PN and 1 ≤ p ≤ q ≤ ∞, 5
1
1
k φ kLqω ≤ cN 6 ( p − q ) k φ kLpω . The next lemma gives another inverse inequality. Lemma 2.2. For any φ ∈ PN , |φ|1,ω ≤
√
2N k φ kω .
Proof. By (2.3), ∂x φ(x) = 2
N X
lφˆl Hl−1 (x).
l=1
Thus (2.4) leads to |φ|21,ω ≤ 4
N −1 X
√ 2l (l + 1)2 l! π φˆ2l+1 ≤ 2N k φ k2ω .
l=0
Some imbedding inequalities are useful in numerical analysis of the Hermite spectral method. We list two of them. Lemma 2.3. For any v ∈ Hω1 (Λ), k xv kω ≤k v k1,ω . Proof. Integrating by parts, we obtain that Z Z 2 xv (x)ω(x)dx = v(x)∂x v(x)ω(x)dx ≤k v kω |v|1,ω . Λ
Λ
2
Thus xv (x)ω(x) → 0 as |x| → ∞. By integrating by parts and the Cauchy inequality, Z Z 1 v 2 (x)ω(x)dx + xv(x)∂x v(x)ω(x)dx k xv k2ω = 2 Λ Λ 1 1 1 ≤ k v k2ω + k xv k2ω + |v|21,ω 2 2 2 1 1 2 2 = k v k1,ω + k xv kω . 2 2 So the desired result follows.
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1070
GUO BEN-YU
Lemma 2.4. If v ∈ Hω1 (Λ), then for any x ∈ Λ, |v(x)| ≤ e Moreover, k e− Proof. We have 2
e−x v 2 (x) =
Z
x2 2
x
x2 2
1
1
k v kω2 (|v|1,ω + k v k1,ω ) 2 .
√ v k4L4ω ≤ 16 π k v k2ω k v k21,ω . 2
−∞ Z x
∂y (e−y v 2 (y))dy
=2 −∞
Z v(y)∂y v(y)ω(y)dy − 2
x
yv 2 (y)ω(y)dy.
−∞
By Lemma 2.3 and the Cauchy inequality, 2
e−x v 2 (x) ≤ 2 k v kω (|v|1,ω + k v k1,ω ). This leads to the first conclusion. Moreover, Z 2 √ − x2 4 2 2 ke v kL4ω ≤ 16 k v kω k v k1,ω ω(x)dx = 16 π k v k2ω k v k21,ω . Λ
The proof is completed. The L2ω (Λ)-orthogonal projection PN : L2ω (Λ) → PN is such a mapping that for any v ∈ L2ω (Λ), (v − PN v, φ)ω = 0, ∀φ ∈ PN , or equivalently, N X vˆl Hl (x). PN v(x) = l=0
Lemma 2.5. For any v ∈ Hωr (Λ) and r ≥ 0, r
k v − PN v kω ≤ cN − 2 k v kr,ω . Proof. We have from (2.4) that k v − PN v
k2ω =
√
π
∞ X
2l l!ˆ vl2 .
l=N +1
According to (2.1), we define the operator A by 2
2
Av(x) = −ex ∂x (e−x ∂x v(x)) = −∂x2 v(x) + 2x∂x v(x). By Lemma 2.3, A is a continuous mapping from Hωβ+2 (Λ) into Hωβ (Λ), where β is any non-negative integer. When r is an even integer, we have from (2.1) and integrating by parts that Z v(x)Hl (x)ω(x)dx Λ Z Z r r 1 Av(x)Hl (x)ω(x)dx = · · · = (2l)− 2 A 2 v(x)Hl (x)ω(x)dx. = 2l Λ Λ Thus (2.6)
− r2
|ˆ vl | = (2l)
k
Hl k−2 ω
Z |
r
Λ
A 2 v(x)Hl (x)ω(x)dx|.
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ERROR ESTIMATION OF THE HERMITE SPECTRAL METHOD
Hence k v − PN v
k2ω
−r
∞ X
k
Hl k−2 ω
Z |
r
A 2 v(x)Hl (x)ω(x)dx|2
≤
(2N )
=
(2N )−r k A 2 v k2ω ≤ cN −r k v k2r,ω .
l=N +1 r
Λ
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When r is an odd integer, we obtain that Z Z r−1 − r−1 2 v(x)Hl (x)ω(x)dx = (2l) A 2 v(x)Hl (x)ω(x)dx Λ ZΛ r−1 r+1 ∂x (A 2 v(x))∂x Hl (x)ω(x)dx. = (2l)− 2 Λ
Finally, we derive from (2.4) and (2.5) that
Z ∞ r−1 1 X l+r+1 r+1 −1 k v − PN v k2ω ≤ √ (2 l l!) ( ∂x (A 2 v(x))∂x Hl (x)ω(x)dx)2 π Λ l=N +1
≤ (2N )−r k ∂x (A
r−1 2
v) k2ω ≤ cN −r k v k2r,ω .
Theorem 2.1. For any v ∈ Hωr (x) and 0 ≤ µ ≤ r, µ
r
k v − PN v kµ,ω ≤ cN 2 − 2 k v kr,ω . Proof. We first consider the case with integer µ. We shall use the induction. Obviously Lemma 2.5 implies the desired result for µ = 0. Assume that it is true for µ − 1. Then k v − PN v kµ,ω ≤k v − PN v kω + k ∂x v − PN ∂x v kµ−1,ω + k PN ∂x v − ∂x PN v kµ−1,ω . We know from Lemma 2.5 that k ∂x v − PN ∂x v kµ−1,ω ≤ cN
µ−r 2
k ∂x v kr−1,ω ≤ cN
µ−r 2
k v kr,ω .
On the other hand, (2.3) leads to PN ∂x v − ∂x PN v = 2(N + 1)ˆ vN +1 HN (x). Using Lemma 2.2 and (2.4), we get that k HN k2µ−1,ω ≤ c2N N µ−1 N !. Moreover by (2.6), |ˆ vN +1 |2 ≤ c(2N +1 N r (N + 1)!)−1 k v k2r,ω . Therefore
k PN ∂x v − ∂x PN v k2µ−1,ω ≤ cN µ−r k v k2r,ω . So the induction is completed. The previous results with space interpolation lead to the conclusion for any r ≥ 0. In order to obtain the optimal error estimation in the Hermite spectral method for partial differential equations, we need the Hω1 (Λ)-orthogonal projection PN1 : Hω1 (Λ) → PN . It means that for any v ∈ Hω1 (Λ), (2.7)
(∂x (v − PN1 v), ∂x φ)ω = 0, ∀φ ∈ PN .
