ON A PROBLEM OF SHALLOW WATER TYPE

Transcription

1 24-Fez conference on Differential Equations and Mecanics Electronic Journal of Differential Equations, Conference 11, 24, pp ISSN: URL: ttp://ejde.mat.txstate.edu or ttp://ejde.mat.unt.edu ftp ejde.mat.txstate.edu (login: ftp) ON A PROBLEM OF SHALLOW WATER TYPE MOHAMED ELALAOUI TALIBI, MOULAY HICHAM TBER Abstract. In tis paper we present an existence teorem for a problem of sallow water kind. We take into account a general friction term depending on water dept and te norm of velocity, wic is te main difficulty. We present also a numerical study in te case wic we consider te above problem as a perturbation of sallow water equations in te non conservative dept-mean velocity form. 1. Introduction and setting of te problem Te two-dimensional sallow water equations (briefly SWE) are deduced by integrating, wit respect to dept, te continuity and te momentum equations of te tree-dimensional incompressible Navier-Stokes system, neglecting te influence of te vertical component of acceleration, te pressure is ten supposed ydrostatic [1]. Tey provide a model allowing to describe te flows of water in domains caracterized by small ratio between vertical and orizontal lengt scales, terefore typical pysical situations modelled are: tidal waves, currents in portual basins, lagoon,..etc. But teir use is surprisingly extended to very different penomena even wit discontinuous beavior, like te dam break problem [11]. Te sallow-water system we are studying in tis work reads u t + (u )u ν 1 u+c() u u + l u + g = f on Ω, (1.1) t ν 2 + (u) = f on Ω, (1.2) were u = (u 1, u 2 ) is te velocity vector and is te dept of studied layer, it can be considered as sum of te bottom topograpy wic is given and te topograpy of te free surface. Ω R 2 is te projection of te domain of te study on te orizontal plane. Γ denotes its boundary. l is te Coriolis force defined by (,, 2ωsin(φ)), were ω is te rotation rate of te eart and φ te latitude. g denotes te acceleration of te gravity. Te bottom friction effect is presented by te term C() u u were C(.) is a continuous function satisfying te condition C(.) < ε wic pysically justified by te Manning-Strickler s formula and by 2 Matematics Subject Classification. 35Q3, 76D3, 76B15, 76M1, 65M25, 65M6. Key words and prases. Sallow water equations; friction in te bottom; existence result; Galerkin metod; finite element metod; caracteristics metod. c 24 Texas State University - San Marcos. Publised October 15,

2 11 M. ELALAOUI TALIBI & M. H. TBER EJDE/CONF/11 te Cezy s one if te free surface elevation remain larger tan minimal level. ν 1, ν 2 are respectively te eddy viscosity and diffusivity coefficients wic we consider as an artificial viscosity taken, numerically, equal to zero to ave te sallow-water equations in te nonconservative dept-mean velocity form. Te rigt-and side terms f and f represent, respectively, te outside stress and te fluid excanges (rain, evaporation, etc.). To solve tese equations we take omogeneous boundary conditions and we set te initial data as (u, ) = (, ) on Γ, (u, )(t = ) = (u, ) in Ω Remark 1.1. To be compatible wit te pysical situation for wic te friction formulation is justified, we assume tat = B min > on Γ and () min > in Ω. However by setting := + L were L is te solution of te problem L t ν 2 L = L () = in Ω L = B on Γ. in Ω (As sown in [8], tis problem as a solution in L 2 (, T, H 1 (Ω)) L (, T, L (Ω)) for B L 2 (, T, H 1 2 (Γ)) L (, T, L (Γ))) we find again te omogeneous boundary conditions modulo a constant in te momentum equation and a linear term in te continuity one canging quit te reasoning done below. Terefore we will consider, for convenience, only te omogeneous case. 2. Notation and variational formulation We introduce te following functional spaces: V 1 = (H 1 (Ω)) 2,H 1 = (L 2 (Ω)) 2, V 2 = H 1 (Ω), H 2 = L 2 (Ω), V = V 1 V 2, H = H 1 H 2. Te norm and semi-norm defined on H 1 (Ω) are equivalent in V 1, V 2, and V. Ten we set u = u V1, = V2 and X = X V for u V 1, V 2, and X V. denotes te norm in L 2 (Ω), 2 denotes te Euclidean norm in R 2, (, ) is te scalar product in H 1, H 2 or H and (, ) 2 te scalar product in R 2. We define a 1 (u, v) = ν 1 ( u, v), a 2 (, β) = ν 2 (, β), a(x, Y ) = a 1 (u, v) + a 2 (, β) wit X = (u, ) and Y = (v, β). Note tat a 1, a 2 and a are bilinear continuous coercive forms, respectively, on V 1, V 2 and V. We denote by ε, ν, A, B, C, λ and θ constants suc tat: C(.) ε a(x, Y ) ν X Y for (X, Y ) V V λ >, C g = constant g, B = 2ν C g λ,

