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Lobachevskii Journal of Mathematics http://ljm.ksu.ru Vol. 16, 2004, 79 – 89

© A. Rashid

Abdur Rashid
THE PSEUDOSPECTRAL METHOD FOR THERMOTROPIC PRIMITIVE EQUATION AND ITS ERROR ESTIMATION
(submitted by A. Lapin)

ABSTRACT. In this paper, a pseudospectral method is proposed for solving the periodic problem of thermotropic primitive equation. The strict error estimation is proved.


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2000 Mathematical Subject Classification. 35Q35, 65M70,65N30..

Key words and phrases. Thermotropic primitive equation, pseudospectral scheme, error estimation.

This work is supported by Gomal University, D.I.Khan, Pakistan..


1. Introduction

Thermotropic primitive equation is governed by the following differential equations[1]:

U t + U U x + V U y + φ x νΔU FV = 0, V t + U V x + V V y + φ y νΔV + FU = 0, φ t + U φ x + V φ y + φ U x + V x = 0, (1.1)

where U,V are the components of the speed in x, y directions respectively, g is the acceleration of gravity, H is the height of the geopotential surface, φ = gH, F is coriolis parameter and ν is the coefficient of friction.
There has been a rapid development in the spectral methods for the last two decades. They have become important tools for numerical solutions of partial differential equations, and have been widely applied to numerical simulations in various fields [2-5]. Although the pseudospectral methods are easier to implement for nonlinear partial differential equations, they are not stable as the spectral ones due to ’aliasing’. Therefore some author proposed the filtering technique [10-11] to remedy the deficiency of instability. Some papers have also been devoted to theoretical study and numerical solutions of (1.1) [6-9].
The aim of this paper is to consider the periodic initial boundary-value problem for thermotropic primitive equation. A pseudospectral scheme with restraint operator in combination with first order time differencing technique is considered for thermotropic primitive equation. The stability and rate of convergence for the approximate problem are proved.

2. The Pseudospectral Scheme

Let Ω = (x,y) π < x,y < π and all functions have the period 2π for the variable x and y. The norm of the space Lq (Ω) is denoted by Lq(Ω). In particular, the scaler product and the norm of L2 (Ω) are denoted by , and L2(Ω) respectively. Let m1,m2 and N be integers and m = m1 2 + m2 2. Define

V N = Span ei(m1x+m2y)m N,N > 0.

Let PN be the orthogonal projection operator, i.e.

PNη,ψ = η,ψ,ψ V N.

For the pseudospectral approximation, we put the nodes

xj1,yj2 = 2πj1 2N + 1, 2πj2 2N + 1 , N j1,j2 N,

and let Pc ˜ be the interpolation operator, i.e. for η(x,y) C(Ω)

Pc ˜η xj1,yj2 = η xj1,yj2 , N j1,j2 N.

Define Pc = PNPc ˜. To weaken the nonlinear instability of computation, we follow the work of [11] to adopt the filtering operator Rγ with γ > 1, i.e. if

η(x,y) = mNηm1,m2ei(m1x+m2y),

then

Rγη(x,y) = mN 1 m N γ η m1,m2ei(m1x+m2y).

Let τ be the mesh spacing of the variable t and define

Sτ = t = kτk = 0, 1, 2,.

ηt(t) = η(t + τ) η(t) τ .

To approximate the nonlinear terms, we define

dα η,u,v = αd(1) η,u,v + (1 α)d(2) η,u,v,0 α 1,

d(1) η,u,v = P c uη x + vη y,

d(2) η,u,v = xPc(uη) + yPc(vη).

Let uN,vN,ϕN be the approximations to U, V and φ respectively, where for all (x,y) Ω and t Sτ,

ηN(x,y,t) = m,nNηm,nN(t)ei(mx+ny),η = u,v,ϕ.