Let vˆl be the coefficients of the Hermite expansion for v(x), and PN1 v(x) =
N X
a ˆl Hl (x).
l=0
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1072
GUO BEN-YU
By (2.3), ∂x PN1 v(x) = 2
N −1 X
(l + 1)ˆ al+1 Hl (x).
l=0
Similarly ∂x v(x) = 2
∞ X
(l + 1)ˆ vl+1 Hl (x).
l=0
By (2.3) and (2.7), we know that a ˆl = vˆl for 0 ≤ l ≤ N . Thus the projection PN1 is exactly the same as PN . 3. Application to Burgers equation on the whole line In this section, we consider the Hermite spectral method for Burgers equation on the whole line. We first change it to a new representation by the similarity transformation, which is suitable for the Hermite approximation. We shall prove the stability and the convergence of the designed scheme strictly. ˜ = {y| − ∞ < y < ∞} and µ > 0 be the kinetic viscosity, while g(y, s) and Let Λ V0 (y, s) are the source term and the initial value, respectively. T is a fixed positive number. We consider the following problem ( 1 ˜ 0 < s ≤ T, ∂s V + ∂y (V 2 ) − µ∂y2 V = g, y ∈ Λ, (3.1) 2 ˜ y ∈ Λ. V (y, 0) = V0 (s), In addition, V and ∂y V satisfy certain conditions at infinity. Let Z ∂y u(y)∂y (v(y)ω(y))dy. aω (u, v) = ˜ Λ
˜ ∩ L∞ (0, T ; L2ω (Λ)) ˜ such A weak formulation of (3.1) is to find v ∈ L2 (0, T ; Hω1 (Λ)) that 1 2 (∂s V (s), v)L2ω (Λ) ˜ − (V (s), ∂y (v(s)ω))L2 (Λ) ˜ 2 ˜ 0 < s ≤ T, +µa (V (s), v) = (g, v) , ∀v ∈ Hω1 (Λ), ˜ ω L2ω (Λ) s = 0. V = V0 , It can be checked that aω (v, v)
= k ∂y v k2L2 (Λ) ˜ −2 ω
= k ∂y v k2L2 (Λ) ˜ ω
Z
yv(y)∂y v(y)ω(y)dy Z + k v k2L2 (Λ) −2 y 2 v 2 (y)ω(y)dy. ˜ Λ
ω
˜ Λ
It is not clear whether the bilinear form aω (v, v) is non-negative or not. Thus the above weak formulation is not suitable for the Hermite spectral method. To remedy this trouble, we try to reform it. Let W (x, t) = V (y, s), g˜(x, t) = g(y, s) and make the similarity transformation (3.2)
y , t = ln(1 + s). x= p 2 µ(1 + s)
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ERROR ESTIMATION OF THE HERMITE SPECTRAL METHOD
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Then (3.1) becomes (3.3) ∂ W − 1 x∂ W + 1 e 2t ∂ (W 2 ) − 1 ∂ 2 W = et g˜, √ t x x 2 4 µ 4 x W = W0 , 2
x ∈ Λ, 0 < t ≤ ln(1 + T ), t = 0.
2
Further let U = ex W and f = ex +t g˜. Then we obtain the following problem (3.4) 1 1 ∂t U + U + x∂x U 2 2 2 1 x2 + t 1 2 2 ∂ (e−2x U 2 ) − e ∂ U = f, x ∈ Λ, 0 < t ≤ ln(1 + T ), + √ x 4 µ 4 x t = 0. U = U0 , In addition, U and ∂x U satisfy some conditions as |x| → ∞. Let 2 1 t B(u, z, v) = − √ e 2 (e−x uz, ∂xv)ω . 4 µ The weak formulation of (3.4) is to find U L∞ (0, ln(1 + T ); L2ω (Λ)) such that (3.5) 1 (∂t U (t), v)ω + 2 (U (t), v)ω + B(U (t), U (t), v) 1 + (∂x U (t), ∂x v)ω = (f (t), v)ω , 4 U = U0 ,
∈ L2 (0, ln(1 + T ); Hω1 (Λ)) ∩
∀v ∈ Hω1 (Λ), 0 < t ≤ ln(1 + T ), t = 0.
As in Maday, Pernaud-Thomas and Vandeven [1], we suppose that V0 and g fulfill some conditions such that for certain α ≥ 0, 2
lim eαy (|V (y, s)| + |∂y V (y, s)|) = 0, 0 ≤ s ≤ T.
|y|→∞
Then
t
lim e4αµe
|x|→∞
x2
(|W (x, t)| + |∂x W (x, t)|) = 0, 0 ≤ t ≤ ln(1 + T )
and so t
lim e(4αµe
−1)x2
|x|→∞
(|U (x, t)| + |∂x U (x, t)|) = 0, 0 ≤ t ≤ ln(1 + T ).
1 , then we have that for all t ≥ 0, 4αµet − 1 > − 21 . By Lemma 2.3, If α > 8µ U ∈ Hω1 (Λ) and so we can use the Hermite approximation for (3.5). The Hermite spectral scheme for (3.5) is to find uN (t) ∈ PN for 0 ≤ t ≤ ln(1+T ), such that (3.6) 1 (∂ u (t), φ)ω + (uN (t), φ)ω + B(uN (t), uN (t), φ) t N 2 1 + (∂x uN (t), ∂x φ)ω = (f (t), φ)ω , ∀φ ∈ PN , 0 < t ≤ ln(1 + T ), 4 u = u =P U , t = 0. N
N,0
N
0
We now consider the stability of (3.6). Since (3.6) is nonlinear, it is not possible to prove the stability in the sense of Courant, Friedrichs and Lewy [13]. But it will be shown that it is still stable in the sense of Guo [14, 15] and Stetter [16].
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1074
GUO BEN-YU
To do this, we assume that f and uN,0 have the errors f˜ and u ˜N,0 , respectively. They induce the error of numerical solution uN , denoted by u ˜N . Then we get the following equation: (3.7) 1 uN (t), φ)ω + B(˜ ˜N (t), φ)ω + (˜ uN (t), u ˜N (t), φ) + 2B(˜ uN (t), uN (t), φ) (∂t u 2 1 + (∂x u ˜N (t), ∂x φ)ω = (f˜(t), φ)ω , ∀φ ∈ PN , 0 < t ≤ ln(1 + T ), 4 ˜N,0, t = 0. u ˜N = u By taking φ = 2˜ uN in (3.7), it follows that
(3.8)
1 d ku ˜N (t) k2ω + k u ˜N (t) k21,ω dt 2 + 2B(˜ uN (t), u ˜N (t), u ˜N (t)) + 4B(˜ uN (t), uN (t), u ˜N (t)) 2 ≤ 2 k f˜(t) k . ω
Using Lemma 2.4, we deduce that for 0 ≤ t ≤ ln(1 + T ), (3.9) |2B(˜ uN (t), u ˜N (t), u ˜N (t))|
≤ ≤
where
p 2 4 π(1 + T )2 . c1 (T ) = √ µ
Furthermore, for any p, q > 0 and P´olya [17] that (3.10)
x2 1 t uN (t)|1,ω k e− 2 u ˜N (t) k2L4ω √ e 2 |˜ 2 µ c1 (T ) k u ˜N (t) kω k u ˜N (t) k21,ω ,
1 p
+ 1q = 1, we know from Hardy, Littlewood and
|ab| ≤
|b|q |a|p + . p q
Thus by using Lemma 2.4 and (3.10), we assert that (3.11) 2 1 t |4B(˜ uN (t), uN (t), u˜N (t))| ≤ √ e 2 |˜ uN (t)|1,ω k e−x uN (t)˜ uN (t) kω µ √ 1 1 1 3 44π t 2 2 ≤ √ e 2 k uN (t) kω2 k uN (t) k1,ω ku ˜N (t) kω2 k u ˜N (t) k1,ω µ 1 ≤ k u˜N (t) k21,ω +c2 (uN , T ) k u ˜N (t) k2ω , 4 where 123 π(1 + T )2 c2 (uN , T ) = k uN k2L∞ (0,ln(1+T );L2ω (Λ)) k uN k2L∞ (0,ln(1+T );Hω1 (Λ)) . µ2
By substituting (3.9) and (3.11) into (3.8), and integrating the resulting inequality, we find that Z t 1 2 ku ˜N (t) kω + ( − c1 (T ) k u ˜N (η) kω ) k u ˜N (η) k21,ω dη 4 0 (3.12) Z t ˜ ku ˜N (η) k2ω dη, ≤ ρ(˜ uN,0 , f , t) + c2 (uN , T ) 0
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ERROR ESTIMATION OF THE HERMITE SPECTRAL METHOD
where ρ(˜ uN,0 , f˜, t) =k u ˜N,0 k2ω +2
Z 0
t
1075
k f˜(η) k2ω dη.
