finite element approximations to the system of shallow water equations

FINITE ELEMENT APPROXIMATIONS TO THE SYSTEM OF SHALLOW WATER EQUATIONS, PART I: CONTINUOUS TIME A PRIORI ERROR ESTIMATES S. CHIPPADAz , C. N. DAWSONz , M. L. MARTINEZy , AND M. F. WHEELERz

TICAM Technical Report TR 95-18 This paper appears in SINUM 35(1998), No.2 pp.692-711

Abstract. Various sophisticated nite element models for surface water ow based on the shallow water equations exist in the literature. Gray, Kolar, Luettich, Lynch and Westerink have developed a hydrodynamic model based on the generalized wave continuity equation (GWCE) formulation, and have formulated a Galerkin nite element procedure based on combining the GWCE with the nonconservative momentum equations. Numerical experiments suggest that this method is robust, accurate and suppresses spurious oscillations which plague other models. We analyze a slightly modi ed Galerkin model which uses the conservative momentum equations (CME). For this GWCE-CME system of equations, we present a continous-time a priori error estimate based on an L2 projection. Key words. shallow water equations, surface water ow, mass conservation, momentum conservation, nite element method, a priori error estimate AMS subject classi cations. 35Q35, 35L65 65N30, 65N15

1. Introduction. In recent years, there has been much interest in the numerical solutions of the shallow water equations. Simulation of shallow water systems can serve numerous purposes. First, it can serve as means for modeling tidal uctuations for those interested in capturing tidal energy for commercial purposes. Second, these simulations can be used to compute tidal ranges and surges such as tsunamis and hurricanes caused by extreme earthquake and storm events. This information can be used in the development planning of coastal areas. Finally, the shallow water hydrodynamic model can be coupled to a transport model in considering ow and transport phenomenon, thus making it possible to study remediation options for polluted bays and estuaries, to predict the impact of commercial projects on sheries, to model freshwater-saltwater interactions, and to study allocation of allowable discharges by municipalities and by industry in meeting water quality controls. The 2-dimensional shallow water equations are obtained by depth (or vertical) averaging of the continuum mass and momentum balances given by the 3-dimensional incompressible Navier-Stokes equations. Shallow water equations can be used to study

ow in uid domains whose bathymetric depth is much smaller than the characteristic length scale in the horizontal direction. We denote by (x; t) the free surface elevation over a reference plane and by hb(x) the bathymetric depth under that reference plane so that H(x; t) = + hb is the total water column (see Figure 1). Also, we denote by u(x; t) = [u(x; t) v(x; t)]T the depth-averaged horizontal velocities. Letting U = uH, the 2-dimensional governing equations, in operator form [12], are the primitive continuity equation (CE) L(; u; hb) @ @t + rU = 0; This work was supported in part by National Science Foundation, Project No. DMS-9408151. Department of Computational and Applied Mathematics-MS 134; Rice University; 6100 Main Street; Houston, TX 77005-1892 z Center for Subsurface Modeling - C0200; Texas Institute for Computational and Applied Mathematics; The University of Texas at Austin; Austin, TX 78712. 1 y

2

Chippada, Dawson, Martnez, Wheeler Fig. 1. De nition of elevation and bathymetry ξ

hb

Bottom sea bed

and the primitive non-conservative momentumequations (NCME), as derived by Westerink et al [26], M(; u; ) @t@ u + (ur) u + bf u + k fc u +gr ? H U ? H1 ws + F = 0; where = (hb ; bf ; fc ; g; ; ws; pa ; ). In particular, bf (; u) is a bottom friction function, k is a unit vector in the vertical direction, fc is the Coriolis parameter, g is acceleration due to gravity, is the horizontal eddy diusion/dispersion coecient, ws is the applied free surface wind stress relative to the reference density of water, and F = (rpa ? gr), where pa (x; t) is the atmospheric pressure at the free surface relative to the reference density of water, and (x; t) is the Newtonian equilibrium tide potential relative to the eective Earth elasticity factor. We will treat bf ; ws; fc ; pa and as data. Moreover, we will treat the diusion coecient as a constant. The primitive conservative momentum equations (CME) are derived from the NCME as Mc H M + uL = 0: The numerical procedure used to solve the shallow water equations must resolve the physics of the problem without introducing spurious oscillations or excessive numerical diusion. Westerink et al [26] note a need for greater grid re nement near land boundaries to resolve important processes and to prevent energy from aliasing. Permitting a high degree of grid exibility, the nite element method is a good candidate. There has been substantial eort over the past two decades in applying nite element methods to the CE coupled with either the NCME or the CME. Early nite element simulations of shallow water systems were plagued by spurious oscillations. Various methods were introduced to eliminate these oscillations through arti cial diffusion [17, 22]. These methods were generally unsuccessful due to excessive damping of physical components of the solution. Recently, Agoshkov et al [2, 4, 3] have investigated a nite element approximation, where the velocity eld is approximated by piecewise linear polynomials and the elevation is approximated by the same functions plus some additional ones. They have studied the eects of various boundary conditions, and proven stability of various time discretization schemes for a continuity equation-momentum equation system. In this paper, we will examine a nite element approximation to a modi ed shallow water model described below. Computational

Finite Element Approximations to a System of Shallow Water Equations

3

and experimental evidence in the literature suggest that this formulation leads to approximate solutions with reduced oscillations. Moreover, these approximate solutions have accurately matched actual tidal data. This modi ed shallow water model is based on a reformulation of the CE, which we now describe.

1.1. Historical Development of the Wave Continuity and Generalized Wave Continuity Equations. In 1979, Lynch and Gray [14] derived the wave con-

tinuity equation (WCE) from the mass and momentum conservation equations, c W(; u; ) @L @t ? rM + L = 0; as a means to eliminate oscillations without resorting to numerical damping. Here, (x; t) is a non-linear friction coecient. In this shallow water formulation, the WCE is then coupled to either the CME or the NCME. The equivalence of this model to the more standard one based on the CE is discussed in [12]. This formulation has led to the development of robust nite element algorithms for depth-integrated coastal circulation models. The WCE approach has motivated a substantial computational and analytical eort [6, 8, 14, 18]. Using Fourier phase/space analysis of the linearized WCE-CME and of the WCE-NCME system of equations, Foreman [8] and Kinnmark [12] prove that the WCE formulation suppresses spurious oscillations of the numerical solution, and is capable of capturing \2x" waves. The WCE formulation has also motivated substantial eld applications; see [9], [10], [11], [15], [16], [19], [20], [21], [24], [25]. These studies have demonstrated the advantage of the WCE formulation for nite element applications in terms of achieving both a high level of computational accuracy and eciency. The generalized wave continuity equation (GWCE) [12] is essentially the same as the WCE except that multiplication of the continuity equation by is replaced with multiplication by some general function that may be independent of time. Westerink and Luettich [13] chose to replace by a time-independent positive constant . Their version of the GWCE is given by @ 2 + @ ? r r 1 U 2 ? ( ? )U + (k f U ) (1) bf c @t2 @t H +Hgr + r @ @t ? ws + H F = 0:

