Lobachevskii Journal of Mathematics
Vol. 16, 2004, 79 – 89
© A. Rashid
THE PSEUDOSPECTRAL METHOD FOR
THERMOTROPIC PRIMITIVE EQUATION AND ITS
(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.
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..
Thermotropic primitive equation is governed by the following differential
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,
, F is coriolis
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
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 and all functions
have the period 2
for the variable x and y. The norm of the space
In particular, the scaler product and the norm of
Let and N be
integers and .
be the orthogonal projection operator, i.e.
For the pseudospectral approximation, we put the nodes
and let be the interpolation
operator, i.e. for
Define . To
weaken the nonlinear instability of computation, we follow the work of  to adopt the
be the mesh spacing of the variable t and define
To approximate the nonlinear terms, we define
Let be the approximations
to U, V and respectively,
where for all
The pseudospectral scheme for solving (1.1) is
3. Some Lemmas
For all ,
are positive constants.
Assume that the following conditions are fulfilled:
are non-negative functions defined on
are nonnegative constants;
(iii) A(x) is a function such that, if ,
In particular, if
then for all
4. Error Estimation
For simplicity, we take
then (1.1) leads to
Then from (1.1) and (2.1), we obtain
We shall use the following notations
be the Sobolev space equipped with the norm
Now we suppose
Theorem 1. Suppose the following conditions are fulfilled
(iii) for suitably small positive constant
Then there exists a positive constant
for all ,
Theorem 2. Assume that the conditions (i), (ii) of Theorem 1 are satisfied. In
being positive constants
depending only on
Now we define
5. The Proof of Theorem 1
be a positive constant which may be different in different cases,
denote an undetermined positive constant and
. Taking the scaler
product (4.2) with ,
Similarly from the second and third formulas of (4.2), we have
Putting (5.1)-(5.3) together, we get
We now estimate .
Because of the Schwarz inequality and embedding theorem, we have
By substituting the above estimates into (5.4), we get
Now let be
suitably small, ,
If , we
and it follows from (5.5) that
If , we
Therefor (5.6) is still holds.
and thus (5.12) holds.
By summing up (5.6) for
, we get
from which and Lemma 5, the proof is completed.
6. The Proof of Theorem 2
We first have
From Lemma 4 and the embedding theorem, we get
It is easy to show that
We have also
Therefore if the conditions of Theorem 1 are fulfilled, then
By combining the above estimations with Theorem 1, we complete the proof
of Theorem 2.
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DEPARTMENT OF MATHEMATICS, GOMAL UNIVERSITY, D.I.KHAN, PAKISTAN.
E-mail address: email@example.com
Received February 8, 2004; revised version May 27, 2004