TICAM REPORT 96-34 August 1996
Finite Element Approximations to the System of Shallow Water Equations, Part II: Discrete Time A Priori Error Estimates
S. Chippada, C. N. Dawson, M. L. Martinez, and M. F. Wheeler FINITE ELEMENT APPROXIMATIONS TO THE SYSTEM OF SHALLOW WATER EQUATIONS, PART II: DISCRETE TIME A PRIORI ERROR ESTIMATES *
S. CHIPPADA!, C. N. DAWSON! M. L. MARTiNEZt. AND M. F. WHEELER!
Abst.ract. Various sophisticated finite element models for surface water flow exist in the lit.- erature. Gray. Kolar. Luet.tich, Lynch and 'vVesterink have developed a hydrodynamic model based on the generalized wave continuit.y equation (GWCE) formulation. and have formulat.ed a Galerkin finit.e element procedure based on combining the G WCE with t.he noncouservat.ive moment.um equa- tions. Numerical experiments suggest that this met.hod is robust .. accurat.e and suppresses spurious oscillat.ions which plague ot.her models. We analyze a Galerkin model which uses Ihe conservat.ive momentum equat.ions (C1VIE). For this GWCE-CME syst.em of equatious. we present.. for discrete time, an a priori error est.imat.e and a st.ability estimat.e based on au C} project.ion.
Key words. shallow water equat.ions. surface flow. mass conservat ion. momentum conservat.ion. finit.e element model. error estimat.e. stability
AMS subject. classificat.ions. 35Q35. 35L65 65N30, 65N15
1. Introduction. In this paper, we derive a-priori error estimates for a discrete- time finite element approximation to a model of shallow water flow. These estimates extend our continuous-time analysis given in [1]. We consider a shallow water model of Gray et al described in a series of papers beginning in [5]; see also [4]. We denote by ~(;r . t) the free surface elevation over a reference plane and by hb(X) the bathymetric depth under that reference plane so that H(;r. t) = (+ lib is the total water column. Also, we denote by U(;r. t), V(;I:. t) the depth-averaged horizontal velocities and let v = H[U \.:V. The formulation we consider consists of deriving a wave equation for the free surface elevation and combining that with the conservative momentum equations for velocities. The generalized wave continuity equation (GWCE) is given by
(1) ~tt+To~t-\7· [\7. (~VV) -ToV +K(v) + Hg\7( + Eh \76 - Tws + HF] = O.
Here, J\.~(v) = k x /"v + Tb.fV, where Tb.f(( U. V) is a bottom friction function, k is a unit vector in the vertical direction, and Ie is a Coriolis function. Additionally, To is a time-independent positive constant, 9 is acceleration due to gravity, Eh is the horizontal eddy diffusion/dispersion (constant) coefficient, Two< is the applied free surface wind stress relative to the reference density of water, and F = (\7pa - g\7I]), where Pa(x. t) is the atmospheric pressure at the free surface relative to the reference density of water, and 1](X, t) is the Newtonian equilibrium tide potential relative to the effective Earth elasticity factor. The GWCE can be coupled to the conservative momentum equations (CME), given by
• This work was supported in part by National Science Foundation. Project No. DMS-9408151. t Department. of Computational and Applied Mathematics - MS 134: Rice University: 6100 Main Street: Houst.on. TX 77005-1892 I Center for Subsurface Modeling - C0200; Texas Institute for Computational and Applied Math- emat.ics: The University of Texas at. Austin; Austin, TX 78712. 2 Chippada. Dawson. Martinez. Wheeler
(2) Vt+\7· (~VV) +K(v)+Hg\7~-Eh\7·\7V-Tw,,+HF = 0, or to the non-conservative momentum equations (NCME). In this paper we will con- sider the GWCE-CME formulation. For technical reasons, we have found that this model lends itself more easily to error analysis. A finite element simulator based on the GWCE-NCME formulation has been developed by Luettich, et al. In [4], it was demonstrated that the approximations generated by this simulator accurately matched tidal data taken from the English Channel and southern North Sea. The temporal discretization scheme we will consider adheres as much as possible to the scheme implemented in this simulator. 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 finite element approximation to the weak solution. In section 4, we derive an (/ priori error estimate based on a discrete £2 projection. We also state a stability estimate.
2. Preliminaries. For the purposes of our analysis, we define some notation used throughout the rest of this paper. Define the partition of J = [0. T] as - k - k ht - {t EJII =k:t::'.I. k=O ..... N} . .6.12:0. N.6.I=T.