3 EJDE/CONF/11 SHALLOW WATER PROBLEM 111 and C = constant C G were C G is te best constant of te Gagliardo-Nirenberg inequality [2]: u 2 L 4 (Ω) 2 C G u u. (2.1) In wat follows we take omogeneous boundary conditions and we write (u. )u = 1 2 grad( u 2 2) + curl(u)α(u) were curl(u) = u2 x 1 u1 x 2 and α(u) = ( u 2, u 1 ). Now we can set te weak formulation of te problem: (V) Find (u, ) L 2 (, T, V ) L (, T, H) suc tat ( u t, v) + a 1 (u, v) + (curl(u)α(u), v) (grad u 2 2, v) (2.2) +(C() u 2 u, v) + (l u, v) g(div(v), ) = (f, v) ( t, β) + a 2 (, β) + (div(u), β) = (f, β) (v, β) V, (2.3) (u, )(t = ) = (u, ). (2.4) 3. Existence teorem Teorem 3.1. Assume tat F = (f, f) L 2 (, T, H), X = (u, ) V L (Ω) 3. Also assume te following conditions are satisfied, (1) B = 2ν C g λ > (2) X < B/C (3) (B/C) 2 > X λ F were constants are defined above. Ten te variational problem (V) admits at last one solution (u, ) in L 2 (, T, V ) L (, T, H). Te proof of te teorem is based on te tree next lemmas. Lemma 3.2. Let X = (u, ) be a classic solution of te problem (V ), on [, T ]. Under te same ypotesis in te teorem, we ave X L (,T,H) + (B C X L (,T,H)) X L2 (,T,V ) 1 λ F L 2 (,T,H) + X (B C X L (,T,H)) >. X L (,T,H) + X L 2 (,T,V ) constant Proof. By writing te energy inequality and using te ypotesis above, we find te result via Green s formula and Gagliardo-Neirenberg inequality. Lemma 3.3. Let (X n ) be a sequence of classic solution of (V ) on [, T ] satisfying X n L (,T,H) + X n L2 (,T,V ) C. (3.1) were C is a constant independent of n. Ten tere exist a subsequence also denoted by X n and X = (u, ) L (, T, H) L 2 (, T, V ) suc tat X n X weakly in L (, T, H), (3.2) X n X weakly in L 2 (, T, V ), (3.3) X n X strongly in L 2 (, T, H). (3.4)

4 112 M. ELALAOUI TALIBI & M. H. TBER EJDE/CONF/11 Proof. Statements (3.2) and (3.3) are immediate consequences of (3.1). On te oter and we can sow tat te sequence X n is uniformly bounded in te set Y = { v L 2 (, t, V ), v t L1 (, T, V ) }. According to [15], te injection of Y into L 2 (, T, H) is compact. Ten we can extract from X n a subsequence also denoted by X n suc tat we ave (3.4) Lemma 3.4. let (u n, n ) be a sequence converging toward (u, ) in L 2 (, T, H) strongly and L 2 (, T, V ) weakly. Ten for any ϕ(t) C 1 (, T ) and (v, β) V L (Ω) 3 we ave 1 2 (C( n ) u n u n, ϕ(t)v)dt (div( m u n ), ϕ(t)β)dt (curl(u n )α(u n ), ϕ(t)v)dt (grad u 2 2, ϕ(t)v)dt 1 2 (C() u u, ϕ(t)v)dt, (div(u), ϕ(t)β)dt, (curl(u)α(u), ϕ(t)v)dt, (grad u 2 2, ϕ(t)v)dt. We can proof tis lemma using Swartz inequality and appropriated Sobolev injections. Proof of te Teorem 3.1. Te proof is based on te construction of sequence of finite dimensional Problems (V n ) of wic te solutions (X n ) (by using lemmas 3.2 and 3.3) converge strongly in H and weakly in V to X (u, ) L 2 (, T, V ) L (, T, H). Ten by tird Lemma we can sow tat X is a solution of te problem. 4. Numerical studies Te goal of tis numerical studies is to know ow te solution of te problem varies wen te included artificial diffusivity coefficients ν 2 tend to zero. Te approac we are using ere is based on te finite elements for te space discretization and on te discretization of te Lagrangian derivative along te caracteristics. Tis metod provides a centred sceme wic ave te advantage of stabilizing te convection and allow large time steps to be taken wen compared to standard time-stepping metods [4]. Similar numerical scemes were considered in [13] for te incompressible Navier- Stokes problem. Witin te framework of te sallow water problems, tis approac combined wit te metod of te fractional steps, is adopted in [1] to simulate transcritical flows, and applied later in TELEMAC project [9]. Temporal discretization. Te caracteristic metods consists in approacing te lagrangian derivative of a function S in time step t n+1 by: ds dt (x, tn+1 ) S(x, tn+1 ) S(X(x, t n+1 ; t n ), t n ) (4.1) were X n = X(x, t n+1 ; t n ) is te position in te time step t n of te particle positioning at te geometrical point x in time step t n+1 and X n (x, t n+1 ; τ) is te