We need the following lemma. Lemma 3.1. Assume that (i) the constants b1 > 0, b2 ≥ 0, b3 ≥ 0 and d ≥ 0, (ii) Z(t) and A(t) are non-negative functions of t, b2 (iii) d ≤ 12 e−b3 t1 for certain t1 > 0, b2 (iv) for all t ≤ t1 , Z t Z t 1 (b1 − b2 Z 2 (η))A(η)dη ≤ d + b3 Z(η)dη. Z(t) + 0
0
Then for all t ≤ t1 , Z(t) ≤ deb3 t . Proof. Consider the function Y (t) satisfying Z t Y (t) = d + b3 Y (η)dη. 0
Then for all t ≤ t1 , Y (t) = deb3 t ≤
b21 . b22
Clearly Z(t) ≤ Y (t) for t ≤ t1 , and so the conclusion is valid. Applying Lemma 3.1 to (3.12), we obtain the following result. Theorem 3.1. Let α >
1 8µ
and uN (t) be the solution of (3.6). If for certian t1 ,
(1 − a)2 −c2 (uN ,T )t1 e ρ(˜ uN,0 , f˜, t1 ) ≤ , a ≥ 0, 16c21 (T ) then for all t ≤ t1 , ku ˜N (t) k2ω +
a 4
Z
t 0
k u˜N (η) k21,ω dη ≤ ρ(˜ uN,0 , f˜, t)ec2 (uN ,T )t .
Theorem 3.1 indicates that the error of the numerical solution is controlled by uN,0 , f˜, t) does the errors of the data uN,0 and f , provided that the average error ρ(˜ not exceed certain critical value. It means that (3.6) is of generalized stability in the sense of Guo [14, 15], and of restricted stability in the sense of Stetter [16]. Next we deal with the convergence of scheme (3.6). Let U be the solution of (3.5), and UN = PN U. We derive from (3.5) that
(3.13)
1 (∂t UN (t), φ)ω + (UN (t), φ)ω + B(UN (t), UN (t), φ) 2 1 + (∂x UN (t), ∂x φ)ω + G(t, φ) = (f (t), φ)ω , 4 ∀φ ∈ PN , 0 < t ≤ ln(1 + T ),
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1076
GUO BEN-YU
where = G1 (t, φ) + G2 (t, φ) + G3 (t, φ), = (∂t U (t) − ∂t UN (t), φ)ω , 1 (U (t) − UN (t), φ)ω , G2 (t, φ) = 2 G3 (t, φ) = B(U (t), U (t), φ) − B(UN (t), UN (t), φ). ˜N = uN − UN . By subtracting (3.13) from Let uN be the solution of (3.6), and U (3.6), we obtain that G(t, φ) G1 (t, φ)
(3.14) ˜N (t), φ)ω + B(U ˜N (t), U ˜N (t), φ) ˜N (t), φ)ω + 1 (U (∂t U 2 ˜N (t), ∂x φ)ω = G(t, φ)ω , ˜N (t), UN (t), φ) + 1 (∂x U + 2B(U 4 ∀φ ∈ PN , 0 < t ≤ ln(1 + T ). ˜N (0) = 0. Comparing (3.14) to (3.7), we can derive a result similar In addition, U ˜N , U ˜N,0 ˜N,0 and f˜ are now replaced by UN , U to that of Theorem 3.1. But uN , u˜N , u ˜ and G(t, φ), respectively. Therefore we only have to estimate the term |G(t, UN (t))|. We first have from Theorem 2.1 that for r ≥ 1, ˜N (t))| ≤ cN − r2 k ∂t U (t) kr,ω k U ˜N (t) kω , |G1 (t, U − r2 ˜ ˜ k U (t) kr,ω k UN (t) kω . |G2 (t, UN (t))| ≤ cN An argument, as in the derivation of (3.9), leads to that for all t < ln(1 + T ), 1 t −x2 ˜N (t))| ≤ √ ˜N (t))ω | e 2 |(e (UN (t) + U (t))(UN (t) − U (t)), U |G3 (t, U 4 µ 1 1 1 1 2 ≤ c1 (T ) k UN (t) + U (t) kω2 k UN (t) + U (t) k1,ω k UN (t) − U (t) kω2 2 1 ˜N (t) k1,ω . × k UN (t) − U (t) k 2 k U 1,ω
By Theorem 2.1, ˜N (t) k21,ω +cc21 (T )N 12 −r k UN (t) k4r,ω . ˜N (t))| ≤ 1 k U |G3 (t, U 8 Hence (3.15) ˜N (t))| |G(t, U
≤
1 ˜ ˜N (t) k2ω k UN (t) k21,ω + k U 8 1 +c(c21 (T ) + 1)N 2 −r (k UN (t) k4r,ω + k ∂t U (t) k2r− 1 ,ω ). 2
Obviously, the last term in (3.15) tends to zero as N goes to infinity. Therefore we obtain the following result. Theorem 3.2. If α >
1 8µ
and r− 12
U ∈ L2 (0, ln(1 + T ); Hωr (Λ)) ∩ H 1 (0, ln(1 + T ); Hω
(Λ))
with r ≥ 1, then for all t ≤ ln(1 + T ), Z t 1 2 k uN (η) − U (η) k21,ω dη ≤ c∗ N 2 −r , k uN (t) − U (t) kω + 0
where c∗ is a positive constant depending only on µ, T and the norms of U in the space mentioned above.