Denote ht = ht - {O. T} and J~t = Jt,t - {OJ . For X, a normed linear space with norm II . IIx, and <": [0, T] -+ X, define C;k = c;(" Ik) and the following norms N 11c;II~p«O.N~t);X) = L IIc;k(x)lli- .6.t, 1 ::; p < 00. k=O
11C;lleOO«O.N~t );X)
11c;II~p((O.T);X) IT''C;(x.t)''~ dt. 1 ::; p < 00,
11c;II~oo((O,T);X) sup 11c;(x.t)ll~ dt . O Furthermore, define Otf(C;k) = (.6.t)-1(C;k+1 - C;k); Otc(C;k) = (2.6.t)-1(C;k+1 - C;k-l); at b ( c;k) = (.6.t )- 1(C;k _ C;k- 1); O;c(C;k) = (.6.tt2(c;Hl - 2C;k + C;k-l); and c;k+~ = (C;k+l +c;k)/2. Finally, we let J{ and (be generic constants not necessarily the same at every occurrence. 2.1. 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 (3) ~(x, t) = 0; U(x. t) = 0; V(x, t) = OJ x Eon. t > O. Syst.em of Shallow Wat.er Equat.iolls 3 and with the compatible initial conditions ~(X, 0) (o(x)j U(x. 0) Uo(;r)j } (4) = = n ~t(X. 0) = 6(:r)j V(x, 0) = VO(X)j ;r E . where 00 is the boundary of 0 C IR2 and n = 0 U 00. By compatible, we mean 0= -\7·vo, where va = Ho[Un VolT = (hb +~o)[Uo 'ioY. The weak form of this system in discrete time that we will consider is the following: For tk E J~t, find e'(;r) E 1-lb(O) and vk(x) E ?t6(0) such that O (5) ~t (o;ee, v) + To (Otbe, 11) + (\7. {~n (v )2} . \7V) - To (va, \7v) + (KO(vO). \711) + (!lhb\7~~, \711) + (~0!l\7~o. \711) + Eh (Otb\7e. \711) l - (Tw,< 0, \711) + (H() FO. \7v) + (pg. v) + To (pi . v) +~ (ghbdg. vv) + Eh (\7p? \711) = O. "Iv E HMO). 2 k k (6) (alee I v) + To (Oti~k. v) + (v. {l:k (v ):?} , vv) - To (v . V v) + (Kk(vk), \711) + (!lhb\7f,k+t. \7v) + (e'g\7e. \711) + Eh (OtiVe'. \7v) k - (Two (~O,v) (~o, v), "Iv E 1-lb(O), (8) (~-l, v) (e, v) - 2.6.t(~I' v). "Iv E Hb(O), (va. '111) = (va, '111), "1'111 E ?t6(0). Observe that (5) is simply (6) at initial time using the initial conditions on elevation. The second initial condition, in (8), on elevation allows us to specially handle the truncation error for the second-order time derivative approximation at time t = 0, in the spirit of [2]. In the above equations, we use the following truncation in time terms which we now define using Taylor's Theorem with integral remainder as follows: a ()O 2 ( 1) 1 r~t 2 P3 = ~tt + .6.t 6 - Otb~ = - (.6.t)2 Jo (.6.t - s) ~ttt ds 4 Chippada. Dawson. Martinez. Wheeler P~ k PI l.' d 1 k S REMARK: It should be noted that, in the simulator, a Crank-Nicolson approach is taken for solving the NCME. And thus, in the simulator, /31 = /32 = ~ and the NCME is centered at A: + ~except for advection terms which are treated explicitly. However, it will be clear that because of the explicit treatement of advective terms (and other forcing terms), eliminating tight coupling between the GWCE and CME, the best that can be achieved is a first-order in time scheme. To that end, we only consider the cases when diffusion terms in the CME are treated explicitly (/31 = 0,/32 = 1) or implicitly (/31 = 1,/32 = 0). 