5 EJDE/CONF/11 SHALLOW WATER PROBLEM 113 solution of dx n dτ (x, tn+1 ; τ) = u n (X n (x, t n+1 ; τ)), for t n τ t n+1, X n (x, t n+1 ; t n+1 ) = x. Using (4.1), te semi-implicit time discretization of (1.1), (1.2) is u n+1 u n X n ν 1 u n+1, (4.2) C( n ) u n u n+1 + l u n + g n+1, = f n n+1 n X n ν 2 u n+1 + n u n+1 = f n, (4.3) were n and u n are te approximations of and u respectively in time step t n. Variational formulation. let us introduce te spaces V 1 φ = { v H 1 (Ω) H 1 (Ω); v = φ on Γ } V 2 η = { H 1 (Ω); = η on Γ }. Multiplying (4.2) and (4.3) by v V 1 and q V 2 respectively, and integrating by part on Ω we obtain (u n+1, v) ( + ν 1 u n+1, v ) + ( C( n ) u n u n+1, v ) g ( n+1, v ) = ( u n X n + f n l u n, v ), (4.4), q) ( + ν 1 u n+1, v ) + ( n u n+1, q ) = ( f n + n X n, q ). (4.5) ( n+1 Ten we write te time-discretized variational formulation as follows: (V) n Find (u n+1, n+1 ) in V 1 φ V 2 η were suc tat e(u n+1, v) + b(v, n+1 ) = (f n, v), v V 1, b(u n+1, n q) + e ( n+1, q)+ = (f n, q), q V 2. e(u, v) = 1 g t (u, v) + ν 1 g ( u, v) + 1 g (C(n ) u n u, v), e (, q) = 1 t (, q) + ν 2(, q), b(v, q) = (q, v), f n := 1 (f n + un X n ) l u n, g f n := f n + n X n. Finite element discretization. Let Vφ 1 and V η 2 (resp V 1 and V 2 ) two finite elements spaces approacing Vφ 1 and V η 2 (resp V 1 and V 2 ) suc tat te LBB condition is satisfied [3]. Ten te discrete problem is written as