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ERROR ESTIMATION OF THE HERMITE SPECTRAL METHOD
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Remark 3.1. In the proof of Theorem 3.1 and Theorem 3.2, we require that U ∈ x2 Hω1 (Λ) and so e− 2 (|U (x, t)| + |∂x U (x, t)|) → 0 as |x| → ∞. A sufficient condition 2 1 , eαy (|V (y, s)| + |∂y V (y, s)|) → 0, as |y| → ∞. It means is that for certain α > 8µ that V (y, s) should decay fast enough. It agrees with the experience in actual computations as described in Funaro and Kavian [7] and other papers. Remark 3.2. In this paper, we use the variable transformation (3.2) and so obtain the error estimations. In fact, a similiar transformation was used in actual computations by Funaro and Kavian [7]. This trick can be generalized to other problems, such as the two-dimensional heat equation and the Navier-Stokes equations. In actual computations, we need to discretize the term ∂t uN in (3.6). We can use Lemmas 2.1–2.4, Theorem 2.1 and an argument as in the proof of Theorems 3.1 and 3.2, to prove the generalized stability and the convergence of a fully discrete scheme, provided that the value of τ N satisfies certain reasonable conditions, where τ is the step size in time t, and N is the number of terms used in Hermite approximations. For instance, by Lemma 2.2, τ N should be bounded in the case of explicite schemes. We can also approximate nonlinear partial differential equations by the base functions √ ˜ l (x) = (2l l! π)− 12 e−ax2 Hl (x), a ≥ 0, l ≥ 0. H ˜ l (x) is an orthogonal system associated with the weight e(2a−1)x2 . For The set of H the application to linear problems with a = 1, we refer to Funaro and Kavian [7]. References [1] Y. Maday, B. Pernaud-Thomas and H. Vandeven, One r´ ehabilitation des m´ ethodes spectrales de type Laguerre, Rech. A´ erospat., 6 (1985), 353-379. [2] O. Coulaud, D. Funaro and O. Kavian, Laguerre spectral approximation of elliptic problems in exterior domains, Comput. Methods Appl. Mech. Engrg., 80 (1990), 451-458. CMP 90:17 [3] D. Funaro, Estimates of Laguerre spectral projectors in Sobolev spaces, in Orthogonal Polynomials and Their Applications, ed. by C. Brezinski, L. Gori and A. Ronveaux, Scientific Publishing Co., 1991, 263-266. MR 95a:41033 [4] V. Iranzo and A. Falqu`es, Some spectral approximations for differential equations in unbounded domains, Comput. Methods Appl. Mech. Engrg., 98 (1992), 105-126. MR 93d:65103 [5] C. Mavriplis, Laguerre polynomials for infinite-domain spectral elements, J. Comp. Phys., 80 (1989), 480-488. MR 90f:65228 [6] K. Black, Spectral elements on infinite domains (unpublished). [7] D. Funaro and O. Kavian, Approximation of some diffusion evolution equations in unbounded domains by Hermite funtions, Math. Comp., 57 (1990), 597-619. MR 92k:35156 [8] J. A. C. Weideman, The eigenvalues of Hermite and rational spectral differential matrices, Numer. Math., 61 (1992), 409-432. MR 92k:65071 [9] C. I. Christov, A complete orthonormal system of funtions in L2 (−∞, ∞) space, SIAM J. Appl. Math., 42 (1982), 1337-1344. MR 84b:42018 [10] J. P. Boyd, Spectral methods using rational basis funtions on an infinite interval, J. Comp. Phys., 69 (1987), 112-142. MR 88e:65093 [11] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. MR 56:9247 [12] R. J. Nessel and G. Wilmes, On Nikolskii-type inequalities for orthogonal expansion, in Approximation Theory II , ed. by G. G. Lorentz, C. K. Chui and L. L. Schumaker, Academic Press, New York, 479-484. ¨ [13] R. Courant, K. O. Friedrichs and H. Lewy, Uber die partiellen differezengleichungen der mathematischen physik, Math. Ann., 100 (1928), 32-74. [14] B. Y. Guo, A certain class of finite difference schemes of two-dimensional vorticity equation viscous fluid, RR. SUST, 1965, Also see Acta Math. Sinica, 17 (1974), 242-258. MR 56:17128
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1078
GUO BEN-YU
[15] B. Y. Guo, Generalized stability of discretization and its applications to numerical solutions of nonlinear differential equations, Contemp. Math., 163 (1994), 33-54. MR 95d:65067 [16] H. J. Stetter, Stability of nonlinear discretization algorithms, in Numerical Solutions of Partial Differential Equations, ed. by J. Bramble, Academic Press, New York, 1966, 111-123. MR 34:5322 [17] G. H. Hardy, J. E. Littlewood and G.P´ olya, Inequalities, 2’ed., Cambridge University Press, Cambridge, 1952. MR 13:727e Department of Mathematics, Shanghai University, Jiading Campus, Shanghai, 201800, China Current address: Department of Mathematics, Shanghai Normal University, Shanghai, China E-mail address: [email protected]
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ERROR ESTIMATION OF HERMITE SPECTRAL METHOD FOR NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS GUO BEN-YU
Abstract. Hermite approximation is investigated. Some inverse inequalities, imbedding inequalities and approximation results are obtained. A Hermite spectral scheme is constructed for Burgers equation. The stability and convergence of the proposed scheme are proved strictly. The techniques used in this paper are also applicable to other nonlinear problems in unbounded domains.
1. Introduction A number of physical problems are set in unbounded domains. Some conditions at infinity are given by certain asymptotic behaviors for solutions. When we use the finite difference method or the finite element method to solve such problems numerically, we often restrict calculations to some bounded domains, and impose certain conditions on artificial boundaries. They cause numerical errors usually. If we use spectral methods associated with some orthogonal systems in unbounded domains, then the above troubles could be avoided. While the spectral methods provide numerical solutions with high accuracies. Maday, Pernaud-Thomas and Vandeven [1], Coulaud, Funaro and Kavian [2], and Funaro [3] used the Laguerre spectral method for several linear partial differential equations. Iranzo and Falqu`es [4] provided some Laguerre pseudospectral schemes and Laguerre tau schemes. Mavriplis [5] and Black [6] developed the Laguerre spectral element method. Also, Funaro and Kavian [7], and Weideman [8] considered the Hermite spectral method and the Hermite pseudospectral method. In particular, Funaro and Kavian [7] proved the convergence of a spectral scheme using the Hermite functions for some linear problems. But so far, there is no paper concerning error estimates of the Hermite spectral method using Hermite polynomials. Another spectral method for partial differential equations in unbounded domains is based on the rational basis functions, see Christov [9], Boyd [10], Iranzo and Falqu`es [4], and Weideman [8]. The purpose of this paper is to study spectral approximation using Hermite polynomials and their applications to nonlinear problems. Some inverse inequalities, imbedding inequalities and approximation results are given, which play important roles in analysis of the Hermite spectral method. We use the Burgers equation as an example showing how to construct Hermite spectral schemes for nonlinear problems. The generalized stability and the convergence of the proposed scheme are proved Received by the editor October 16, 1997 and, in revised form, January 2, 1998. 1991 Mathematics Subject Classification. Primary 65N30, 76D99. Key words and phrases. Hermite approximation, Burgers equation, error estimations. c
1999 American Mathematical Society
1067
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1068
GUO BEN-YU
strictly. The main idea and techniques used in this paper are also applicable to various nonlinear problems arising in fluid dynamics, quantum mechanics and other fields. 2. Hermite approximation 2
Let Λ = {x| − ∞ < x < ∞} and ω(x) = e−x . For 1 ≤ p ≤ ∞, set Lpω (Λ) = {v|v is measurable and k v kLpω (Λ) < ∞}, where k v kLpω (Λ) =
Z 1 ( |v(x)|p ω(x)dx) p ,
1 ≤ p < ∞,
Λ
ess sup |v(x)|,
p = ∞.