2.2. Some Assumptions. Our analysis requires that we make certain physi- cally reasonable assumptions about the solutions and the data. First, we assume for (x, t) E f2 X J~t. Al the solutions (~, v) to (6)-(8) exist and are unique, A2 ::3 positive constants H. and H' such that H* ::; H(x, t) ::; H*, A3 the velocities U(x, t). V(x, t) are bounded, A 4 \7 hb (x) is bounded. Now, as defined in [4], the bottom friction function is given by 2 c U V) _ JU + V2 = Ilv/ HIIl2 (9) Tb.f (<,.,. - c.f H - cf H ' where c.f is a friction coefficient. And, the Coriolis function is given by (10) le(x) - 2wsinQ', where w is the angular velocity of the earth in its daily rotation and Q'is the degrees latitude. Thus, A5 ::3 non-negative constants T. and T' such that T. ::; Tbf (~, U, V) ::; T*. A6 Ie (x) is bounded. Additionally, we assume that for (x, t) En x J~t' A 7 Eh is a positive constant, AS Pa(x, t) and its first spatial derivatives are bounded, A9 1](x, t) and its first spatial derivatives are bounded. Finally, we make the following smoothness assumptions on the initial data and on the solutions: Syst.em of Shallow Wat.er Equat.ions 5 A10 ~o(x) E 7-{A(O), All ~1(X) E 7-{(\(0), A12 vo(x) E 1t6(0), A13 H(x, t) E 7-{6(0) n 7-{i(O), t E ht, A14 v(x, t) E 1t6(0) n 1ti(O). t E ht, A15 ~(x. t) E 7-{2 ((0, T); 7-{1(0)) n 7-{4 ((0. T); .c2(0)), A16 ~ttt(x, t) E .coo ((0, .6.t); .c2(0)), A17 (tt(x.t) E.coo ((O . .6.t);7-{I(O)), A18 ~t(x.t) E.coo ((O,.6.t)j7-{I(O)), A19 v(x. t) E 7-{2 ((0. T); 7-{1(0)), where the integer e is defined in the next section. 3. Finite Element Approximation. 3.1. The Discrete-Time Galerkin Approximation. Let T be a triangula- tion of 0 into elements Ei• i = 1.. ,., m. with diam( Ei) = hi and II = max; hi, Let Sh denote a finite dimensional subspace of 'H6(O) defined on this triangulation consisting of piecewise polynomials of degree (051-1). Define 7-{(0) = 7-{A(O) n7-{i(O), and assume Sh satisfies the standard approximation property (11 ) and the inverse assumption o (12) 11 At time t = tk, let rr = hb + 2. Also, let f(Y) = kxl"Y + 'h.fY. where , IIY/rrlli" (13) Tb.f = cf rr . We define the discrete-time Galerkin approximations to ~A: (x), vk(x) to be the map- pings 2k (x) E Sh, yk (x) E Sh satisfying 1 (14) ~t (Otb21, v) + To (Otb21. v) + (ghb \72!. \7v) + Eh (Otb\72 • \7v) O = ~t ((1. v) - (\7. {~o(Y )2} , \7v) + To (yO, \7v) - (to(yO). \7v) - (20g\72°, \7v) + (TwsO. \7v) - (rrOFO, \7v), \;fv E Sh(O)j (15) (oic2k, v) + To (Oti2k, v) + (\7. {~k (yk f} ,\7v) - To (yk, \7v) + (,ek(yk), \7v) + (ghb \72k+!, \7v) + (2kg\72k, \7v) +Eh (Oti\72k, \7v) - (Tw/, \7v) + (rrkFk. \7v) = O. Vv E Sh(O), k 2: Ij (16) (Otbyk+1, w) + (\7. {;k (yk) 2} .w) + (tk(yk+~), w) + ( rr k g\7:::.,w~k) +Eh (13I\7Y k + 1 +132\7Y,\7'Wk) - ( Tws k ,W ) + (rrkFk,w) = 0, Vw E Sh(O), k:::: 0 6 Chippada. Dawson. Mart.inez. Wheeler with boundary conditions (17) 3k(;r.) = 0, o. x E Em, ~~> 0; and with initial conditions (~o. 11), '1v E Sh(O), (18) (31,11) - 2.6.t(~I' 11). '1v E Sh(O). (vo.1lJ), '1111 E Sh(O). As before, equation (14) arises from considering (15) at k = 0 and using the fictitious value 3-1 defined in (18). We also need to make certain assumptions about the discrete-time Galerkin ap- proximation similar to those made about the solution to (6)-(8). In particular, we assume that B 1 :3 positive constants I1* and I1* such that I1* ::; I1(x. I) ::;I1*, B2 Y is bounded. Observe that Bl and B2, in definition (13), imply B3 :3 non-negative constants f, and f* such that f* ::;fb.f ::; f' . 4. A-Priori Discrete-time Error Estimate. Following the analysis in [1], we will comp~re our finite element approximations (3. Y) satisfying (15)-(18) to .c2 pro- jections (~. v) satisfying ((e- - ~'k),p) o. '1VESh.k2:-l. (19) ((vk - vA'). 111) o. For the purpose of succinctness in the rest of the paper, we define Ok = e- - P' . 