This choice of yields a system of time-independent matrices when the GWCE is discretized in time using a three-level implicit scheme for linear terms. (Here and in the equations below we have used tensor notation reviewed in Appendix A.) The GWCE can be coupled to the CME, given by 1 @ U (2) @t + r H U 2 + bf U + (k fc U ) + Hgr ? U ? ws + H F = 0; or to the NCME. A nite element simulator based on the GWCE-NCME, which uses same-order polynomials to approximate elevation and velocity unknowns, has been developed by Luettich, et al. In [13], it was demonstrated that the approximations generated by this simulator accurately matched tidal data taken from the English Channel and southern North Sea. To date, no formal convergence analysis of nite element approximations to the WCE or GWCE combined with either the NCME or the CME exists in the literature. In this paper, we analyze the coupled GWCE-CME system of equations for a continuous time Galerkin nite element approximation.

4

Chippada, Dawson, Martnez, Wheeler

The rest of this paper is outlined as follows. In section 2 we detail the assumptions we will need in our analysis. We also introduce the weak formulation associated with the GWCE-CME system of equations. In section 3, we introduce the nite element approximation to the weak solution. In the derivation of an error estimate, we investigated various types of projections, such as the L2; elliptic, and parabolic projections. Because of the highly nonlinear nature of the coupled system of equations (1)-(2), we found these projections all led to suboptimal estimates. To that end, we present, in section 4, the simplest derivation of an a priori error estimate based on an L2 projection.

2. Preliminaries. 2.1. Notation and De nitions. For the purpose of our analysis, we de ne some notation used throughout the rest of this paper. Let be a bounded polygonal domain in IR2 and x= (x1; x2) 2 IR2: The L2 inner product is denoted Z by ' ! dx; '; ! 2 [L2( )]n; ('; !) =

where \" here refers to either multiplication, dot product, or double dot product as appropriate. We denote the L2 norm by jj'jj = jj'jjL2 ( ) = ('; ')1=2 : In IRn; = (1; : : :; n) is an n-tuple with nonnegative integer components, @ 1 @ n D = D11 Dnn = @x 1 @xnn 1 P n and jj = i=1 i : For ` any nonnegative integer, let H` f' 2 L2( ) j D ' 2 L2 ( ) for jj `g be the Sobolev space with norm 0 11=2 X jj'jjH` ( ) = @ jjD jj2L2 ( ) A : jj`

Additionally, H10 ( ) denotes the subspace of H1 ( ) obtained by completing C01 ( ) with respect to the norm jjjjH1 ( ) ; where C01 ( ) is the set of in nitely dierentiable functions with compact support in : Moreover, let W1` f' 2 L1 ( ) j D ' 2 L1( ) for jj `g be the Sobolev space with norm jj'jjW1` ( ) = jmax jjD ' jjL1 ( ) : j` For relevant properties of these spaces, please refer to [1]. Observe, for instance, that H` are spaces of IR-valued functions. Spaces of IRn valued functions will be denoted in boldface type,2 butPtheir norms will not be distin2( )]n has norm jj'jj = n jj'i jj2; H1 ( ) = [H1 ( )]n guished. Thus, L2 ( ) = [ L i=1 P P has norm jj'jj2H1 ( ) = ni=1 jj1 jjD 'i jj2; etc. For X, a normed space with norm jj jjX and a map f: [0; T] ! X, de ne


jjf jj2L2((0;T);X) =

jjf jjL1((0;T);X)

ZT

5

jjf(; t)jj2X dt; 0 = sup jjf(; t)jjX : 0tT

Finally, we let K; Ki ; (i = 0; 1; 2; ::) and be generic constants not necessarily the same at every occurrence.

2.2. Variational Formulation. We will consider the coupled system given by the GWCE-CME described in Section 1, with the following homogeneous Dirichlet boundary conditions for simplicity (x; t) = 0; x 2 @ ; t > 0; (3) u(x; t) = 0; and with the compatible initial conditions 9 (x; 0) = 0 (x); = @ (x; 0) = 1 (x); x 2 ; (4) @t ; u(x; 0) = u0(x); where @ is the boundary of IR2 and = [@ . Extensions to more general land and sea boundary conditions will be treated in a later paper. As noted in Kinnmark [12], the condition necessary for the solution of the GWCE-CME system of equations to be the same as the solution of the primitive form is that 1(x) = ?rU (x; 0): The weak form of this system is the following: For t 2 (0; T], nd (x; t) 2 H10 ( ) and U (x; t) 2 H10 ( ) such that @2 @ 1 U 2 ; rv + ((bf ? )U ; rv) (5) @t2 ; v + @t ; v + | r H {z } (a)

@ + (k fc U ; rv) + (Hgr; rv) + r @t ; rv ? (ws ; rv) + (H ; rv) = 0; F

(6)

8v 2 H10 ( ); t > 0;

@U 1 2 @t ; w + | r H{zU ; w } + (bf U ; w) + (k fc U ; w) + (Hgr; w)

(b) + (rU ; rw) ? (ws; w) + (H F ; w) = 0;

8w 2

with initial conditions ((x; 0); v) = (0 (x); v); 8v 2 H10 ( ); 1 ( @ (7) @t (x; 0); v) = (1 (x); v); 8v 2 H0 ( ); (U (x; 0); w) = (U 0 (x); w); 8w 2 H10 ( ): Here, we have set U 0 = u0 H0:

H

1 ( ); t > 0; 0

6


2.3. Some Assumptions. Our analysis requires that we make certain physically reasonable assumptions about the solutions and the data. First, we assume for (x; t) 2 (0; T] A1 the solutions (, U ) to (5)-(7) exist and are unique, A2 9 positive constants H and H such that H H(x; t) H , A3 the velocities u(x; t); v(x; t) are bounded, A4 @x@ i hb (x) is bounded. Assumption A2 is obvious from Figure 1. Dimensional analysis as explained in [23] accounts for assumption A3. Second, we assume that for (x; t) 2 (0; T] A5 9 positive constants and such that ghb (x) ; A6 9 non-negative constants and such that bf ; A7 (bf ? ) is bounded, A8 9 non-negative constants f and f such that f fc f , A9 is a positive constant, A10 rpa(x; t) and r(x; t) are bounded. Finally, we make the following smoothness assumptions on the initial data and on the solutions: A11 0(x); 1(x)12 H10 ( ), A12 U 0(x) 2 H0( ), A13 H(x; ) 2 H101( ) \ H`( ) \ W11 ( ); t 2 (0; T), A14 U (x; ) 2 H0 ( ) \ H` ( ) \ W11 ( ); t 2 (0; T), where ` is a positive integer de ned below. 3. Finite Element Approximation. 3.1. The Continuous-Time Galerkin Approximation. Let T be a quasiuniform triangulation of into elements Ei; i = 1; : : :; m; with diam(Ei ) = hi and h = maxi hi : Let S h denote a nite dimensional subspace of H10 ( ) de ned on this triangulation consisting of piecewise polynomials of degree at most (s1 ? 1). De ne H( ) = H10 ( ) \H` ( ), and assume S h satis es the standard approximation property (8)

inf jjv ? 'jjHs0 ( ) K0 h`?s0 jjvjjH` ( ) ; v 2 H( );