x∈Λ
In particular, L2ω (Λ) is a Hilbert space with the inner product Z (u, v)L2ω (Λ) = u(x)v(x)w(x)dx. Λ
Further, let ∂x v =
∂v ∂x ,
and for any non-negative integer m,
Hωm (Λ) = {v|∂xk v ∈ L2ω (Λ), 0 ≤ k ≤ m}. The semi-norm and the norm of Hωm (Λ) are given by |v|
m (Λ) Hω
=k
∂xm v
k
L2ω (Λ)
, kvk
m (Λ) Hω
m X 1 =( |v|2Hωk (Λ) ) 2 . k=0
For any real r ≥ 0, we define the space Hωr (Λ) with the norm k v kHωr (Λ) by the space interpolation as in Adams [11]. For simplicity, we denote the inner product (u, v)L2ω (Λ) , the semi-norm |v|Hωr (Λ) , the norms k v kHωr (Λ) and k v kLpω (Λ) , by (u, v)ω , |v|r,ω , k v kr,ω and k v kLpω , respectively. In particular, k v kω =k v k0,ω . Besides, let c denote a generic positive constant in this paper. The Hermite polynomial of degree l is defined by 2
2
Hl (x) = (−1)l ex ∂xl (e−x ). It is the l-th eigenfunction of a singular Liouville problem (2.1)
2
2
∂x (e−x ∂x v(x)) + λe−x v(x) = 0, x ∈ Λ.
The corresponding eigenvalue λl = 2l. Clearly H0 (x) = 1 and H1 (x) = 2x. The Hermite polynomials satisfy the recurrence relations (2.2)
Hl+1 (x) − 2xHl (x) + 2lHl−1 (x) = 0, l ≥ 1,
and ∂x Hl (x) = 2lHl−1 (x), l ≥ 1.
(2.3)
The set of Hermite polynomials is an orthogonal system with the weight function ω(x) on the whole line Λ, namely, Z √ (2.4) Hl (x)Hm (x)ω(x)dx = 2l l! πδl,m . Λ
By (2.3), the set of ∂x Hl (x) is also an orthogonal system with the same weight, i,e., Z √ (2.5) ∂x Hl (x)∂x Hm (x)ω(x)dx = 2l+1 ll! πδl,m . Λ
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ERROR ESTIMATION OF THE HERMITE SPECTRAL METHOD
1069
For any function v ∈ L2ω (Λ), v(x) =
∞ X
vˆl Hl (x),
l=0
where vˆl is the Hermite coefficient, vˆl (x) =
1 √ 2l l! π
Z Λ
v(x)Hl (x)ω(x)dx, l ≥ 0.
We now consider the Hermite approximation. Let N be any positive integer and PN be the set of polynomials of degree at most N . In numerical analysis of the Hermite spectral method, we need some inverse inequalities. The first is due to Nessel and Wilmes [12], stated in the following lemma. Lemma 2.1. For any φ ∈ PN and 1 ≤ p ≤ q ≤ ∞, 5
1
1
k φ kLqω ≤ cN 6 ( p − q ) k φ kLpω . The next lemma gives another inverse inequality. Lemma 2.2. For any φ ∈ PN , |φ|1,ω ≤
√
2N k φ kω .
Proof. By (2.3), ∂x φ(x) = 2
N X
lφˆl Hl−1 (x).
l=1
Thus (2.4) leads to |φ|21,ω ≤ 4
N −1 X
√ 2l (l + 1)2 l! π φˆ2l+1 ≤ 2N k φ k2ω .
l=0
Some imbedding inequalities are useful in numerical analysis of the Hermite spectral method. We list two of them. Lemma 2.3. For any v ∈ Hω1 (Λ), k xv kω ≤k v k1,ω . Proof. Integrating by parts, we obtain that Z Z 2 xv (x)ω(x)dx = v(x)∂x v(x)ω(x)dx ≤k v kω |v|1,ω . Λ
Λ
2
Thus xv (x)ω(x) → 0 as |x| → ∞. By integrating by parts and the Cauchy inequality, Z Z 1 v 2 (x)ω(x)dx + xv(x)∂x v(x)ω(x)dx k xv k2ω = 2 Λ Λ 1 1 1 ≤ k v k2ω + k xv k2ω + |v|21,ω 2 2 2 1 1 2 2 = k v k1,ω + k xv kω . 2 2 So the desired result follows.
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1070
GUO BEN-YU
Lemma 2.4. If v ∈ Hω1 (Λ), then for any x ∈ Λ, |v(x)| ≤ e Moreover, k e− Proof. We have 2
e−x v 2 (x) =
Z
x2 2
x
x2 2
1
1
k v kω2 (|v|1,ω + k v k1,ω ) 2 .
√ v k4L4ω ≤ 16 π k v k2ω k v k21,ω . 2
−∞ Z x
∂y (e−y v 2 (y))dy
=2 −∞
Z v(y)∂y v(y)ω(y)dy − 2
x
yv 2 (y)ω(y)dy.
−∞
By Lemma 2.3 and the Cauchy inequality, 2
e−x v 2 (x) ≤ 2 k v kω (|v|1,ω + k v k1,ω ). This leads to the first conclusion. Moreover, Z 2 √ − x2 4 2 2 ke v kL4ω ≤ 16 k v kω k v k1,ω ω(x)dx = 16 π k v k2ω k v k21,ω . Λ
The proof is completed. The L2ω (Λ)-orthogonal projection PN : L2ω (Λ) → PN is such a mapping that for any v ∈ L2ω (Λ), (v − PN v, φ)ω = 0, ∀φ ∈ PN , or equivalently, N X vˆl Hl (x). PN v(x) = l=0
Lemma 2.5. For any v ∈ Hωr (Λ) and r ≥ 0, r
k v − PN v kω ≤ cN − 2 k v kr,ω . Proof. We have from (2.4) that k v − PN v
k2ω =
√
π
∞ X
2l l!ˆ vl2 .
l=N +1
According to (2.1), we define the operator A by 2
2
Av(x) = −ex ∂x (e−x ∂x v(x)) = −∂x2 v(x) + 2x∂x v(x). By Lemma 2.3, A is a continuous mapping from Hωβ+2 (Λ) into Hωβ (Λ), where β is any non-negative integer. When r is an even integer, we have from (2.1) and integrating by parts that Z v(x)Hl (x)ω(x)dx Λ Z Z r r 1 Av(x)Hl (x)ω(x)dx = · · · = (2l)− 2 A 2 v(x)Hl (x)ω(x)dx. = 2l Λ Λ Thus (2.6)
− r2
|ˆ vl | = (2l)
k
Hl k−2 ω
Z |
r
Λ
A 2 v(x)Hl (x)ω(x)dx|.