The pseudospectral scheme for solving (1.1) is

utN + R γd12 Rγ(uN + δτu tN),uN,vN + xϕN νΔ(uN + στu tN) F(vN + δτv tN) = 0, vtN + R γd12 Rγ(vN + δτv tN),uN,vN + yϕN νΔ(vN + στv tN) + F(uN + δτu tN) = 0, ϕtN + R γd0 Rγ(ϕN + δτϕ tN),uN,vN + A(ϕN + δτϕ tN,uN + δτu tN,vN + δτv tN) = 0, (2.1)

where 0 δ 1,0 σ 1 and A(η,ξ,η) = P c η ξ x + η x .

3. Some Lemmas

Lemma 1. [1]. For all η(x,y,t)

2 η(t),ηt(t) = η(t) 2 t τ ηt(t) 2.

Lemma 2. [5]. For all η(x,y,t) V N, then

η x2 N2 η(t) 2, η y2 N2 η(t) 2.

Lemma 3. [10]. For all η(x,y,t) V N and ξ(x,y,t) V N, then

η(t)ξ(t)2 (2N + 1)2 η(t) 2 ξ(t) 2.

Lemma 4. [15]. For all η(x,y,t) Hβ(Ω) and ξ(x,y,t) V N, then

PNη(t) η(t) HS(Ω) C1NSβ η(t) Hβ(Ω),0 s β, PCη(t) η(t) HS(Ω) C2NSβ η(t) Hβ(Ω),0 s β,β > 1, Rγξ(t) ξ(t) HS(Ω) C3NSβ ξ(t) Hβ(Ω),0 s β,γ > β s, RγPNη(t) η(t) HS(Ω) C4NSβ η(t) Hβ(Ω),0 s β,γ > β s,

where C1 C4 are positive constants.

Lemma 5. [9]. Assume that the following conditions are fulfilled:
(i) ξ(t) and η(t) are non-negative functions defined on Sτ;
(ii) ρ,a,M1,M2, and M3 are nonnegative constants;
(iii) A(x) is a function such that, if x M3, then A(x) 0;
(iv) ξ(t) ρ + τ t=0tτ M 1ξ(t) + M 2Naξ2(t) + A(ξ(t))η(t) ;
(v) ρe(M1+M2)T min M 3, 1 Na, ξ(0) ρ, t T.
Then

ξ(t) ρe(M1+M2)t.

In particular, if M2 = 0 and A(ξ(t)) = 0, then for all ρ and n.

ξ(t) ρeM1t.

4. Error Estimation

For simplicity, we take δ = 0, let UN = P NU,V N = P NV, and φN = P Nφ, then (1.1) leads to

UtN + R γd12 RγUN,UN,V N + xφN νΔ(UN + στU tN) FV N = 0, V tN + R γd12 RγV N,UN,V N + yφN νΔ(V N + στV tN) + FUN = 0, φtN + R γd0 RγφN,UN,V N + A(φN,UN,V N) = 0, (4.1)

where

G1N = U tN UN t , G2N = R γd12 RγUN,UN,V N P N U U x + V U y , G3N = νστΔU tN, G4N = V tN V N t , G5N = R γd12 RγV N,UN,V N P N U V x + V V y , G6N = νστΔV tN, G7N = φ tN φN t , G8N = R γd0 RγφN,UN,V N P N U φ x + V φ y, G9N = A φN,UN,V N P N φU x + φV x .

Put

u˜ = u UN,v˜ = v V N,ϕ˜ = ϕ φN.

Then from (1.1) and (2.1), we obtain

u˜tN + ξ 1N + ξ 2N + xϕN νΔ u˜N + στu˜ tN Fv˜N = l=13G lN, v˜tN + ξ 3N + ξ 4N + yϕN νΔ v˜N + στv˜ tN + Fu˜N = l=46G lN, ϕ˜tN + k=59ξ kN = l=79G lN, (4.2)

where

ξ1N = R γd12 Rγu˜N,UN,vN + R γd12 RγUN,u˜N,v˜N , ξ2N = R γd12 Rγu˜N,u˜N,v˜N , ξ3N = R γd12 Rγv˜N,UN,V N + R γd12 RγV N,u˜N,v˜N , ξ4N = R γd12 Rγv˜N,u˜N,v˜N , ξ5N = R γd0 Rγϕ˜N,UN,V N + R γd0 RγφN,u˜N,v˜N , ξ6N = R γd0 Rγϕ˜N,u˜N,v˜N , ξ7N = A ϕ˜N,u˜N,v˜N , ξ8N = A ϕ˜N,UN,V N , ξ9N = A φN,u˜N,v˜N .