'2S h

and the inverse assumptions jj'jjH`?s0 ( ) K0 jj'jjL2 ( ) h?(`?s0 ) ; 9 = jj'jjL1 ( ) K0 jj'jjL2 ( ) h?1 ; (9) ' 2 S h ( ); ; ? 1 jjr'jjL1 ( ) K0 jjr'jjL2 ( ) h ; for 0 s0 k; s0 ` s1 ; with k de ned from 0 k (s1 ? 1); and where K0 is a constant independent of h and v. We de ne the continuous-time Galerkin approximations to ; U to be the mappings h (x; t) 2 S h ; U h (x; t) 2 S h for each t > 0 satisfying @2 @ 1 h h (10) h 2 ; rv + ((bf ? )U h ; rv) @t2 ; v + @t ; v + | r Hh U {z } (a0 )

+ (k fc U h ; rv) + (Hh grh ; rv) + r @@th ; rv ? (ws; rv) + (Hh F ; rv) = 0; 8v 2 S h ( );


@U

(11)

@t

7

1 r H U h 2 ; w + (bf U h ; w) + (k fc U h ; w) + (Hh grh ; w) h{z } | 0

h;w +

(b ) + (rU h ; rw) ? (ws; w) + (Hh F ; rv) = 0;

8w 2 h ( ); S

with boundary conditions ( x; t) = 0; h (12) x 2 @ ; t > 0; U h (x; t) = 0; and with initial conditions (h (x; 0); v) = (0 (x); v); 8v 2 S h ( ); (13) ( @@th (x; 0); v) = (1 (x); v); 8v 2 S h ( ); (U h (x; 0); w) = (U 0 (x); w); 8w 2 S h ( ): Here, Hh (x; t) = hb (x) + h (x; t).

4. A Priori Error Estimate. We will compare our nite element approximations h and U h , satisfying (10){(13), to L2 projections e and Ue satisfying

e ( ? )( ; t); v = 0; e

(14)

(U ? U )(; t); w

8v 2 S h ; t 0; 8w 2 h ; t 0:

= 0;

S

For the purpose of succinctness in the rest of the paper, we de ne (15)

(

= ? e; e; = U ?U

= =

h ? e;

e

Uh ? U:

Clearly, ? h = ? and U ? U h = ? : We shall call and the projection

errors and we shall call and the ane errors.

The following results are standard. Lemma 4.1. Let 0 s0 k; s0 ` s1 ; 0 k (s1 ? 1); and H( ) = 1 H0 ( ) \ H` ( ): Let 2 L2 ((0; T); H( )) and U 2 L2 ((0; T); H( )) and let (e; Ue ) be the corresponding L2 projections de ned by (14). And let and be de ned as above. If for some integer | 0 then and

@ | 2 L2 ((0; T); H( )) ; @ | U ; 2 L2 ((0; T); H( )) ; @t| @t|

@ | e 2 L2 ?(0; T); S h ( ) ; @ | Ue 2 L2 ?(0; T); S h ( ) ; @t| @t|

? @ | @t L ((0;T);Hs( )) K0hq?s ? @t@ | L ((0;T);Hq ( )) ; ? @ | @t L ((0;T);Hs( )) K0hq?s ? @t@ | q L ((0;T);Hq ( )) 2

2

2

2

8


for some constant K0 independent of ; q; h; `; where q = min(`; s1 ):

We will also need the following result. e and their rst-order spatial derivatives are bounded above in Lemma 4.2. e; U 1 1 L ((0; T); L ( )) by a positive constant K . Proof. See [5] Corollary 4.8.9. Before proceeding, it will be necessary to make certain assumptions about the Galerkin approximations. We employ an argument similar to that made in [7] to handle nonlinearities. In particular, we will assume that the Galerkin approximations are bounded by some constant in order to derive the a priori error estimate. Then we will show for suciently small h, in the case of polynomials of degree at least two (s1 3), that we can remove the estimates' dependence on the assumed bound of the approximations, being dependent instead on a smaller bound on the comparison projections. To that end, we assume that, given K de ned in Lemma 4.2, 9 positive constants C H2 and C 2K such that B1 C Hh(x; t) C ; and B2 jjU h jjL1 ((0;T);L1 ( )) C : The following lemma will be needed when we bound the right-hand sides of (18), (19), and (20). Lemma 4.3. Let Assumptions A2, B1 hold. There exists constants K1 (C ); K2(C ) such that

Proof .

U U ? h K1 (jjjj + jj jj) + K2 (jjjj + jjjj) : H Hh

U ? U h = U (Hh ? H) + (U ? U h) H H Hh 1 U HHh jjHh ? H jj +

Hh L1 ( ) jjU ? U h jj

HHh L1 ( ) = K1 jjh ? jj + K2 jjU ? U h jj : Assumptions A2, B1 are used to get the rst part of the inequality and assumption B1 is used to get the second part of the inequality.

4.1. Error Estimate. In order to obtain an error estimate for ( ? h ) and (U ? U h ), we must rst obtain an estimate on the ane error terms (h ? e) and (U h ? qe). Then, with the approximation result stated in Lemma 4.1 and with the estimate on the ane error to be obtained in the proof of Theorem 4.4, an application of the triangle inequality will yield an estimate for ( ? h ) and (U ? U h ).