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ERROR ESTIMATION OF THE HERMITE SPECTRAL METHOD
Hence k v − PN v
k2ω
−r
∞ X
k
Hl k−2 ω
Z |
r
A 2 v(x)Hl (x)ω(x)dx|2
≤
(2N )
=
(2N )−r k A 2 v k2ω ≤ cN −r k v k2r,ω .
l=N +1 r
Λ
1071
When r is an odd integer, we obtain that Z Z r−1 − r−1 2 v(x)Hl (x)ω(x)dx = (2l) A 2 v(x)Hl (x)ω(x)dx Λ ZΛ r−1 r+1 ∂x (A 2 v(x))∂x Hl (x)ω(x)dx. = (2l)− 2 Λ
Finally, we derive from (2.4) and (2.5) that
Z ∞ r−1 1 X l+r+1 r+1 −1 k v − PN v k2ω ≤ √ (2 l l!) ( ∂x (A 2 v(x))∂x Hl (x)ω(x)dx)2 π Λ l=N +1
≤ (2N )−r k ∂x (A
r−1 2
v) k2ω ≤ cN −r k v k2r,ω .
Theorem 2.1. For any v ∈ Hωr (x) and 0 ≤ µ ≤ r, µ
r
k v − PN v kµ,ω ≤ cN 2 − 2 k v kr,ω . Proof. We first consider the case with integer µ. We shall use the induction. Obviously Lemma 2.5 implies the desired result for µ = 0. Assume that it is true for µ − 1. Then k v − PN v kµ,ω ≤k v − PN v kω + k ∂x v − PN ∂x v kµ−1,ω + k PN ∂x v − ∂x PN v kµ−1,ω . We know from Lemma 2.5 that k ∂x v − PN ∂x v kµ−1,ω ≤ cN
µ−r 2
k ∂x v kr−1,ω ≤ cN
µ−r 2
k v kr,ω .
On the other hand, (2.3) leads to PN ∂x v − ∂x PN v = 2(N + 1)ˆ vN +1 HN (x). Using Lemma 2.2 and (2.4), we get that k HN k2µ−1,ω ≤ c2N N µ−1 N !. Moreover by (2.6), |ˆ vN +1 |2 ≤ c(2N +1 N r (N + 1)!)−1 k v k2r,ω . Therefore
k PN ∂x v − ∂x PN v k2µ−1,ω ≤ cN µ−r k v k2r,ω . So the induction is completed. The previous results with space interpolation lead to the conclusion for any r ≥ 0. In order to obtain the optimal error estimation in the Hermite spectral method for partial differential equations, we need the Hω1 (Λ)-orthogonal projection PN1 : Hω1 (Λ) → PN . It means that for any v ∈ Hω1 (Λ), (2.7)
(∂x (v − PN1 v), ∂x φ)ω = 0, ∀φ ∈ PN .
Let vˆl be the coefficients of the Hermite expansion for v(x), and PN1 v(x) =
N X
a ˆl Hl (x).
l=0
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1072
GUO BEN-YU
By (2.3), ∂x PN1 v(x) = 2
N −1 X
(l + 1)ˆ al+1 Hl (x).
l=0
Similarly ∂x v(x) = 2
∞ X
(l + 1)ˆ vl+1 Hl (x).
l=0
By (2.3) and (2.7), we know that a ˆl = vˆl for 0 ≤ l ≤ N . Thus the projection PN1 is exactly the same as PN . 3. Application to Burgers equation on the whole line In this section, we consider the Hermite spectral method for Burgers equation on the whole line. We first change it to a new representation by the similarity transformation, which is suitable for the Hermite approximation. We shall prove the stability and the convergence of the designed scheme strictly. ˜ = {y| − ∞ < y < ∞} and µ > 0 be the kinetic viscosity, while g(y, s) and Let Λ V0 (y, s) are the source term and the initial value, respectively. T is a fixed positive number. We consider the following problem ( 1 ˜ 0 < s ≤ T, ∂s V + ∂y (V 2 ) − µ∂y2 V = g, y ∈ Λ, (3.1) 2 ˜ y ∈ Λ. V (y, 0) = V0 (s), In addition, V and ∂y V satisfy certain conditions at infinity. Let Z ∂y u(y)∂y (v(y)ω(y))dy. aω (u, v) = ˜ Λ
˜ ∩ L∞ (0, T ; L2ω (Λ)) ˜ such A weak formulation of (3.1) is to find v ∈ L2 (0, T ; Hω1 (Λ)) that 1 2 (∂s V (s), v)L2ω (Λ) ˜ − (V (s), ∂y (v(s)ω))L2 (Λ) ˜ 2 ˜ 0 < s ≤ T, +µa (V (s), v) = (g, v) , ∀v ∈ Hω1 (Λ), ˜ ω L2ω (Λ) s = 0. V = V0 , It can be checked that aω (v, v)
= k ∂y v k2L2 (Λ) ˜ −2 ω
= k ∂y v k2L2 (Λ) ˜ ω
Z
yv(y)∂y v(y)ω(y)dy Z + k v k2L2 (Λ) −2 y 2 v 2 (y)ω(y)dy. ˜ Λ
ω
˜ Λ
It is not clear whether the bilinear form aω (v, v) is non-negative or not. Thus the above weak formulation is not suitable for the Hermite spectral method. To remedy this trouble, we try to reform it. Let W (x, t) = V (y, s), g˜(x, t) = g(y, s) and make the similarity transformation (3.2)
y , t = ln(1 + s). x= p 2 µ(1 + s)
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ERROR ESTIMATION OF THE HERMITE SPECTRAL METHOD
1073
Then (3.1) becomes (3.3) ∂ W − 1 x∂ W + 1 e 2t ∂ (W 2 ) − 1 ∂ 2 W = et g˜, √ t x x 2 4 µ 4 x W = W0 , 2
x ∈ Λ, 0 < t ≤ ln(1 + T ), t = 0.
2
Further let U = ex W and f = ex +t g˜. Then we obtain the following problem (3.4) 1 1 ∂t U + U + x∂x U 2 2 2 1 x2 + t 1 2 2 ∂ (e−2x U 2 ) − e ∂ U = f, x ∈ Λ, 0 < t ≤ ln(1 + T ), + √ x 4 µ 4 x t = 0. U = U0 , In addition, U and ∂x U satisfy some conditions as |x| → ∞. Let 2 1 t B(u, z, v) = − √ e 2 (e−x uz, ∂xv)ω . 4 µ The weak formulation of (3.4) is to find U L∞ (0, ln(1 + T ); L2ω (Λ)) such that (3.5) 1 (∂t U (t), v)ω + 2 (U (t), v)ω + B(U (t), U (t), v) 1 + (∂x U (t), ∂x v)ω = (f (t), v)ω , 4 U = U0 ,
∈ L2 (0, ln(1 + T ); Hω1 (Λ)) ∩
∀v ∈ Hω1 (Λ), 0 < t ≤ ln(1 + T ), t = 0.
As in Maday, Pernaud-Thomas and Vandeven [1], we suppose that V0 and g fulfill some conditions such that for certain α ≥ 0, 2
lim eαy (|V (y, s)| + |∂y V (y, s)|) = 0, 0 ≤ s ≤ T.
|y|→∞
Then
t
lim e4αµe
|x|→∞
x2
(|W (x, t)| + |∂x W (x, t)|) = 0, 0 ≤ t ≤ ln(1 + T )
and so t
lim e(4αµe
−1)x2
|x|→∞
(|U (x, t)| + |∂x U (x, t)|) = 0, 0 ≤ t ≤ ln(1 + T ).