1/.,k 3k - p. , (20) A-,k k -k k yk-k { 'P = V -v X = -v . Clearly, e- - 3k = Ok - 1/..k and vk - yk = ~k - Xk. We shall call Ok and ~k the projection errors at time tk, and we shall call 'lj;k and X" the affine errors at time tk. Consequently, we can prove the following result. THEOREM 4.1. Let 0 ::; So ::; fI ::; Sl and e. Sl 2: 2. Let (e:, vk) be the solution to (6)-(8) and Sllppose that assumptions A2-A19 hold. Let (3", yk) be the Galerkin approximations to (e-, vk). If3k E Sh(O), yk E Sh(O) for each ~;:then. :3 a constant C = C(T. Sl. 1') such that 2 N+l_,;:;,N+lll -';:;'.. II k_,;:;,k 11 II ~ ~ + II ~ ~ Ile2((O,N~t);1tl(!1)) + Otb(~ ~) i2«(O.N~t);.[2(!1)) N 1 i 1 + II v + - yN+l II + II \7v - \7Y IlcClo.(N+l)M);.c1(!1)) ::; c: (h - + .6.t). for.6.t ::;min{l. "'1. "'2}, where /'i:l = 4/To. "'2 = h2 (7* h2 + CEh{32r1 . Observe that the time step restriction is less severe when the diffusion term in the momentum equation is completely implicit ({32 = 0). In particular, the time step dependence on the mesh size is nullified. Proof In order to obtain an error estimate for (e- - 3k) and (vk - yk), we must first obtain an estimate on affine error terms (3 - [k) and (yk - vk). Then, with the System of Shallow Water Equations 7 standard approximation results (stated in Lemma 4.1 in [1]) and with the estimate on the affine error to be obtained in the proof of Theorem 4.1, an application of the triangle inequality will yield an estimate for (~k - 3k) and (vk - yk). It will be useful to employ the following expansion of the advective terms in (5)-(7), at time t = tk, 1 ~} \7. { HV Similarly, the expansion of the advective terms in (14)-(16) gives, at time t = tk, Y) y Y 11·\7Y (\7.Y) 11 - (\711·Y) 11 " ( + 2 Subtract (5) from (14), (6) from (15), and (7) from (16), using the fact that we can write ~'\7(Y-V)- (;.\7v) + (~'\7V) (~'\7x) - (;.\7~) - ([; - ~] '\7V); \7.(y - v) Y - (\7·v) ~ + (\7.v) Y 11 H 11 = (\7·X)- Y - (\7.~)-v - (\7·v)_ [v- - -Y] . 11 H H 11' -)Y v - Y (\7(3-~)·Y 112 -(\7~·v) H2 + (\7(·Y) 112 = (\7~·Y) ~ - (\78·v) ;2 - \7[. { [; r-[~r}j and I1g\7(3 - ()- - H g\7( + I1g\7~- - - = I1g\7~ - H g\78 - 8g\7~ + ~g\7~. Consequently, we obtain the following GWCE-CME error equations: (21) (8;clj;k,v)+To(8ti~k.v)+ (~:'\7Xk,\7V) + ((\7'Xk)~:,\7V) + ((\7lj;k.Yk)(~k)2' \7V) - To (Xk, \7v) + (k.k(Xk). \7v) + (ghb \7~k+!, \7v) + (3kg\7~k, \7v) + Eh (Oti \7~,k, \7v) + (1/i Fk. \7v) k v k V -A, k v = ( HA'k .\7~ , \7v ) + ([ Hkk - Y11k] ·\7v, \7v ) + ( (\7.~) Hk'k) \7v 8 Chippacla. Dawson, Mart.inez. Wheeler k yA,] ) ( {( k) 2 (Yk ) 2} ) + ( [ ~k - Ilk (\7'l}), \7v + \7hb· ~k - Ilk, ' \7v + ("0' .v') (I~:)"",,) + ("f' {(~:)'- (~:n",,) k k -To (¢k. \7v) + (Kk(¢k), \7v) - (UC - Kk)(v ), \7v) + ((H g\70)k+~ . \7v) + ((k g\70k. \7v) + (Ok g\7[k, \7v) - (11}g\7[k. \7v) k + E h (ati \7 Ok, \7 v) + (OkF , \7 v) + (p~, v) + To (pi:, v) +~ (9hbd~. \7v) + Eh (\7pi:. \7v). "Iv E Sh(O), k 21; (22) (atbXk+1,w) + (~:'\7Xk.w) + ((\7oXk)~.w) + ((\71jl.y) (I1~)2' w) + (K:k(Xk+~), w) + (I1kg\7lj>k. w) +Eh (,81 \7Xk+1 + ,82\7Xk, \7w) + (1,k Fk. w) V k V -k k V' = -·yA..·.Wk ) + ([ ---.k yk ] ·\7V.W ) + ( (\70A.. )-,wk) ( Hk 'P Hk Ilk 'P Hk k v yk k vk - Y" + ([ Hk - Ilk ] (\7.V ),w ) + ( \7hb· { (Hk) .,- ( Ilk,')2} .w ) + ("Ok.v'),;:),,"') + ("f.