Convergence of the finite-element scheme for the equation of internal waves
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CONVERGENCE OF THE FINITE-ELEMENT SCHEME FOR THE EQUATION OF INTERNAL WAVES M. N. Moskalkova and D. Utebaevb
UDC 517.5:519.6
Abstract. The Dirichlet boundary-value problem for the equation of internal waves is considered. To solve it, the finite-element method for both space and time variables is selected. This allows obtaining a highly accurate solution. The accuracy of the method is estimated making certain assumptions on the smoothness of the solutions to the differential problem. If piecewise-cubic finite elements are used, the order of accuracy equals three. Keywords: finite-element method, difference scheme, estimates of accuracy.
The motion of internal waves that propagate in a stratified fluid that rests or uniformly rotates is mathematically modeled by partial differential equations with additional terms that cause strong dispersion of these waves. Such equations are applied in geophysics, oceanology, and atmospheric physics [1]. We will consider the Dirichlet boundary-value problem for the nonstationary three-dimensional equation of internal waves and estimate the accuracy of the finite-element method. 1. PROBLEM STATEMENT Let us consider the problem [1] ¶2 ¶t 2
( L1 u ) + L2 u = f ( x, t ), ( x, t ) Î QT = {x Î W , t Î ( 0, T ]}, u = 0, x Î G = ¶W , t Î [ 0, T ] , u( x, 0) = u 0 ( x ),
where L1 u = D 3 u º
¶2u ¶ x12
+
¶2u ¶ x 22
+
(1)
¶u ( x, 0) = u1 ( x ), ¶t
æ ¶2u ¶2u ö ÷ ; w is the Brunt–V&&ais&&al&&a frequency; and , L2 u = w 20 D 2 u º w 20 ç + 0 2 2 ÷ 2 ç ¶ x3 è ¶ x1 ¶ x 2 ø ¶2u
W = {0 < x k < lk , k = 1, 2 , 3}. Let us present the generalized formulation of problem (1). By the generalized solution of problem (1) we will mean a o ¶2u function u( x, t ) that for each t Î ( 0, T ] belongs to space H = W 1 (W ) , has the derivative Î W21 (W ), and satisfies the 2 2 ¶t following relations almost everywhere for all t Î ( 0, T ] "J( x ) Î H : a
National Aviation University, Kyiv, Ukraine, [email protected]. bBerdakh Karakalpak State University, Nukus, Uzbekistan, [email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 146–152, May–June 2011. Original article submitted February 24, 2010. 1060-0396/11/4703-0459
©
2011 Springer Science+Business Media, Inc.
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æ d 2 u( t ) ö du a1 ç , J ÷ + a 2 ( u( t ), J ) = ( f ( t ), J ), u( 0) = u 0 Î H , ( 0) = u1 Î H , ç dt 2 ÷ dt è ø where
a1 ( w( t ), J ) = ò
3
¶w ¶J dx ; a 2 ( u( t ), J ) = w 20 ò ¶ x k k
å ¶x
W k =1
2
(2)
¶u ¶ J dx ; k ¶ xk
å ¶x
W k =1
o
u( t ) is a function of abstract argument t Î[ 0, T ] with values in H = W 1 (W ) . The existence, smoothness, and uniqueness 2
of the solution of this problem are discussed in [1]. Denote by | u | m = a m ( u, u ) , m = 1, 2 , energetic seminorms in H that correspond to the bilinear forms a m ( u, J ). o
Energetic space H A 1 generated by seminorm | u | 1 is equivalent to the space H = W 1 (W ) [2]; therefore, the estimates 2
c1 || u || 1 £ | u | 1 £ c 2 || u || 1 , c1 > 0, are true where || u || 1 is a norm in H. For the other energetic seminorm, the
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