1 , then we have that for all t ≥ 0, 4αµet − 1 > − 21 . By Lemma 2.3, If α > 8µ U ∈ Hω1 (Λ) and so we can use the Hermite approximation for (3.5). The Hermite spectral scheme for (3.5) is to find uN (t) ∈ PN for 0 ≤ t ≤ ln(1+T ), such that (3.6) 1 (∂ u (t), φ)ω + (uN (t), φ)ω + B(uN (t), uN (t), φ) t N 2 1 + (∂x uN (t), ∂x φ)ω = (f (t), φ)ω , ∀φ ∈ PN , 0 < t ≤ ln(1 + T ), 4 u = u =P U , t = 0. N
N,0
N
0
We now consider the stability of (3.6). Since (3.6) is nonlinear, it is not possible to prove the stability in the sense of Courant, Friedrichs and Lewy [13]. But it will be shown that it is still stable in the sense of Guo [14, 15] and Stetter [16].
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1074
GUO BEN-YU
To do this, we assume that f and uN,0 have the errors f˜ and u ˜N,0 , respectively. They induce the error of numerical solution uN , denoted by u ˜N . Then we get the following equation: (3.7) 1 uN (t), φ)ω + B(˜ ˜N (t), φ)ω + (˜ uN (t), u ˜N (t), φ) + 2B(˜ uN (t), uN (t), φ) (∂t u 2 1 + (∂x u ˜N (t), ∂x φ)ω = (f˜(t), φ)ω , ∀φ ∈ PN , 0 < t ≤ ln(1 + T ), 4 ˜N,0, t = 0. u ˜N = u By taking φ = 2˜ uN in (3.7), it follows that
(3.8)
1 d ku ˜N (t) k2ω + k u ˜N (t) k21,ω dt 2 + 2B(˜ uN (t), u ˜N (t), u ˜N (t)) + 4B(˜ uN (t), uN (t), u ˜N (t)) 2 ≤ 2 k f˜(t) k . ω
Using Lemma 2.4, we deduce that for 0 ≤ t ≤ ln(1 + T ), (3.9) |2B(˜ uN (t), u ˜N (t), u ˜N (t))|
≤ ≤
where
p 2 4 π(1 + T )2 . c1 (T ) = √ µ
Furthermore, for any p, q > 0 and P´olya [17] that (3.10)
x2 1 t uN (t)|1,ω k e− 2 u ˜N (t) k2L4ω √ e 2 |˜ 2 µ c1 (T ) k u ˜N (t) kω k u ˜N (t) k21,ω ,
1 p
+ 1q = 1, we know from Hardy, Littlewood and
|ab| ≤
|b|q |a|p + . p q
Thus by using Lemma 2.4 and (3.10), we assert that (3.11) 2 1 t |4B(˜ uN (t), uN (t), u˜N (t))| ≤ √ e 2 |˜ uN (t)|1,ω k e−x uN (t)˜ uN (t) kω µ √ 1 1 1 3 44π t 2 2 ≤ √ e 2 k uN (t) kω2 k uN (t) k1,ω ku ˜N (t) kω2 k u ˜N (t) k1,ω µ 1 ≤ k u˜N (t) k21,ω +c2 (uN , T ) k u ˜N (t) k2ω , 4 where 123 π(1 + T )2 c2 (uN , T ) = k uN k2L∞ (0,ln(1+T );L2ω (Λ)) k uN k2L∞ (0,ln(1+T );Hω1 (Λ)) . µ2
By substituting (3.9) and (3.11) into (3.8), and integrating the resulting inequality, we find that Z t 1 2 ku ˜N (t) kω + ( − c1 (T ) k u ˜N (η) kω ) k u ˜N (η) k21,ω dη 4 0 (3.12) Z t ˜ ku ˜N (η) k2ω dη, ≤ ρ(˜ uN,0 , f , t) + c2 (uN , T ) 0
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ERROR ESTIMATION OF THE HERMITE SPECTRAL METHOD
where ρ(˜ uN,0 , f˜, t) =k u ˜N,0 k2ω +2
Z 0
t
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k f˜(η) k2ω dη.
We need the following lemma. Lemma 3.1. Assume that (i) the constants b1 > 0, b2 ≥ 0, b3 ≥ 0 and d ≥ 0, (ii) Z(t) and A(t) are non-negative functions of t, b2 (iii) d ≤ 12 e−b3 t1 for certain t1 > 0, b2 (iv) for all t ≤ t1 , Z t Z t 1 (b1 − b2 Z 2 (η))A(η)dη ≤ d + b3 Z(η)dη. Z(t) + 0
0
Then for all t ≤ t1 , Z(t) ≤ deb3 t . Proof. Consider the function Y (t) satisfying Z t Y (t) = d + b3 Y (η)dη. 0
Then for all t ≤ t1 , Y (t) = deb3 t ≤
b21 . b22
Clearly Z(t) ≤ Y (t) for t ≤ t1 , and so the conclusion is valid. Applying Lemma 3.1 to (3.12), we obtain the following result. Theorem 3.1. Let α >
1 8µ
and uN (t) be the solution of (3.6). If for certian t1 ,
(1 − a)2 −c2 (uN ,T )t1 e ρ(˜ uN,0 , f˜, t1 ) ≤ , a ≥ 0, 16c21 (T ) then for all t ≤ t1 , ku ˜N (t) k2ω +
a 4
Z
t 0
k u˜N (η) k21,ω dη ≤ ρ(˜ uN,0 , f˜, t)ec2 (uN ,T )t .