It will be useful to employ the following expansion of terms (a){(b) in (5){(6): U r H1 U 2 = UH rU + (rU ) H ? (rhb U ) HU2 ? (r U ) HU2 : Similarly, the expansion of terms (a'){(b') in (10){(11) gives U 1 h 2 r H U h = H rU h + (rU h ) UHh ? (rhbU h ) HU h2 ? (rh U h ) HU h2 : h h h h h


Subtract (5) from (10) and (6) from (11), using the fact that we can write

Uh

U Uh e e rU Hh r(U h ? U ) ? H rU + H h h r ? U r(U ? U e ) ? U ? U h rUe = U H H H H

U h U U = Hh r ? H r ? H h

h

? UHh rUe ; h

U h + r Ue U r (U h ? Ue ) UHh ? (r U ) H Hh h U Uh U Uh e = (r ) H ? (r ) H ? (r U ) H ? H ;

h

h

r(h ? e) U h HU h2 ? (r U ) HU2 + (re U h ) HU h2 h (h 2 2) U ? UHh ; = (r U h ) U h2 ? (r U ) HU2 ? re H

Hh

h

and

Hh gr(h ? e) ? Hgr + Hh gre = Hh gr ? Hgr( ? e) ? (H ? Hh )gre = Hh gr ? Hgr ? gre + gre:

Consequently, we obtain the following GWCE-CME error equations: (16)

@2

@ U h r ; rv + (r) U h ; rv ; v + ; v + @t 2 Hh Hh @t + (r U h ) U h2 ; rv + ((bf ? ); rv) + (k fc ; rv) + (Hh gr ; rv) @ Hh + r @t ; rv + ( F ; rv) U U U = H r ; rv + H ? Hh rUe ; rv + (r) U ; r v H h

( 2 2) ! Uh U e ? ; rv + (r U ) H ? H ; rv + rhb H Hh h ( 2 2) ! U U + (r U ) H 2 ; rv + re H ? UHh ; rv + ((bf ? ); rv) e h e + (k fc ; rv) + (Hgr; rv) + gr ; rv ? gr ; rv @

U

Uh

+ r @t ; rv + (F ; rv);

8v 2 S h ( ); t > 0:

9

10 (17)


@ U

Uh Uh h @t ; w + Hh r ; w + (r) Hh ; w + (r U h ) Hh 2 ; w + ( ; w) + (Hh gr ; w) + (r; rw) + ( F ; w) bf ; w) +(k fc U Uh U e = U H r ; w + H ? H rU ; w + (r) H ; w

U

h

Uh

+ (rUe ) H ? H ; w + rhb h

U

+ re

( 2 2) ! Uh U H

? H h

( 2 2) ! Uh U

;w

; w + (bf ; w) H ? Hh + (k fc ; w) + (Hgr; w) + gre; w ? gre; w + (r; rw) + (F ; w) 8w 2 S h ( ); t > 0: + (rU ) H 2 ; w

4.1.1. Choice of Test Functions and their Manipulation. We now choose special test functions to obtain Rthe ane error estimate. Let rRbe a positive constant to be chosen. We let v1 (; t) = tT e?rs (; s)ds and v2 (; t) = tT e?rs @t@ (; s)ds be the test functions in (16). And, we let w = be the test function in (17). First, we will investigate the use of v1 and v2 as the test functions in (16) followed by temporally integrating over (0; T]. Note that v1 (; T)=0, recall 2v2(; T)=0. ?Also ? 1 1 d d @& rt that given &, the following relations hold: @t ; & = 2 dt jj& jj and 2 dt e jj& jj2 = 1 e?rt d jj& jj2 ? r e?rt jj& jj2 : 2 dt 2 Now, consider the rst two terms of (16). When v = v1 ; we obtain, upon integrating by parts, Z T @ @v1 @ T Z T @ Z T @2 dt = ? ; v ; @t dt + @t ; v1 = e?rt @t ; dt 1 2 @t @t 0 0 0 Z0T d Z T r 1 = 2 dt e?rt jj jj2 dt + 2 e?rt jj jj2 dt 0 0 Z T = 21 e?rT jj (; T)jj2 + 2r e?rt jj jj2 dt 0 and ZT Z T @v1 Z T @ ; v1 dt = ? ; @t dt = e?rt jj jj2 dt: 0 0 0 @t We also have from by parts in time: diusion term upon integrating Z Tthe ZT @ r @t ; rv1 dt = e?rt jjr jj2 dt: 0

0

We are also able to manipulate part of the eighth term in (16) by using the de nition of v1 as follows: Z ZT T d (ghb rv1; rv1) dt (ghb r ; rv1) dt = ? 21 ert dt 0

0

ZTd Z T p 2 2 p r 1 rt = ? 2 dt e ghb rv1 dt + 2 ert ghb rv1 dt 0 0

11


ZT r

2 2 jjrv1 (x; 0)jj + 2 ert jjrv1jj2dt: 0 Similarly, using v = v2 as the test function in (16) followed by temporally integrating over (0; T] yields

Z T @2

@t2 ; v2 dt = ?

0

Z T @

Z T @ @v2 Z T @ 2 ?rt @t ; @t dt = e @t dt; 0

0

Z T @v2 ZT ; @t dt = e?rt ; @@t dt 0 0 Z T = 2 e?rT jj (; T)jj2 + 2r e?rt jj jj2dt:

@t ; v2 dt = ?

0

0

The rst equality below follows from the de nition of v2 : Z T @ ZT d r @t ; rv2 dt = ? 2 ert dt (rv2 ; rv2) dt 0 Z0T d ZT rt jjrv2jj2 dt + r ert jjrv2jj2 dt e = ?2 2 0 0 dt Z T 2 rt = 2 jjrv2 (x; 0)jj2 + r 2 0 e jjrv2jj dt:

The temporal integration of terms @@t ; ; (bf ; ) ; (r; r) in (17) are straightforward. R

4.1.2. Bounding the GWCE Error Equations. Using v1(; t)= tT e?rs (; s)ds

as the test function in (16), integrating in time over (0; T], and using the relations above yields ZT (r + 2 ) 1 ? rT 2 (18) 2 e jj (; T)jj + 2 e?rt jj jj2 dt + 2 jjrv1(x; 0)jj2 0

ZT ZT

2 rt +r 2 e jjrv1jj dt + e?rt jjr jj2dt 0 0 Z T U h

Z T

Uh r ; rv1 dt ? ( r ) Hh Hh ; rv1 dt 0 0 ZT Z T + (r U h ) U h2 ; rv1 dt ? ((bf ? ); rv1) dt H

?

ZT 0

0

?

h

(k fc ; rv1) dt ?

0

0

Z T U

(h gr ; rv1) dt +

ZT 0

(

F

; rv1) dt

e + H r ; rv1 dt + 0 H ? Hh rU ; rv1 dt 0 Z T U Uh Z T e + ; r v ? H ; rv1 dt dt + (r ) U ( r U ) 1 H H 0 0 ( 2 2) ! Z Th ZT U Uh U Z T U

ZT

+

0

rhb

Uh

H ? Hh

; rv1 dt +

0

(rU ) H 2 ; rv1 dt

12 + +

?


ZT

re

0

ZT

H ? Hh

(k fc ; rv1) dt +

Z0T

!