We shall use the following notations

E = U,V,φ,EN = UN,V N,φN ,E˜N = u˜N,v˜N,ϕ˜N ,

E ˜ N (t)2 = u˜N(t) 2 + v˜N(t) 2 + ϕ˜N(t) 2,
E ˜ tN(t)2 = u˜ tN(t) 2 + v˜ tN(t) 2 + ϕ˜ tN(t) 2,
E ˜ N (t) 12 = u˜N(t) 12 + v˜N(t) 12 + ϕ˜N(t) 12.

Let Hβ(Ω) be the Sobolev space equipped with the norm Hβ(Ω). In particular L2(Ω) = H0(Ω), we define E1 = U,V,φ

E1Hβ(Ω)2 = U(t) Hβ(Ω)2 + V (t) Hβ(Ω)2 + φ(t) Hβ(Ω)2,

E1β = max 0tτ E1(t) Hβ(Ω).

Now we suppose

ρ(t) = E˜N(0)2)+ντ(σ+q 2)E˜N(0) 12+τ t=0tτG lN(t)2,l = 1,..., 9

E˜(t) = E˜N(t)2 + ντ(σ + q 2)E˜N(0) 12 + τ t=0tτr 1τ E˜tN(t)2 + ν(2 7ɛ)E˜N(t) 12 .

Theorem 1. Suppose the following conditions are fulfilled
(i) δ = 0,τN2 < ,
(ii) σ > 12orτN2 < 2 v(12σ),
(iii) for suitably small positive constant M1 and all t T , such that ρ(T) M1 N2 .
Then there exists a positive constant M2 such that for all t Sτ, t T, we have

E˜(t) ρ(t)eM2t.

Theorem 2. Assume that the conditions (i), (ii) of Theorem 1 are satisfied. In addition E C2(0,T; m 00(Ω)),E 1 C(0,T; H52+r(Ω) m0B+1(Ω)), r > 0,β 1, then

E˜(t) M3(τ2 + N2β)eM4t.

M being positive constants depending only on E15 2+r and ν.

Now we define

η(t)mrβ = max 0sq 1 4π2 a+b=0β Ω a+bη(x,y,t) xayb 2dxdy12,

ηCq(0,T;mrβ(Ω)) = max 0sq max 0tT sη(t) ts mrβ(Ω),

Cq 0,T; m rβ(Ω) = η(x,y,t)η Cq(0,T;mrβ(Ω)) < .

5. The Proof of Theorem 1

Let c be a positive constant which may be different in different cases, q denote an undetermined positive constant and ɛ > 0. Taking the scaler product (4.2) with 2u˜N + qτu˜ tN, we have

(u˜N(t)2) t + τ(q 1 ɛ)u˜tN(t)2 + (2u˜N(t) + qτu˜ tN(t),ξ˜ 1N(t) + ξ˜ 2N(t) + φ˜N x (t) + Fv˜N(t)) + 2νu˜N(t) 12 + ντ(σ + q 2)(u˜(t)12) t + ντ2(σq σ q 2)u˜tN(t) 12 cu˜N(t)2 + c(1 + q2τ 4ɛ ) l=13G lN(t)2. (5.1)

Similarly from the second and third formulas of (4.2), we have

(v˜N(t)2) t + τ(q 1 ɛ)v˜tN(t)2 + (2v˜N(t) + qτv˜ tN(t),ξ˜ 3N(t) + ξ˜ 4N(t) + φ˜N y (t) + Fu˜N(t)) + 2νv˜N(t) 12 + ντ(σ + q 2)(v˜(t)12) t + ντ2(σq σ q 2)v˜tN(t) 12 cv˜N(t)2 + c(1 + q2τ 4ɛ ) l=46G lN(t)2 (5.2)
(ϕ˜N(t)2) t + τ(q 1 ɛ)ϕ˜tN(t)2 + (2ϕ˜N(t) + qτϕ˜ tN(t), l=59ξ˜ lN(t)) cϕ˜N(t)2 + c(1 + q2τ 4ɛ ) l=79G lN(t)2. (5.3)