{(;~',)'-(~:)'}",) + (Kk(¢k+~), w) - ((K:k - Kk)(vk+~), w) + (Hkg\70k, w) + (Okg\7[k,w) - (¢kg\7[k,w) +Eh (,81\7¢k+1+,82\7¢k,\7w) k + (OkF , w) + (q~, w) + ~ (Kk(d~), w) +Eh,81 (5k, \7w), Vw E Sh(O), k 2 o. For reasons that will be clear later in the proof, we write (21) separately at k = 0 as follows: Syst.em of Shallow Water Equat.ions 9 O 0 -To (4)0. \7v) + (KO(4>0), \7v) - ((A~O - KO)(v ), \7/1) + (H°,q\70 , \7v) + ((Og\700, vv) + (OOg\7[O. \7v) + Eh (ali \70°, \7v) + (oUf!, vv) + (pg. v) + To (p~, v) + ~(ghbdg. \7v) + Eh (Vp~. \7v). \;/v E Sh(O); We have used in (23) the fact that, from (8),(18), and (19), 1J'0 = 0,1/,-1 = 1/.1. XO = O. 4.0.1. Choice of Test Functions. We now choose the test functions employed to obtain the affine error estimate. Let l' be a positive constant to be determined. Let v = vi+l = 2:f=Hl e-rj~t'1jJj.6.t and v = v;+l = 2:f=Hl e-"j~t Otb1/.,i.6.t be the test functions in (21). Let v = 1/.,1and v = Otb1;} be the test functions in (23). And, let U7 = XHI be the test function in (22). First, multiply (21) and (22) by .6.t and sum from k = 0 to k = N using our the test functions above. Then manipulate (23) to derive upper bounds for 111/.,III7-lllf!) and for II0tb'1jJIII7-lI(f!) . Finally, after manipulating certain terms in (21) and (22), using the derived bounds on 111/..lII7-lI(f!) and on IIOtb'1jJIII7-l1(f!)' adding the resulting inequalities, ap- plying a generalized discrete Gronwall's inequality (GDGI), and taking bounds above and below, we will obtain a relation giving an estimate of the affine error. REMARK: In order to complete our estimate, we will need to define a function 2 k 2 Ak = e-d,~t 111/.,k11 + IILe-rj~'1/,j.6.t j=O 7-l' (f!) Moreover, we will need to add IIJahb v?W to both sides of the inequality obtained l from (21), using v = vi+ , after multiplying by .6.t and summing over k. We will then, after some work, obtain AN in which 2:~=0Ak.6.t will be hidden in the spirit of the GDGI (see Heywood and Rannacher [3]). 4.0.2. Bounding the first time-step ofthe GWCE Error Equation. First, use v = lp1 in (23) to obtain (24) Here, Pi denotes a projection error term. Terms PI - Pg and Pl1 - Pl5 are treated similarly to the corresponding terms in continuous time (using test function VI with r = 0) investigated thoroughly in [1]. We now investigate the remaining terms. In estimating PIa. recall (9) and use Lemma 4.2 and Lemma 4.3 in [1] to obtain PlO = (Ch/ - Tb"o)Vo, \7'1jJl) "yo/noli2 _ IIvO/HOlli2] -0 \7./,1) c.f ([ no HO v, 'f' O o H IIYo /nolli2 - nO Ilv / HOlli2] -0 1) C.f ([ nOHO v,\71/' 10 Chippada, Dawson. Martinez. Wheeler O O (H - ITO) II yO /IT lIi2 ~ 'J ([ II"H' II" (1Ir'/II"II" -livO/flOII,,)] va, \71jJl) + ITOHO (1jJ0-OO)IIYOlli2 ) ([ < c.f (ITO)2HO II all II¢Olli + IIxOlli"] va. \71/,1 (1/,0_00) v i2+ 2ITOHO + ITO(H°)2 ., < (11\71/,1 112 + J"IIOOI12\ + J<:lI¢oW, In estimating PIG, consider 0112 (.6.1.)2 112 (25) IIP3 ::; -S11 ~ttt COO((O.~t);£2(I1))' so that, In estimating Pl7 and PIg, consider, for instance, 2 0112 .6.t 2 IIPI < 311(ttllcoo((o.~t);C'(I1))' so that, l P17 = To (p~.1/)I) ::; (lI1jJ W + J<:(.6.t)211(ttll~oo«0,~t);£2(I1))' 1 Pl9 = Eh (\7p~, \71jJl) ::; (11\71jJ 112 + J<:(.6.t)211\7~ttll~oo«o.~t);£2(I1))' Finally, in estimating P18, consider so that, Consequently, we observe that the right hand side of (24) is bounded above by 2 2 0 2 0 (0 114,111 + (11\71/,111 + J<: 1100W + J<:11\70 11 + J{ 1I0ti \70 W +J{ 1I¢0112 + J{(.6.t)211~tttll~oo(0,~t);£2(I1)) + J{(.6.t)211\7~tll~oo«0,~t)P(I1)) +J{ (.6.t)2 II~tt II~oo«o,~t);£2(I1)) + J{ (.6.t)2 11\7~tt II~oo«(0,~t);£2(11)) . Let (0 = ~T; so that.6.t < T~ = /i:l· Now, let (To = e+l;2~t) - Eo> 0 and 'Y (Tl = 2' - ( > O. Thus, 1 (To 1I1jJ 1l2 + (Tl 1I\71jJl W + ~~11\71/,1W 0 2 < J{ IIOOW + !