( 2 2) Uh U

; rv1 dt +

ZT

ZT 0

(Hgr; rv1) dt +

((bf ? ); rv1) dt

Z T

gre; rv1 dt

Z0 T @ Z0 T gre; rv1 dt + r @t ; rv1 dt + ( ; rv1) dt

0

0

F

0

= (A1 + + A7) + (P1 + + P14) :

Here, A denotes an ane error term and P denotes a projection error term. We will explore the nonlinear terms appearing in the right-hand side of (18) in more detail. It will be implicitly understood that we use either the the Holder Inequality or the Arithmetic Geometric Mean Inequality or both in the treatment of the ane and projection error terms. We will explicitly mention additional justi cations in the derivation of bounds for these terms. From Assumptions A9, B1 B2, there exists K3 = K3 (; C; C ) and K4 = K4 (; C; C ) with which we obtain the following bounds on A1, A2 and on A3 :

Z T U h

ZT

ZT

0

0

r ; rv1 dt e?rt jjrjj2dt + K3 ert jjrv1jj2 dt; H h Z T0 Z T0 Z0T Uh 2 A2 = ? (r ) H ; rv1 dt e?rt jjrjj dt + K3 ert jjrv1jj2 dt; A1 = ?

A3 =

Z T0 0

(r

U h)

h

Uh

Hh

2 ; rv1 dt

ZT

ZT e?rt jjr jj2dt + K4 ert jjrv1jj2 dt:

0

0

From Assumption A7, there exists K5 = K5 jjbf ? jjL1 ((0;T);L1 ( )) such that we obtain a bound on A4 :

A4 = ?

ZT

ZT

ZT

0

0

0

((bf ? ); rv1) dt K

e?rt jjjj2dt + K5

ert jjrv1jj2 dt:

From Assumption A8, there exists K6 = K6 (f ) such that we obtain the following bound on A5 :

A5 = ?

ZT

ZT

ZT

0

0

0

(k fc ; rv1) dt K

e?rt jjjj2 dt + K6

ert jjrv1 jj2 dt:

From Assumptions A9, B1, there exists K7 = K7 (; C ) such that bound for A6 is as follows:

A6 = ?

ZT

ZT

0

0

(h gr ; rv1 ) dt

e?rt jjr

jj2dt + K

7

ZT 0

ert jjrv1jj2 dt:

From Assumption A10, there exists K8 = K8 ; jjF jjL1 ((0;T);L1 ( )) such that the bound for A7 is as follows:

ZT

ZT

ZT 2 ? rt A7 = ( F ; rv1) dt e jj jj dt + K8 ert jjrv1jj2 dt: 0 0 0


13

From Assumptions A2, A3, there exists K9 = K9 UH L1 ((0;T);L1 ( )) so that we obtain the following upper bound on the projection error terms P1 and P3:

Z T U

ZT

2 2 rt H r ; rv1 dt K9 jjrjjL2 ((0;T);L2 ( )) + K 0 e jjrv1jj dt; 0 ZT Z T U ; rv1 dt K9 jjrjj2L2 ((0;T);L2 ( )) + K ert jjrv1jj2 dt: P3 = (r ) H

P1 =

0

0

From Lemma 4.3, and Lemma 4.2, there exists K10 = K10 ( ; K1 ; K2; K ) such that the bounds for P2 and P4 are as follows: ZT ZT Z T U U h 2 2 ? rt ?rt e P2 = H ? H rU ; rv1 dt K e jjjj dt + e jj jj dt 0

+K

P3 =

h

ZT

Z T

0

e?rt jjjj2 dt + K

0

ZT 0

e?rt jjjj2dt + K10

0

ZT 0

ert jjrv1 jj2dt;

Z

T Uh 22 dt K jj jj ; r v ? e?rt jj jj2 dt (rUe ) U 2 ( )) + 1 L ((0;T); L H H h 0 0 ZT 2 2 +K jjjjL2 ((0;T);L2 ( )) + K jjjjL2 ((0;T);L2 ( )) + K10 ert jjrv1jj2 dt: 0

From Lemma4.3, Lemma 4.2 and Assumptions A2, A3, A4, B1, B2, there exists K11 = K11 ; K1; K2; jjrhbjjL1 ((0;T);L1 ( )) ; C; C such that the bound for P5 is as follows: ( ) !

P5 =

ZT 0

Z T

rhb

U 2 U 2 h H ? Hh

U

U

; rv1 dt

U

? U U h ; rv dt = rhb H ? H H + H + h UHH 1 h h h 0 ZT K jjjj2L2 ((0;T);L2 ( )) + e?rt jj jj2dt + K jjjj2L2 ((0;T);L2 ( )) 0 ZT +K jjjj2L2 ((0;T);L2 ( )) + K11 ert jjrv1jj2dt: 0 Uh

Uh

u From Assumption A2,A3, there exists K12 = K12 H L1 ((0;T);L1 ( )) such that we obtain the following upper bound on the projection error term P6 : Z T ZT U 2 2

P6 =

0

(rU ) H 2 ; rv1 dt K jjrjj2L2 ((0;T);L2 ( )) + K12

0

ert jjrv1jj2 dt:

From Lemma 4.3, Lemma 4.2, and Assumptions A2, A3, B1, B2, there exists K13 = K13 ( ; K1 ; K2; C(; C ; K ) such that ) the!bound for P7 is given by

P7 = =

ZT 0

Z T 0

re

re

U 2 U h 2 H ? Hh

U

Uh

H ? Hh

U

; rv1 dt Uh

U

H + Hh +

hU ? U U h HHh

; rv1 dt

14

K


ZT

e?rt jjjj2dt +

0

+K

ZT

ZT 0

e?rt jjjj2 dt + K13

0

jj2dt + K

e?rt jj

ZT

ZT 0

e?rt jjjj2dt

ert jjrv1jj2 dt:

0

Term P8 is treated similarly to A4 and term P9 is treated similarly to A5 if, for both, we treat like . From Assumption A2, there exists K14 = K14 jjHgjjL1 ((0;T);L1 ( )) such that we obtain the the following upper bound on the projection error term P10: ZT ZT 2 P10 = (Hgr; rv1 ) dt K jjrjjL2 ((0;T);L2 ( )) + K14 ert jjrv1 jj2dt: 0

0

From Lemma 4.2, there exists K15 = K15 (K ) and K16 = K16 ( ; K ) such that we obtain the following bounds on P11 and on P12 : ZT ZT Z T 2 ? rt e gr ; rv1 dt K e jjjj dt + K15 ert jjrv1jj2dt P11 = 0

P12 = ?