Putting (5.1)-(5.3) together, we get

(E˜N(t)2) t + τ(q 1 ɛ)E˜tN(t)2 + 2νE˜N(t) 12 + ντ(σ + q 2)(E˜(t)12) t + ντ2(σq σ q 2)E˜tN(t) 12 + l=16M lN(t) cE˜N(t)2 + c(1 + q2τ 4ɛ ) l=19G lN(t)2, (5.4)

where

M1N(t) = (2u˜N(t) + qτu˜ tN(t),ξ 1N(t)) + (2v˜N(t) + qτv˜ tN(t),ξ 3N(t)) + (2ϕ˜N(t) + qτϕ˜ tN(t),ξ 5N(t)) qτ(u˜ tN(t),Fv˜N(t)) + qτ(v˜tN(t),Fu˜N(t)), M2N(t) = (2u˜N(t) + qτu˜ tN(t),ξ 2N(t)) + (2v˜N(t) + qτv˜ tN(t),ξ 4N(t)) + (2ϕ˜N(t) + qτϕ˜ tN(t),ξ 6N(t)), M3N(t) = (2u˜N(t) + qτu˜ tN(t), xϕN(t)) + (2v˜N(t) + qτv˜ tN(t), yϕ˜N(t)), M4N(t) = (2ϕ˜N(t) + qτϕ˜ tN(t),ξ 7N(t)), M5N(t) = (2ϕ˜N(t) + qτϕ˜ tN(t),ξ 8N(t)), M4N(t) = (2ϕ˜N(t) + qτϕ˜ tN(t),ξ 9N(t)).

We now estimate MlN(t). Because of the Schwarz inequality and embedding theorem, we have

M1N(t) ɛνE˜N(t) 12 + ɛτE˜ tN(t)2 + c ɛν(1 + τq2N2)E 1N 5 2+rE˜N(t)2, M2N(t) ɛνE˜N(t) 12 + ɛτE˜ tN(t)2 + cN2 ɛν (E˜N(t)2 + E˜N(t) 12 + E 1N 5 2+r)E˜N(t)2, M3N(t) ɛτE˜ tN(t) 12 + ɛνE˜N(t) 12 + c ɛν(1 + τq2N2)E˜N(t)2, M4N(t) ɛτE˜ tN(t) 12 + ɛνE˜N(t) 12 + c ɛν(1 + τq2N2)E˜N(t)2, M5N(t) ɛτE˜ tN(t) 12 + ɛνE˜N(t) 12 + cN2 ɛν (1 + τq2N2)E˜N(t)4, M6N(t) ɛτE˜ tN(t) 12 + ɛνE˜N(t) 12 + c ɛνE15 2+rE˜N(t)2. By substituting the above estimates into (5.4), we get
(E˜N(t)2) t + τ(q 1 7ɛ)E˜N(t)2 + ν(2 6ɛ)E˜N(t) 12 + ντ(σ + q 2)(E˜N(t) 12) t + ντ2(σq σ q 2)E˜N(t) 12 HN(t) + A 1E˜N(t)2 + B 1E˜N(t)4 + B 2E˜N(t) 12, (5.5)

where

A1 = c 1 + c ɛν + c ɛν(1 + τq2N2) E 152+r,

B1 = cN2 ɛν (1 + τq2N2) + cN2 ɛν ,

B2 = cN2 ɛν E˜N(t)2,

HN(t) = c 1 + q2τ 4ɛ l=19G lN2.

Now let ɛ be suitably small, r1 > 0, and

q1 = max 1 + r1 + 7ɛ, 2σ 2σ 1 , q2 = r1 + 1 + 7ɛ + ντN2 2 , q3 = 2r1 + 2 + 14ɛ + 2σντN2 2 ντN2(1 2σ) 1.