\'II\70 11 + J{ 1I0ti \700 W + J{ II¢OW +J( (.6.t)2 II~ttt II~oo«o,~t );L2(I1)) + J{ (.6.t)2 11\7~t lI~oo«0,~t);C2(11)) +!{ (.6.t)2 II(tt II~oo«0,~t);£2(11)) + !{ (.6.t)2 11\7~tt lI~oo((O,~t)P(I1)) . System of Shallow Wat.er Equat.ions 11 Finally, we have (26) Now, use v = Otb~)l in (23) to obtain (27) < i\ + .. ·p19, After some work similar to that above, we determine that the right hand side of (27) is bounded above by 2 0 2 f IIOtb1/>lll~ + f II0tb \71;:-I11 + !{ IIOOW + K II\70 W + [( II0ti \70°11 0 2 +J{ 114> 11+ K(.6.tf II(tttll~oo«o.~t);.c2(!1)) + I\'-(.6.f)211\7~tll~oc«(0,~t);.c2(!1)) + J{ (.6.f)2 II~tt lI~oc«0,~t);.c2(!1)) + J{ (.6.t)2 11\7~tt II~oo«0,M);.c2(!1)) , Let CY2= To - f > 0 and CY3= Eh - f > 0, Then, 2 (2 + ::.6.t) IIOtb~,1112 + 2';.t 11\71/>111+ CY3/IOtb\7~11112 2 2 < !{ IloOW +!{ 11\70011 +!( 1100\700112 + J{ 114>011 +J{(.6.t)211(tttll~oc«0~t);.c2(!1)) + [((.6.t)211\7~tll~oc«O.~t);C2(!1)) +K(.6.t)211(ttll~oc«o.~t);C2(!1)) + !\(.6.t)211\7~ttll~oc«o.~t);.c2(!1)) . Finally, we have (28) IIOtb4,11l~'(!1) + 11\74,11l~2(!1) ::; !( (h2(i-l) + .6.e) . 4.0.3. Summation by Parts and other Tools. Before proceeding in the treat- ment of (23), we introduce tools to be used in the next sections. First investigate the use of v~+I and v~+I as the test functions in (21) when we multiply by .6.t and sum over k. For a generic c; E 1{1(O), consider the following relations: 2 N . 1 N (e±rk~t IIc;'"11 _ e±r(k-l)~t IIc;k-1112) (A) Le±rk~t (OtbC;k.C;"').6.t = '2 L .6.t .6.t k=O k=O N N N +~ Le±"k~t IIc;k112+ ~ Le±"(k-l)~t IIc;A,-11l2 - Le±l'k~t (C;k-l.C;k) k=O k=O k=O N N = ~LOtb (e±d,~t IIc;kln.6.t + ~ Le±r(k-l)~t(l_ e±,·~t) IIc;k-1112 k=O k=O N 2 +~ L e±rk~t Il0tbC;k 11 (.6.t)2 k=O N N 2 = ~ L Otb (e±r"'~t IIc;k I n .6.t + ~ L e±rk~t(l - e±r~t) IIc;k 11 k=O k=O 12 Chippada. Dawson. Martinez. Wheeler +~e±r(-I)~t(1_ e±r~t) 11c;-IW _ ~e±rN~t(1_ e±I·~t) IIc;NI12 N 2 +~ L e±rk~t Il0tbC;k11 (.6.t)2 k=O N 2 = ~e±r(N+l)~t IIc;NI12 _ ~ 11c;-1112 =F ~eB Le±rk~t IIc;kI1 .6.t k=O N k 2 +~ Le±I'k~t IIOtbC; 11 (.6.t)2, () E (0. ±'I'.6.t). k=O where the last equality results from an application of the Mean Value Theorem. N . 1 N (e±r(k+l)~t II k+1112 _ e±d'~t II k112) (B) Le±rk~t (OtfC;k,C;k+l).6.t = 2' L c;.6.t C;.6.t k=O k=O N N -~ Le±r(k+l)~t lIc;k+1112 + ~Le±rk~t IIc;kW k=O k=O N N + L e±rk~t IIc;k+1112 - L e±rk~t (C;k.C;k+l) k=O k=O N N = ~ LOti (e±rk~t lIc;kln .6.t+ ~ Le±rMt(l- e±r~t) 11c;k+1112.6.t k=O k=O N 2 +~ L e±rMt Il0tiC;k 11 (.6.tf k=O 2 N = ~e±r(N+l)~t IIc;N+1112 _ ~ 11c;011 =F ~ee L e±rk~t IIc;k+1112 .6.t k=O N 2 +~ L e±rMt Il0tiC;k 11 (.6.t)2. () E (0, ±I'.6.t). k=O where the last equality follows from the Mean Value Theorem. And finally, N N N 2 (C) L e±rk~t (Oti C;k,C;k).6.t = ~ LOti (e±rk~t IIc;k' W) .6.t =F ~ee L e±"k~t 11c;k+111 .6.t k=O k=O k=O N -~ Le±rk~t IIOtiC;kI12(.6.t)2, () E (O,±r.6.t). k=O We understand the following to be true: v~'>N = 0, v~>N = O. Now, consider the first two terms of (21). When v = v~+l, we obtain N N L(o;c1/i,v~+I) .6.t - L(Otb1f'k,OtiV~').6.t- (OtblfJO,V~) k=O k=O System of Shallow Wat.er Equat.ions 13 N (~) ~e-r(N+l)~t II7/;NI12 _ ~ 117,l,-1112 + ~eBo Le-rk~t II7/;kI12.6.t 2 2 k=O N + ~ Le-"k~t IIOtb7/;k112(.6.tY - (Otblj},v?), (Jo E (-r.6.t,O) k=O N 2 1 2 2 - ~e-r(N+l )~t II7/;N 11 _ ~ 117/; 11 + ~eBo L e-"k~t II7/;k 11 .6.t k=O N + ~ L e-rk~t 1 I ()tb7,l,kW (.6.t)2 + (Otb7,l)I,v?). (JOE (-I'.6.t, O)j k=O N N N ') To "(::lL.J Uti .f.k"', 'VIk+l) utA -To L(7,lJk.Otbv~+I).6.t == To Le-rk~t II7/;kW.6.t. k=O k=O k=O The first equalities above result from temporal summation by parts. We also have from the diffusion term upon summing by parts in time: N N l 2 Eh L (Oti \77/;k, \7vi+ ).6.t = Eh L e-rk~t 11\77,l)k 11 .6.t. k=O k=O We are also able to manipulate the eighth term in (21) by using the definition of VI as follows: 1 N N 2' L(ghb \77/;k, \7v~+l).6.t -~ Lerk~t (ghb Otb(\7v~+I), \7v~'+1 ).6.t k=O k=O (!!) -~ tOti (erk~t 11~\7v~W).6.t k=O N 2 +~eB L e,'k~t IIJgh; \7v~+111 .6.t k=O 1 N ? -4 Lerk~t II0ti(~\7vnW .6.t2, (J E (O./'.6.t) k=O N B > : II\7v~112 + 1'~. e LerMt II\7v~+1112.6.t k=O 1 N 2 -4Le-rk~tll~\71/>kll.6.t2, (JE(O,r.6.t); k=O 1 N N l l 1 " rk~t ( h ::l ( k+l) n k) A 2 L(ghb \77/;k+ , \7vi+ ).6.t -2 L.Je 9 b Utb \7v1 ' v VI ut k=O k=O 14 Chippada, Dawson. Mart.inez, Wheeler (~) -~ tOti (er!:~t 11~\7v~W).6.1 k=O N ? +~e8 L er'k~t II~ \7v~+1 W .6.1 k=O 1 N 2 +4LeI'MtIIOti(~\7vi')1I .6./~. BE (O.I·.6.t) k=O N 8 > : II\7v~112+ 1'~'* e Lerk~t II\7v~+1112.6.1 k=O 1 N 2 +4 Le-I'k~t 11Jgh;;"\71jJkll .6.t2. BE (0.1'.6./). k=O Similarly, using v = v;+l as the test function in (21), we obtain N N L(O;c1j,k.v;+I).6.t = - L(OtbV,k,OtiV;).6.t- (Otb1jJo.v;) k=O k=O N 2 L e-rk~t IIOtb1jJk11 .6.1 + (Otb1jJl, Vn ' k=O where the first equality above results from summation by parts. N N To L (Oti 1jJk, v;+I).6.t -To Ler(k+l)~t (Otbv;+2, v;+I).6.t k=O .,=0 N -To Lerk~t (Otbv;+l, vn.6.t - To (Otb1j,O,vg).6.t k=O N (~) ~ IlvgW + 1';0 eOL er!:~t Ilv;+1112 .6.t + To (Otb1j)I, vg) .6.t !:=O N +~ L e-I'Mt IIOtb1jJkW (.6.t)2, BE (0, r.6.t); k=O N N 8 Eh L(Oti(\71jJk), \7v;+I).6.t ~h II\7vg W + r ~h e L el'k~t II\7v;+1112 .6.t + Eh (Otb \7'fP, \7vg) .6.t k=O k=O N + ~h L e-rk~t IIOtb(\71jJk)112 (.6.t)2, BE (0, 1'.6.t). k=O The two terms above are manipulated using the definition of the test function followed by an application of (C). Finally, algebraic manipulation of the following terms from (22) yields: Syst.em of Shallow \-Yat.er Equat.ions 15 N N 2 L (OtbXk+l'XHl).6.t = ~ L (1Ixk+lW -llxkl1 + IIOtbXk+1112 (.6.t)2) k=O k=O 2 N = ~ IlxN+111 + ~L IIOtbXk+1112 (.6.t)2; k=O t, (7'b/Xk+~.Xk+l).6.t = ~ t,lIvh/ xk+lW.6.t N +~ L ( VTb/ (XA,+1 - (Xk+l - Xk)), VTb/ Xk+l) .6.t k=O > ~t Ilvh/ XHlW .6.1 - ~ t IlvTb/ OtbXHlW (.6.1)3; k_O k_1I N N HI k k Eh L (131\7X + 132\7X , \7X + 1).6.t > Eh (131 + ~2 ) L IIvXHll12 .6.t k=O k=O N - E~2 L IIOtb(\7XHl)112 (.6.t)3 . k=O 4.0.4. Bounding the GWCE-Error Equations. Using v = v~+l as the test function in (21), multiplying by .6.t and2 summing from k = 0 to k = N, using the relations above, and adding 11..;gJlb v~ 11 to both sides of the resulting inequality yields N 2 (28) ~e-r(N+l)~tll1jJNI12 + (To + ~e80) Le-rMt 1I1/,kI1 .6.t + ~ lI\7v~112 k=O N N 2 + r~, e8 L erk~t II\7v~+1112 .6.t + Eh L e-rk~t 11\71/,k11 .6.t k=O k=O N 2 2 +~ L e-rk~t Il0tb1jJk 11 (.6.t)2 + 1* Ilv~ 11 k=O ::; ~ 114)1 W - (Otb1jJl, vn - t,((~:.\7Xk) , \7V~+I).6.t N( k Y k k+l ) N( k k Y k k+l ) -t; (\7·X ) Ilk' \7v1 .6.t - t; (\71jJ .y )(IIk)2' \7v1 .6.t N N k k +To L(X . \7v~+I).6.t - L(tk(X ), \7v~+l).6.t k=O k=O N N - L(=:kg\71/,k+!. \7v~+I).6.t - L(1jJkFk" \7v~+I).6.t k=O k=O 16 Chippada. Dawson. Mart.inez. Wheeler + t((;"k'\7¢k) ,\7v~'+I).6.t+ t([;: - ~:] .