Z T

0

ZT

gre; rv1 dt e?rt jj 0

0

jj2 dt + K

0

ZT

16

0

ert jjrv1jj2 dt:

Obtaining a bound on P13 is straightforward: @ 2 ZT Z T @ + K17 ert jjrv1jj2 dt: P13 = r @t ; rv1 dt K r @t 2 0 0 L ((0;T);L2 ( )) Finally, term P14 is treated similarly to A7 :

R

Using v2 (; t)= tT e?rs @t@ (; s)ds as the test function in (16), integrating in time over (0; T], and using the relations above yields Z T 2 ZT (19) e?rt @@t dt + e?rT 2 jj (; T)jj2 + r 2 e?rt jj jj2 dt 0 0 Z T + 2 jjrv2 (x; 0)jj2 + r 2 ert jjrv2jj2 dt 0

Z T U h

Z T

Uh r ; rv2 dt ? ( r ) Hh Hh ; rv2 dt 0 0 ZT Z T + (r U h ) U h2 ; rv2 dt ? ((bf ? ); rv2) dt H

=?

ZT 0

0

?

h

(k fc ; rv2) dt ?

Z T U

ZT

0

0

Z T U

(Hh gr ; rv2) dt +

ZT 0

(

F

; rv2) dt

Uh e r ; rv2 dt + ? H H Hh rU ; rv2 dt 0 0 Z T U Uh Z T e + ; r v dt + (r ) U ( r U ) H 2 H ? H ; rv2 dt

+

( 2 02) ! Z Th ZT U Uh U + rhb H ? H ; rv2 dt + (rU ) H 2 ; rv2 dt h 0 0 0

15


+ +

ZT

H ? Hh

0

ZT

!

( 2 2) Uh U

re

(k fc ; rv2) dt +

; rv2 dt +

ZT

Z0T

0

ZT 0

(Hgr; rv) dt +

((bf ? ); rv2) dt

Z T 0

gre; rv2 dt

Z T @ Z T ? gre; rv2 dt + r @t ; rv2 dt + ( ; rv2) dt 0 0 0 : = Ac1 + + Ac7 + Pc1 + + Pd 14 F

The terms on the right-hand side of the inequality are handled as in (18). Note that term Ac6 diers from A6 by one term. From Assumptions A2, A9, there exists K18 = K18 (C Z) such on Ac6 is as follows ZT T that the the upperZ bound T Ac6 = ? (Hh gr ; rv2) dt e?rt jjr jj2 dt + K18 ert jjrv2jj2dt: 0

0

0

4.1.3. Bounding the CME Error Equation. Using w = as the test function in (17) followed by integration in time over (0; T], yields (20) 21 jj(; T)jj2 + jjrjj2L2 ((0;T);L2 ( )) + pbf 2L2 ((0;T);L2 ( ))

Z T U h

Z T

Uh r ; dt ? ( r ) Hh Hh ; dt + 0 0 ZT ZT ? (Hh gr ; ) dt + ( ; ) dt

?

0

Z T

U h)

Uh

Hh

2 ; dt

e + H r ; dt + 0 H ? Hh rU ; dt 0 Z T U Uh ZT U e + ; dt (r U ) ? (r ) H ; dt + 0 ) ! H Hh 0 ( U 2 U 2 ZT Z T U h Z T U

F

0

0

(r

Z T U

+ +

0

ZT

rhb re

0

+

Uh

H ? Hh

; dt +

( 2 2) ! Uh U

ZT

H ? Hh

(k fc ; ) dt +

Z0T

; dt +

ZT

(Hgr; ) dt +

(rU ) H 2 ; dt

0

ZT 0

(|{z; }) j dt

0 by (14)

Z T

gre; dt

Z0 T Z 0T gre; dt + (r; r) dt + ( ; ) dt ? 0 0 0 = Af1 + Af2 + Af3 + Af6 + Af7 + Pf1 + + Pf7 + Pf9 + + Pg 14 : F

The terms on the right-hand side of the inequality are handled as in (18). Again, observe that term Af6 diers from A6 by one component and is thus treated similarly to Ac6 in (19). Also, the treatment of term Pg 13 diers slightly from the treatment of the corresponding terms in the previous equations:

Pg 13 =

ZT 0

(r; r) dt K19 jjrjj2L2 ((0;T);L2 ( )) + jjrjj2L2 ((0;T);L2 ( )) :

16


4.1.4. Bounding the Sum of the Error Equations. We observe that the

right-hand-side of (18) can be bounded by (21)

ZT

ZT ZT e?rt jj jj2dt + e?rt jjr jj2 dt + Kv1 ert jjrv1jj2 dt + K jjjj2L2 ((0;T);L2 ( )) 0 0 0 + jjrjj2L2 ((0;T);L2 ( )) + K jjjj2L2 ((0;T);L2 ( )) + K jjrjj2L2 ((0;T);L2 ( )) 2 2 2 +K r @ @t L2 ((0;T);L2 ( )) + K jjjjL2 ((0;T);L2 ( )) + K jjrjjL2 ((0;T);L2 ( ))

where Kv1 = Kv1 (K1 ; : : :; K17) : The right-hand-side of (19) can be bounded by (22)

ZT

ZT ZT 2 2 ? rt ? rt e jj jj dt + e jjr jj dt + Kv2 ert jjrv2jj2 dt + K jjjj2L2 ((0;T);L2 ( )) 0 0 0 +K jjrjj2L2 ((0;T);L2 ( )) + K jjjj2L2 ((0;T);L2 ( )) + K jjrjj2L2 ((0;T);L2 ( )) @ 2 + K jjjj2L2 ((0;T);L2 ( )) + K jjrjj2L2 ((0;T);L2 ( )) +K r @t L2 ((0;T);L2 ( ))

where Kv2 = Kv2 (K1 ; : : :; K6 ; K8; : : :; K18) : Finally, the right-hand-side of (20) can be bounded by (23)

ZT 0

ZT ZT e?rt jj jj2 dt + e?rt jjr jj2dt + Kw ert jjjj2dt

0 0 2 2 +K jjjjL2 ((0;T);L2 ( )) + jjrjjL2 ((0;T);L2 ( )) + K jjjj2L2 ((0;T);L2 ( )) +K jjrjj2L2 ((0;T);L2 ( )) + K jjjj2L2 ((0;T);L2 ( )) + K jjrjj2L2 ((0;T);L2 ( ))

where Kw = Kw (K1 ; : : :; K4; K8 ; : : :; K16; K18; K19) : Now choose r = max f2Kv1 = ; 2Kv2 =g ; such that r1 = r 2 ? Kv1 0

and

r2 = r 2 ? Kv2 0: Then, summing (18), (19), and (20), using the above choice for r; and using bounds (21), (22), and (23) yields ZT (24) r(2+ 1) e?rt jj jj2 dt + 2+ 1 e?rT jj (; T)jj2 Z T 0 @ 2 Z T + e?rt @t dt + 2 e?rt jjr jj2dt + 2 jjrv1(x; 0)jj2 0 0 {z } | A