If σ > 12, we put q = q1, and it follows from (5.5) that

(E˜N(t)2) t + r1τE˜N(t)2 + ν(2 6ɛ)E˜N(t) 12 + ντ(σ + q 2)(E˜N(t) 12) t HN(t) + A 1E˜N(t)2 + B 1E˜N(t)4 + B 2E˜N(t) 12. (5.6)

If σ = 12, we put q = q2, and so

τ(q 1 7ɛ)E˜tN2 + ντ2(σq σ q 2)E˜tN(t) 12 r 1τE˜tN(t)2.

Therefor (5.6) is still holds.
If σ < 12, τN2 < 2 ν(12σ), we put q = q3, and thus (5.12) holds.
By summing up (5.6) for t Sτ , we get

(E˜N(t)2) t + ντ(σ + q 2)(E˜N(t) 12) t + τ t=0tτr 1τE˜tN(t)2 + ν(2 7ɛ)E˜N(t) 12 ρ(t) + τ t=0tτA 1E˜1N(t) + B 1E˜12(t) + B2E˜2N(t) 12, (5.7)

where

ρ(t) = E˜N(0)2)+ ντ(σ + q 2)E˜N(0) 12 + τ t=0tτHN(t)

from which and Lemma 5, the proof is completed.

6. The Proof of Theorem 2

We first have

GlN(t) cτE C2(0,T;m00(Ω)),l = 1, 4, 7.

From Lemma 4 and the embedding theorem, we get

GlN(t) cE 152+rNβE 1m0β+1(Ω),l = 2, 5, 8.

It is easy to show that

GlN(t) cτ E t (t) m20(Ω) + E t (t) m20(Ω) ,l = 3, 6.

We have also

G9N(t) cNβE 152+rE(t)m0β+1(Ω),E˜N(0) cNβE(0) m0β(Ω).

Therefore if the conditions of Theorem 1 are fulfilled, then

ρ(t) c τ2 + N2β .

By combining the above estimations with Theorem 1, we complete the proof of Theorem 2.

References

[1]   B. Y. Guo, Difference Methods for Partial Equations, Science Press, Beijing, 1988.

[2]   R. Peyret, Spectral Methods in Incompressible Viscous Flow, Springer-Verlag, New York, 2002.

[3]   B. Fornberg, A practical Guide to Pseudospectral Methods, Cambridge University press, Cambridge, 1996.

[4]   J. P. Boyd, Chebyshev and Fourier Spectral Methods, Second Edition(Revised), Dover Publication Inc. Mineola New York, 2001.

[5]   C. Canuto., M. Y. Hussaini., A. Quarteroni and T. A. Zang, Spectral Methods in Fluid Dynamics, Springer-Verlag, Berlin, 1988.

[6]   H. Wengle and J. H. Seifeld, Pseudospectral solution of atmospheric diffusion problems, J. Comput. Phys., 26 (1978), 87-106.

[7]   A. Robert., T. L. Yee and H. Ritchic, A semi Lagrangian and semi implicit numerical integration scheme for multilevel atmospheric models, Monthly Weather Review, 113 (1985), 338-394.

[8]   B. Y. Guo, and J. D. Zheng. Pseudospectral Difference Method for Baroclinic Primitive Equation and its Error Estimation, Scince in China (Series A), 35 (1992), 1-13.

[9]   B. Y. Guo, Difference method for fluid dynamics numerical solution of Primitive equation, Scientia Sinica, 24 (1981), 297-312.

[10]   A. Rashid and Li. Yuan, Convergence Analysis of Pseudospectral Method with Restraint operator for Fluid Flow with Low Mach Number Appl. Comput. Math., 2 (2003),135-142.

[11]   H. Vendeven, Family of spectral filters for discontinous problems, J. Sci. Comput. 6 (1991), 159-192.

DEPARTMENT OF MATHEMATICS, GOMAL UNIVERSITY, D.I.KHAN, PAKISTAN.

E-mail address: rashid_himat@yahoo.com

Received February 8, 2004; revised version May 27, 2004