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 = g H

, 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 2

π

for the variable x and y. The norm of the space

L^{q} (Ω)

is denoted by

{∥\cdot∥}_{L^{q} (Ω)}

. In particular, the scaler product and the norm of

L^{2} (Ω)

are denoted by

(\cdot, \cdot)

and

{∥\cdot∥}_{L^{2} (Ω)}

respectively. Let

m_{1}, m_{2}

and N be integers and

m = \sqrt{m_{1}^{2} + m_{2}^{2}}

. Define

and let

\tilde{P_{c}}

be the interpolation operator, i.e. for

η (x, y) \in C (Ω)

Define

P_{c} = P_{N} \tilde{P_{c}}

. To weaken the nonlinear instability of computation, we follow the work of [11] to adopt the filtering operator

R_{γ}

with

γ > 1

, i.e. if

Let

u^{N}, v^{N}, ϕ^{N}

be the approximations to U, V and

φ

respectively, where for all

(x, y) \in Ω

and

t \in S_{τ},

where

0 \leq δ \leq 1, 0 \leq σ \leq 1

and

A (η, ξ, η^{*}) = P_{c} [η (\frac{\partial ξ}{\partial x} + \frac{\partial η^{*}}{\partial x})] .

3. Some Lemmas

Lemma 4. $[15] .$ For all $η (x, y, t) \in H^{β} (Ω)$ and $ξ (x, y, t) \in V_{N}$ , then

\begin{array}{l} {∥P_{N} η (t) - η (t)∥}_{H^{S} (Ω)} & \leq C_{1} N^{S - β} {∥η (t)∥}_{H^{β} (Ω)}, 0 \leq s \leq β, \\ {∥P_{C} η (t) - η (t)∥}_{H^{S} (Ω)} & \leq C_{2} N^{S - β} {∥η (t)∥}_{H^{β} (Ω)}, 0 \leq s \leq β, β > 1, \\ {∥R_{γ} ξ (t) - ξ (t)∥}_{H^{S} (Ω)} & \leq C_{3} N^{S - β} {∥ξ (t)∥}_{H^{β} (Ω)}, 0 \leq s \leq β, γ > β - s, \\ {∥R_{γ} P_{N} η (t) - η (t)∥}_{H^{S} (Ω)} & \leq C_{4} N^{S - β} {∥η (t)∥}_{H^{β} (Ω)}, 0 \leq s \leq β, γ > β - s, \end{array}

Lemma 5. $[9] .$ Assume that the following conditions are fulfilled:
(i) $ξ (t)$ and $η (t)$ are non-negative functions defined on $S_{τ};$
(ii) $ρ, a, M_{1}, M_{2},$ and $M_{3}$ are nonnegative constants;
(iii) A(x) is a function such that, if $x \leq M_{3}$ , then $A (x) \leq 0;$
(iv) $ξ (t) \leq ρ + τ \sum_{t^{'} = 0}^{t - τ} [M_{1} ξ (t^{'}) + M_{2} N^{a} ξ^{2} (t^{'}) + A (ξ (t^{'})) η (t^{'})];$
(v) $ρ e^{(M_{1} + M_{2}) T} \leq min (M_{3}, \frac{1}{N^{a}})$ , $ξ (0) \leq ρ$ , $t \leq T$ .
Then

ξ (t) \leq ρ e^{(M_{1} + M_{2}) t} .

In particular, if $M_{2} = 0$ 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

U^{N} = P_{N} U, V^{N} = P_{N} V,

and

φ^{N} = P_{N} φ

, then (1.1) leads to

${∥{\tilde{E}}^{N} (t)∥}^{2} = {∥{\tilde{u}}^{N} (t)∥}^{2} + {∥{\tilde{v}}^{N} (t)∥}^{2} + {∥{\tilde{ϕ}}^{N} (t)∥}^{2},$
${∥{\tilde{E}}_{t}^{N} (t)∥}^{2} = {∥{\tilde{u}}_{t}^{N} (t)∥}^{2} + {∥{\tilde{v}}_{t}^{N} (t)∥}^{2} + {∥{\tilde{ϕ}}_{t}^{N} (t)∥}^{2},$
${|{\tilde{E}}^{N} (t)|}_{1}^{2} = {|{\tilde{u}}^{N} (t)|}_{1}^{2} + {|{\tilde{v}}^{N} (t)|}_{1}^{2} + {|{\tilde{ϕ}}^{N} (t)|}_{1}^{2} .$

Let

H^{β} (Ω)

be the Sobolev space equipped with the norm

{∥\cdot∥}_{H^{β} (Ω)}

. In particular

L^{2} (Ω) = H^{0} (Ω)

, we define

E_{1} = (U, V, φ)

Theorem 1. Suppose the following conditions are fulfilled
(i) $δ = 0, τ N^{2} < \infty$ ,
(ii) $σ > 1 ∕ 2 o r τ N^{2} < \frac{2}{v (1 - 2 σ)},$
(iii) for suitably small positive constant $M_{1}$ and all $t \leq T$ , such that $ρ (T) \leq \frac{M_{1}}{N^{2}} .$
Then there exists a positive constant $M_{2}$ such that for all $t \in S_{τ}$ , $t \leq T$ , we have

\tilde{E} (t) \leq ρ (t) e^{M_{2} t} .

Theorem 2. Assume that the conditions (i), (ii) of Theorem 1 are satisfied. In addition $E \in C^{2} (0, T; m_{0}^{0} (Ω)), E_{1} \in C (0, T; H_{\frac{5}{2} + r} (Ω) ⋂ m_{0}^{B + 1} (Ω)), r > 0, β \geq 1,$ then

\tilde{E} (t) \leq M_{3} (τ^{2} + N^{- 2 β}) e^{M_{4} t} .

$M_{ℓ}$ being positive constants depending only on $∥ ∣ E_{1} ∥ ∣_{\frac{5}{2} + r}$ and $ν$ .

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

2 {\tilde{u}}^{N} + q τ {\tilde{u}}_{t}^{N}

, we have

We now estimate

∣ M_{l}^{N} (t) ∣

. Because of the Schwarz inequality and embedding theorem, we have

Therefor (5.6) is still holds.
If

σ < 1 ∕ 2

τ N^{2} < \frac{2}{ν (1 - 2 σ)}

, we put

q = q_{3}

, and thus (5.12) holds.
By summing up (5.6) for

t \in S_{τ}

, 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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