ZT ZT + 2 jjrv2 (x; 0)jj2 +r1 ert jjrv1jj2 dt +r2 ert jjrv2jj2 dt | {z } | 0 {z } | 0 {z } B

C

D


+ 21 jj(; T)jj2 + pbf 2L2 ((0;T);L2 ( )) + 2 jjrjj2L2 ((0;T);L2 ( )) @ 2 2 2 K jjjjL2 ((0;T);L2 ( )) + K jjrjjL2 ((0;T);L2 ( )) + K r @t L2 ((0;T);L2 ( )) 2 2 2 +K jjjjL2 ((0;T);L2 ( )) + K jjrjjL2 ((0;T);L2 ( )) + K20 jjjjL2 ((0;T);L2 ( ))

17

where K20 = Kw erT + K. From (24), we have jj(; T)jj2 2K jjjj2L2 ((0;T);L2 ( )) + 2K jjrjj2L2 ((0;T);L2 ( )) (25) 2 2 +2K r @ @t L2 ((0;T);L2 ( )) + 2K jjjjL2 ((0;T);L2 ( )) +2K jjrjj2L2 ((0;T);L2 ( )) + 2K20 jjjj2L2 ((0;T);L2 ( )) : Applying Gronwall's Lemma n to2 (25) yields 2 (26) jj(; T)jj 2K21 K jjjjL2 ((0;T);H1 ( )) ) @ 2 2 + K jjjjL2 ((0;T);H1 ( )) ; +K r @t L2 ((0;T);L2 ( )) where K21 = e2K20 T : We now return to (24) to bound it above and below. Let 1 = minf; r( + 1)g; observe that terms and use (26) to obtain, " A; B; C; D are all non-negative, 2 (27) 21 e?rT ( + 1) jj (; T)jj2 + 2 @@t + 1 jj jj2L2 ((0;T);H1 ( )) L2 ((0;T);L2 ( )) i p 2 2 + jj(; T)jj + 2 bf L2 ((0;T);L2 ( )) + jjrjj2L2 ((0;T);L2 ( ))

n ) @ 2 2 + jjjjL ((0;T);H ( )) : + r @t L ((0;T);L ( ))

(1 + 2K20 K21K)K jjjj2L2 ((0;T);H1 ( )) 2

2

2

1

Now, letting = min f ( + 1); 2; 1; 1; g and multiplying through by 2erT ; we obtain @ 2 2 (28) jj (; T)jj + @t + jj jj2L2 ((0;T);H1 ( )) L2 ((0;T);L2 ( )) 2 + jj(; T)jj + pbf 2L2 ((0;T);L2 ( )) + jjrjj2L2 ((0;T);L2 ( )) ( ) @ 2 2 2 K22 jjjjL2 ((0;T);H1 ( )) + r @t + jjjjL2 ((0;T);H1 ( )) L2 ((0;T);L2 ( ))

rT with K22 = 2e (1+2K20K21 K)K : Observe that K22 depends on r; s1; T, and on K ; C; C .

Use the approximation stated in Lemma 4.1, result @ ; jjjjL2 ((0;T);H1 ( )) K0 h`?1 ; jjjjL2 ((0;T);H1 ( )) ; r @t L2 ((0;T);L2 ( ))

18


to obtain

@ (29) jj (; T)jj + @t + jj jjL ((0;T);H ( )) L ((0;T);L ( )) + jj(; T)jj + pbf L ((0;T);L ( )) + jjrjjL ((0;T);L ( )) K23 h`?1 2

2

2

2

p

1

2

2

2

where K23 K0 K22 . Finally, applying the triangle inequality to the projection error and to the ane error (29) yields the following error estimate. Theorem 4.4 (A Priori Error Estimate). Let 0 s0 k; s0 ` s1 ; 0 k (s1 ? 1); and let H = H10 ( ) \ H` ( ). Let (; U ) be the solution to (5)-(7). 1 ( );, Let (h ; U h ) be the Galerkin approximations to (; U ). If (t) 2 H( ) \ W1 1 h h U (t) 2 H( ) \ W1 ( ); for each t; if h (t) 2 S ( ), U h (t) 2 S ( ) for each t; and suppose that assumptions A2{A10 and B1, B2 hold; then, 9 a constant s1 ; r; K ; C; C ) such that K = K(T; (30)

@ ( ? h ) + jj ( ? h )(; T) jj + jj ? h jjL ((0;T);H ( )) @t L ((0;T);L ( )) + jj (U ? U h )(; T) jj + pbf (U ? U h ) L ((0;T);L ( )) 2

2

2

2

1

2

`?1: + jjrU ? rU h jjL2 ((0;T);L2 ( )) Kh

Moreover, for h suciently small and s1 3 then

jjh jjL1 ((0;T);L1 ( )) + jjU h jjL1 ((0;T);L1 ( )) < 2K C ;

and

h > H2 C: Thus, the dependence of K on C ; C is removed.

4.1.5. Boundedness of Approximations. The proof of the theorem is now complete in the case of linears since we assumed that B1, B2 holds for this case. We now complete the proof of the theorem in the case of at least quadratic polynomials (s1 3). From the inverse assumptions, the boundedness of the L2 projection and ane error estimate (29), we obtain jjU h jjL1 ((0;T);L1 ( )) U h ? Ue L1 ((0;T);L1 ( )) + Ue L1 ((0;T);L1 ( ))

K0 h?1 U h ? Ue L1 ((0;T);L2 ( )) + K K0 h?1K23 h`?1 + K

= K24h`?2 + K

For h suciently small, viz, h`?2 < KK24 , we get

jjU h jj2L1 ((0;T);L1 ( )) < 2K C :

The upper bound for Hh (x; t) is shown similarly.


19

To get the lower bound for Hh (x; t), use Assumption A2, inverse assumptions, and estimates on the ane and projection errors. Hh = H ? (H ? Hh ) = H ? + H ? K25h`?2 : For h suciently,small, viz. h`?2 < 2KH25 ; we get Hh > H2 C:

Thus for the case of quadratics and higher, there exists a K bounded above independent of C ; C: This completes the proof of the theorem. 5. Conclusions. We have analyzed a full nonlinear coupled GWCE-CME system of equations. Making physically-realistic assumptions, we derived an a priori error estimate for the Galerkin nite element approximation to the solution of GWCE-CME system of equations, in weak form, by using an L2 projection. This led to a suboptimal estimate. That is, if we use continuous, piecewise polynomials of degree s1 ? 1 to approximate the elevation and velocity unknowns on a mesh with grid-spacing h, then the approximations tend to the solutions of the weak form like hs1 ?1. To our knowledge, our error analysis of a system of shallow water equations is the rst of its kind. 6. Acknowledgments. The authors thank Martha Carey and Amr Elbakry for their valuable feedback, as well as the referees for their helpful comments.