6 114 M. ELALAOUI TALIBI & M. H. TBER EJDE/CONF/11 (V) n Find (un+1, n+1 ) in Vφ 1 V η 2 suc tat e(u n+1, v ) + b(v, n+1 ) = (f n, v ), v V 1, b(u n+1, n q ) + e ( n+1, q )+ = (f n, q ), q V 2. Te value X m (x) is approximated by X((n + 1) t, x), te solution of te problem dx dτ = un (X(τ), τ), X((n + 1) τ) = x, terefore, at eac time step we ave to solve te linear system ( ) (U ) ( ) A B FU B = D H F H were A and D are two definite positive matrices, and B, B are two matrices approacing operator of divergence type. We can sow easily (see for example [14] [7]) tat te problem (V) n (resp (V) n ) is well posed if n (resp n ) remain larger tan one level ξ >. Moreover in [6] and [1], a preconditionner of Caouet-Cabard kind [5] are proposed for te linear system. Numerical results. Te studied domain is a square wit 1km in lengt wit mean water elevation of 1m. We suppose tere is no excange wit te external medium and te surface stress is reduced to te wind stress tensor defined by f wind = 1 ρ water a wind u wind 2 u wind, ρ air were ρ water, ρ air are te density of te water and te air respectively, and a wind is an adimensional empiric coefficient. On te oter and, if we coose te Manning- Strickler s formula for te bottom friction we obtain were n is te Manning coefficient. C() = gn2 4 3 Table 1. Pysical parameters g(m/s 2 ) ρ water (kg/m 3 ) ρ air (kg/m 3 ) n ν 2 (m 2 /s) w (rad/s) φ () ν 1 (m 2 /s) (s) a wind (4.6) Conclusions. Altoug te continuous problem (2.2) requires a condition on it, we can take te diffusion coefficient of continuity equation ν 2 numerically as small as we want, witout any explosion of te solution (see figure 1). Ten for ν 2 = and f = we find te sallow water equations establised in [1]. Moreover for tis case we can prove formally by caracteristics tat te free surface elevation remain larger tan minimal level if te initial one it is. Terefore te coice of Manning- Strickler s formula (4.6) is justified and te numerical results are satisfactory. Acknowledgement. Tis work was supported by te Projet WADI (5t PCRD - Programmes IST et INCO-MED) and Action integrée CMIFM AI MA/4/94.

7 EJDE/CONF/11 SHALLOW WATER PROBLEM ν 2 EVO H ν2= ν2=.1 ν2=1 ν2= Figure 1. Te section (x,25m) for different values of ν Figure 2. Water elevation for ν2 = in t = 1s References [1] C. Bernardi and O. Pironneau; On te sallow water equations at low Reynolds number, Comm. Partial Differential Equations, 16 (1991), pp [2] H. Brezis; Analyse fonctionnelle.teorie et Application, Masson, Paris, [3] F. Brezzi, M. Fortin (1991), Mixed and ybrid finite element metod. Springer Series in Computational Matematics, Springer Verlag New York (1991). [4] C. N. Dawson, M. L. Martínez-Canales; Caracteristic-Galerkin approximation to a system of sallow water equation. Numer. Mat. 86 (2), p.p [5] Caouet, Cabart; Some fast 3-d finite element solvers for generalized Stokes problem, Rapport EDF HE/41/87-3 (1987). [6] F. Dabagi, A. El Kacimi, and B. Nakle; Quelques Préconditionneurs par Blocs pour des Problèmes de Point-Selle Issus d un Modèle Numérique d Eau Peu Profonde, TAMTAM 23 EMI, MAROC.

8 116 M. ELALAOUI TALIBI & M. H. TBER EJDE/CONF/ Figure 3. Velocity field for ν2 = in t = 1s [7] F. Dabagi, A. El Kacimi, and B. Nakle (23); A Priori error analysis of te caracteristicsmixed finite element metod for sallow water equations. IASTED International Conference on Modelling, Identification and Control, (MIC 23). [8] C. Fabre, J. P. Puel, and E. Zuazua; Approximate controlability for semilinear eart equation, Prov. Roy. Soc. Edinburg Sect. A, 124 (1994), pp [9] J. C. Galland, N. Goutal, J. M. Hervouet (1991); TELEMAC: A new numerical model for solving sallow water equations. Adv. Water ressources, 14 (3), p.p [1] N. Goutal, Résolution des équations de Saint-Venant en régime transcritique par une métode d éléments finis. Application aux bancs découvrants. Tèse de doctorat de l université de Paris VI (1987). [11] M. J-M Hervouet, Hydrodynamique des écoulements à surface libre, modélisation numérique avec la métode des éléments finis. T`se abilitation à diriger des recerces, Université de Caen/ Basse-Normandie (21). [12] J.-L. Lions, Equations diffrentielles oprationnelles, Springer-Verlag, New York, [13] O. Pironneau, On te transport-diffusion algoritm and its application to te Navier-Stokes equations. Numer. Mat. 83, p.p (1982). [14] O. Pironneau, Métodes des elements finis pour les fluides, Masson (1988). [15] R. Temam, Navier-Stokes Equations, Nort-Holland, Amsterdam, Departement de Matématiques, Faculté des Sciences Semlalia, Maroc address, M. ElAlaoui: elalaoui@ucam.ac.ma address, M. H. Tber: tber.icam@eudoramail.com