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]:

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

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

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

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

3. Some Lemmas

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) ≤ C1NS−β η(t) Hβ(Ω),0 ≤ s ≤ β, PCη(t) − η(t) HS(Ω) ≤ C2NS−β η(t) Hβ(Ω),0 ≤ s ≤ β,β > 1, Rγξ(t) − ξ(t) HS(Ω) ≤ C3NS−β ξ(t) Hβ(Ω),0 ≤ s ≤ β,γ > β − s, RγPNη(t) − η(t) HS(Ω) ≤ C4NS−β η(t) Hβ(Ω),0 ≤ s ≤ β,γ > β − s,

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′=0t−τ 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) \leq ρ e (M 1 + M 2) t .

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

ξ (t) \leq ρ e M 1 t .

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

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

Theorem 1. Suppose the following conditions are fulfilled
(i) δ = 0,τN2 < ∞,
(ii) σ > 1∕2orτN2 < 2 v(1−2σ),
(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.

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

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

Therefor (5.6) is still holds.
If σ < 1∕2, τN2 < 2 ν(1−2σ), we put q = q3, and thus (5.12) holds.
By summing up (5.6) for t ∈ Sτ , we get

6. The Proof of Theorem 2

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

References

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