20


A. Review of Tensor Notation. Let '; 2 IR2.

The dyadic product is de ned as (' )ij = 'i j : Thus, '2 ' ' 1 2 1 '2 = '2 '1 '22 : The dot product P of a tensor with a vector is the usual matrix-vector multiplication result [M $ ]i = j Mij $j : scalar product (or double-dot product) of two tensors is de ned as S:T = P The T S ij ij : i;j j The gradient of a vector is de ned as fr'gij = @' @xi : For example,

r' =

@'1 @x1 @'1 @x2

@'2 @x1 @'2 @x2

!

:

P

The divergence of a tensor is de ned as [rS]j = i @x@ i Sij : Thus,

0 r r' = @

! @ 2'21 + @ 2 '21 1 4'1 @x1 @x2 A = @ 2'22 + @ 2 '22 4'2 : @x1 @x2 2 2 2 2 Observe that (r':r') = @x@ 1 '1 + @x@ 2 '1 + @x@ 1 '2 + @x@ 2 '2 and 2 2 ('2 '1 ) + @x@2@x ('1 '2) + @x@ 222 '22: that r(r'2) = @x@ 221 '21 + @x@1@x 2 1 REFERENCES [1] R. A. Adams, Sobolev Spaces, vol. 65 of Pure and Applied Mathematics, Academic Press, New York, 1978. [2] V. I. Agoshkov, E. Ovchinnikov, A. Quarteroni, and F. Saleri, Modi ed nite element approximation to shallow water equations and stability results, in Finite Elements in Fluids: New Trends and Applications, K. Morgan, E. O~nate, J. Periaux, J. Peraire, and O. C. Zienkiewicz, eds., Centro Internacional de Metodos Numericos en Ingeniera, Wales, 1993, Pineridge Press, pp. 1020{1025. , Recent developments in the numerical simulation of shallow water equations II: Tem[3] poral discretization, Mathematical Models and Methods in Applied Sciences, 4 (1994), pp. 533{556. [4] V. I. Agoshkov, A. Quarteroni, and F. Saleri, Recent developments in the numerical simulation of shallow water equations I: Boundary conditions, Applied Numerical Mathematics, 15 (1994), pp. 175{200. [5] S. C. Brenner and L. R. Scott, The Mathematical Theory of Element Methods, vol. 15 of Texts in Applied Mathematics, Springer-Verlag, New York, 1994. [6] J. Drolet and W. G. Gray, On the well-posedness of some wave formulations of the shallow water equations, Advances in Water Resources, 11 (1988), pp. 84{91. [7] R. E. Ewing and M. F. Wheeler, Galerkin methods for miscible displacement problems in porous media, SIAM Journal of Numerical Analysis, 17 (1980), pp. 351{365. [8] M. G. G. Foreman, An analysis of the \wave equation model" for nite element tidal computations, Computational Physics, 52 (1983), pp. 290{312. , A comparison of tidal models for the southwest coast of Vancouver Island, in Compu[9] tational Methods in Water Resources: Proceedings of the VII International Conference, MIT, USA, June 1988, M. A. C. et al, ed., Amsterdam, 1988, Elsevier. [10] W. G. Gray, A nite element study of tidal ow data for the North Sea and English Channel, Advances in Water Resources, 12 (1989), pp. 143{154.


21

[11] W. G. Gray, J. Drolet, and I. P. E. Kinnmark, A simulation of tidal ow in the southern part of the North Sea and the English Channel, Advances in Water Resources, 10 (1987), pp. 131{137. [12] I. P. E. Kinnmark, The Shallow Water Wave Equations: Formulation, Analysis and Applications, vol. 15 of Lecture Notes in Engineering, Springer-Verlag, New York, 1985. [13] R. A. Luettich, J. J. Westerink, and N. W. Scheffner, ADCIRC: An advanced threedimensional circulation model for shelves, coasts, and estuaries, Tech. Report 1, Department of the Army, U.S. Army Corps of Engineers, Washington, D.C. 20314-1000, December 1991. [14] D. R. Lynch and W. G. Gray, A wave equation model for nite element tidal computations, Computers and Fluids, 7 (1979), pp. 207{228. [15] D. R. Lynch and F. E. Werner, Three-dimensional hydrodynamics on nite elements, part II: Nonlinear time-stepping model, International Journal for Numerical Methods in Fluids, 12 (1991), pp. 507{534. [16] D. R. Lynch, F. E. Werner, J. M. Molines, and M. Fornerino, Tidal dynamics in a coupled ocean/lake system, Estuarine, Coastal and Shelf Science, 31 (1990), pp. 319{343. [17] P. W. Partridge and C. A. Brebbia, Quadratic nite elements in shallow water problems, ASCE Journal of Hydraulic Engineering, 102 (1976), pp. 1299{1313. [18] R. A. Walters, Numerically induced oscillations in nite element approximations to the shallow water equations, International Journal for Numerical Methods in Fluids, 3 (1983), pp. 591{604. , A model for tides and currents in the English Channel and southern North Sea, Ad[19] vances in Water Resources, 10 (1987), pp. 138{148. [20] , A nite element model for tides and currents with eld applications, Communications in Numerical Methods in Engineering, 4 (1988), pp. 401{441. [21] R. A. Walters and F. E. Werner, A comparison of two nite element models of tidal hydrodynamics using a North Sea data set, Advances in Water Resources, 12 (1989), pp. 184{193. [22] J. D. Wang and J. J. Connor, Mathematical modeling of near coastal circulation, Tech. Report 200, MIT Parsons Laboratory, Cambridge, MA, 1975. [23] T. Weiyan, Shallow Water Hydrodynamics, vol. 55 of Elsevier Oceanography Series, Elsevier, Amsterdam, 1992. [24] F. E. Werner and D. R. Lynch, Field veri cation of wave equation tidal dynamics in the English Channel and southern North Sea, Advances in Water Resources, 10 (1987), pp. 115{130. , Harmonic structure of English Channel/Southern Bight tides from a wave equation [25] simulation, Advances in Water Resources, 12 (1989), pp. 121{142. [26] J. J. Westerink, R. A. Luettich, A. M. Baptista, N. W. Scheffner, and P. Farrar, Tide and storm surge predictions in the Gulf of Mexico using a wave-continuity equation nite element model, in Estuarine and Coastal Modeling: Proceedings of the 2nd International Conference, e. a. Malcolm L. Spaulding, ed., New York, 1992, American Society of Civil Engineers.

finite element approximations to the system of shallow water equations

finite element approximations to the system of shallow water equations

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