Theorem 3.1 indicates that the error of the numerical solution is controlled by uN,0 , f˜, t) does the errors of the data uN,0 and f , provided that the average error ρ(˜ not exceed certain critical value. It means that (3.6) is of generalized stability in the sense of Guo [14, 15], and of restricted stability in the sense of Stetter [16]. Next we deal with the convergence of scheme (3.6). Let U be the solution of (3.5), and UN = PN U. We derive from (3.5) that
(3.13)
1 (∂t UN (t), φ)ω + (UN (t), φ)ω + B(UN (t), UN (t), φ) 2 1 + (∂x UN (t), ∂x φ)ω + G(t, φ) = (f (t), φ)ω , 4 ∀φ ∈ PN , 0 < t ≤ ln(1 + T ),
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1076
GUO BEN-YU
where = G1 (t, φ) + G2 (t, φ) + G3 (t, φ), = (∂t U (t) − ∂t UN (t), φ)ω , 1 (U (t) − UN (t), φ)ω , G2 (t, φ) = 2 G3 (t, φ) = B(U (t), U (t), φ) − B(UN (t), UN (t), φ). ˜N = uN − UN . By subtracting (3.13) from Let uN be the solution of (3.6), and U (3.6), we obtain that G(t, φ) G1 (t, φ)
(3.14) ˜N (t), φ)ω + B(U ˜N (t), U ˜N (t), φ) ˜N (t), φ)ω + 1 (U (∂t U 2 ˜N (t), ∂x φ)ω = G(t, φ)ω , ˜N (t), UN (t), φ) + 1 (∂x U + 2B(U 4 ∀φ ∈ PN , 0 < t ≤ ln(1 + T ). ˜N (0) = 0. Comparing (3.14) to (3.7), we can derive a result similar In addition, U ˜N , U ˜N,0 ˜N,0 and f˜ are now replaced by UN , U to that of Theorem 3.1. But uN , u˜N , u ˜ and G(t, φ), respectively. Therefore we only have to estimate the term |G(t, UN (t))|. We first have from Theorem 2.1 that for r ≥ 1, ˜N (t))| ≤ cN − r2 k ∂t U (t) kr,ω k U ˜N (t) kω , |G1 (t, U − r2 ˜ ˜ k U (t) kr,ω k UN (t) kω . |G2 (t, UN (t))| ≤ cN An argument, as in the derivation of (3.9), leads to that for all t < ln(1 + T ), 1 t −x2 ˜N (t))| ≤ √ ˜N (t))ω | e 2 |(e (UN (t) + U (t))(UN (t) − U (t)), U |G3 (t, U 4 µ 1 1 1 1 2 ≤ c1 (T ) k UN (t) + U (t) kω2 k UN (t) + U (t) k1,ω k UN (t) − U (t) kω2 2 1 ˜N (t) k1,ω . × k UN (t) − U (t) k 2 k U 1,ω
By Theorem 2.1, ˜N (t) k21,ω +cc21 (T )N 12 −r k UN (t) k4r,ω . ˜N (t))| ≤ 1 k U |G3 (t, U 8 Hence (3.15) ˜N (t))| |G(t, U
≤
1 ˜ ˜N (t) k2ω k UN (t) k21,ω + k U 8 1 +c(c21 (T ) + 1)N 2 −r (k UN (t) k4r,ω + k ∂t U (t) k2r− 1 ,ω ). 2
Obviously, the last term in (3.15) tends to zero as N goes to infinity. Therefore we obtain the following result. Theorem 3.2. If α >
1 8µ
and r− 12
U ∈ L2 (0, ln(1 + T ); Hωr (Λ)) ∩ H 1 (0, ln(1 + T ); Hω
(Λ))
with r ≥ 1, then for all t ≤ ln(1 + T ), Z t 1 2 k uN (η) − U (η) k21,ω dη ≤ c∗ N 2 −r , k uN (t) − U (t) kω + 0
where c∗ is a positive constant depending only on µ, T and the norms of U in the space mentioned above.
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ERROR ESTIMATION OF THE HERMITE SPECTRAL METHOD
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Remark 3.1. In the proof of Theorem 3.1 and Theorem 3.2, we require that U ∈ x2 Hω1 (Λ) and so e− 2 (|U (x, t)| + |∂x U (x, t)|) → 0 as |x| → ∞. A sufficient condition 2 1 , eαy (|V (y, s)| + |∂y V (y, s)|) → 0, as |y| → ∞. It means is that for certain α > 8µ that V (y, s) should decay fast enough. It agrees with the experience in actual computations as described in Funaro and Kavian [7] and other papers. Remark 3.2. In this paper, we use the variable transformation (3.2) and so obtain the error estimations. In fact, a similiar transformation was used in actual computations by Funaro and Kavian [7]. This trick can be generalized to other problems, such as the two-dimensional heat equation and the Navier-Stokes equations. In actual computations, we need to discretize the term ∂t uN in (3.6). We can use Lemmas 2.1–2.4, Theorem 2.1 and an argument as in the proof of Theorems 3.1 and 3.2, to prove the generalized stability and the convergence of a fully discrete scheme, provided that the value of τ N satisfies certain reasonable conditions, where τ is the step size in time t, and N is the number of terms used in Hermite approximations. For instance, by Lemma 2.2, τ N should be bounded in the case of explicite schemes. We can also approximate nonlinear partial differential equations by the base functions √ ˜ l (x) = (2l l! π)− 12 e−ax2 Hl (x), a ≥ 0, l ≥ 0. H ˜ l (x) is an orthogonal system associated with the weight e(2a−1)x2 . For The set of H the application to linear problems with a = 1, we refer to Funaro and Kavian [7]. References [1] Y. Maday, B. Pernaud-Thomas and H. Vandeven, One r´ ehabilitation des m´ ethodes spectrales de type Laguerre, Rech. A´ erospat., 6 (1985), 353-379. [2] O. Coulaud, D. Funaro and O. Kavian, Laguerre spectral approximation of elliptic problems in exterior domains, Comput. Methods Appl. Mech. Engrg., 80 (1990), 451-458. CMP 90:17 [3] D. Funaro, Estimates of Laguerre spectral projectors in Sobolev spaces, in Orthogonal Polynomials and Their Applications, ed. by C. Brezinski, L. Gori and A. Ronveaux, Scientific Publishing Co., 1991, 263-266. MR 95a:41033 [4] V. Iranzo and A. Falqu`es, Some spectral approximations for differential equations in unbounded domains, Comput. Methods Appl. Mech. Engrg., 98 (1992), 105-126. MR 93d:65103 [5] C. Mavriplis, Laguerre polynomials for infinite-domain spectral elements, J. Comp. Phys., 80 (1989), 480-488. MR 90f:65228 [6] K. Black, Spectral elements on infinite domains (unpublished). [7] D. Funaro and O. Kavian, Approximation of some diffusion evolution equations in unbounded domains by Hermite funtions, Math. Comp., 57 (1990), 597-619. MR 92k:35156 [8] J. A. C. Weideman, The eigenvalues of Hermite and rational spectral differential matrices, Numer. Math., 61 (1992), 409-432. MR 92k:65071 [9] C. I. Christov, A complete orthonormal system of funtions in L2 (−∞, ∞) space, SIAM J. Appl. Math., 42 (1982), 1337-1344. MR 84b:42018 [10] J. P. Boyd, Spectral methods using rational basis funtions on an infinite interval, J. Comp. Phys., 69 (1987), 112-142. MR 88e:65093 [11] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. MR 56:9247 [12] R. J. Nessel and G. Wilmes, On Nikolskii-type inequalities for orthogonal expansion, in Approximation Theory II , ed. by G. G. Lorentz, C. K. Chui and L. L. Schumaker, Academic Press, New York, 479-484. ¨ [13] R. Courant, K. O. Friedrichs and H. Lewy, Uber die partiellen differezengleichungen der mathematischen physik, Math. Ann., 100 (1928), 32-74. [14] B. Y. Guo, A certain class of finite difference schemes of two-dimensional vorticity equation viscous fluid, RR. SUST, 1965, Also see Acta Math. Sinica, 17 (1974), 242-258. MR 56:17128
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GUO BEN-YU
[15] B. Y. Guo, Generalized stability of discretization and its applications to numerical solutions of nonlinear differential equations, Contemp. Math., 163 (1994), 33-54. MR 95d:65067 [16] H. J. Stetter, Stability of nonlinear discretization algorithms, in Numerical Solutions of Partial Differential Equations, ed. by J. Bramble, Academic Press, New York, 1966, 111-123. MR 34:5322 [17] G. H. Hardy, J. E. Littlewood and G.P´ olya, Inequalities, 2’ed., Cambridge University Press, Cambridge, 1952. MR 13:727e Department of Mathematics, Shanghai University, Jiading Campus, Shanghai, 201800, China Current address: Department of Mathematics, Shanghai Normal University, Shanghai, China E-mail address: [email protected]
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