\7v",\7vi+l).6.1 ,,=0 k=O V k+l -" 1) Y k+l + {;N ( (\7.¢) Hk'k \7vl ) .6.t + {;N ( (\7·V) [kHk ~ Ilk"] ,\7vl ) .6.t N ( {[ V k]2' [k]2}Y k+l ) N ( k k 1) k k+l ) + {; \7hb• Hk - II"' , \7vl .6.t + {; (\70 'V ) (Hk)2' \7vl .6.t + {;N ( \7f'. {[ ;k]k 2 - [k]~k 2} .\7v~+1 ) .6.t~To{;(¢k.\7v~+I).6.tN N N + L (K"( ¢k). \7v~'+I).6.t - L ((,ICk - A:k)(vk), \7v~'+I).6.1 k=O k=O N N N + L (H" g\70k+~, \7vi+l).6.t + L (eg'VOk. \7v~:+I).6.t + L (Okg\7fk, \7v~+I).6.1 k=O k=O k=O N N N - L (1jJkg\7fk. \7vi'+I).6.t + Eh L (Oti \70", \7vi+1).6.t + L (OkFk. \7vi'+I).6.t k=O k=O k=O N N N "'( k k+l)A "'( k k+l)A 1",( I dk '\""7 k+l)A + L..- P2' VI ut + ToL..- PI' VI ut + 2 L..- g Ib 0' v VI ut k=O k=O k=O N +EhL(\7pr\7vi+l).6.t+'Y* Ilv?112 k=O = 51 + 52 + (AI + ... + A7) + (PI + . . + P2I) , Here, Ai denotes an affine error term and 5i denotes a term resulting from summation by parts. Terms Al - A7, PI - pg, and Pn - Pl6 are treated similarly to their continuous-time analogs which are investigated thoroughly in [1]. Again, we now investigate the remaining terms. The treatment of terms 51 and 52 is straightforward. From (26) and (28), 51 _ ~111jJ1112::;I\(h2(e-l)+.6.e); 52 _(Otb1jJl,vn ::;K(h2Ci-l)+.6.t2) +c1H112, The treatment of PIa is similar to that given in the previous section: N P-10 - L..-'" (('Tb.f k - Tb.f k)-kV, '\""7v VI'k+l) utA k=O System of Shallow Water Equations 17 N N N 2 < (Le-rk~t II1/i112 .6.t+KLllgkIl .6.t+KLllxk+1112.6.t k=O k=O k=O N ') N +]{LII¢kW .6.t+ [{Lerk~t II\7v~+11l2 .6.t, k=O k=O In estimating Pl8 and P20, consider, for instance, k 1 1 pi ::; 1- ~tl (i: +(tk+l_S)2dS) 2 (i:k+(~tt)2dS) 2 (~t) 'I' (['''''l'd') j so that 2 k+' IIpi'1I ::; .6.t it II~tt112ds . 3 tk Thus, N - _ "'( k k+l) A P 18 - To ~ PI' VI ut k=O < ,11,,111'+ l{ ~ lit,,-'ja,¢, llt N P20 = EhL(\7pi,\7vi+l).6.t k=O < '11'7,,111' + Ii ~ lit,'-".''7¢' llt The treatment of Pl9 follows similarly N l P19 = ~ L(ghbd~, \7vi+ ).6.t < ( II\7v~ W + ]{ {;N IIf;k e-"j ~t \71//:.6.t k=O 18 Chippada, Dawson. Mart.inez, Wheeler Finally, algebraic manipulation yields P2l = ,* Ilv~112 = ,* (t.e-'·Mt1j>k.6.f.t.e-rk~t1/,k.6.t) '> < f IIt.e-'·k~t1/,A'.6.tW + J{ t.111'-rk~t1/>kI12.6.1 N = fllv~112+J{Llle-rk~t1/,kI12 .6.t, k=O Using v = V;+1 as the test function in (21), multiplying by .6.t and summing from k = 0 to k = N, and using the relations above yields N N 2 (29) L e-rMt 110/b1/·kW .6.t + ~ L e-rk~t IIOtb1/,A:11 (.6.t)2 k=O k=O N 2 + ~' Le-rk~t IIOtb\71/.kIl (.6.t)2 + To Ilv~W + ~h II\7v~112 k=O N N +1';0I'BLerk~t Ilv;+1112 .6.t+ r~"eBLerMt II\7v~+IIl:!.6.t k=O k=(J ::; - (Otb1/>I,v~) - To (Otb1/>I,v~).6.t - E" (dtb\71/>I, \71J~).6.t k - LN (( ykIlk ·\7X ) , \7v;+1 ) .6.1 - L,N ( (\7'Xk) ykIlk' \7v;+1 ) .6.1 k=O k=O -" N( (\7.I,k.yk)~ k \7V!:+1) .6.1 + ~ "(vA,N \7vk+l).6.t ~ 'P (lIk)2' 2 '0 ~ A.' 2 ' k=O k=O N N N - L (kk(Xk), \7v;+1 ).6.t - L (lIkg\71/>k+&, \7v;+I).6.t - L (1/,kFk. \7v;+1).6.t k=O k=O k=O k + t.((~:.\7q)k) , \7v;+I).6.t + t.([~:-~:].\7v , \7v;+I).6.t k + t.((\7'q)) ~:' \7V~+l).6.t + t.((\7.V ) [ ~: - ~:] , \7v;+I).6.t N ( {[ v k] 2 [Yk] 2} , k+l ) N ( k, k V k k+l ) + {; \7hb• Hk - Ilk ,\7v2 .6.t + {; (\70 'V ) (Hk)2' \7v2 .6.t +{;N ( \7[". {[ ~kk] 2 - [YkIlk ] 2 } ,\7V;+1 ) .6.t-To{;(q)k,\7v;+I).6.tN N N + L (Kk( q)k), \7v;+I).6.t - L ((kk - 1\/')(vk), \7v;+I).6.t k=O k=O N N N + L (Hkg\70k+&, \7v;+I).6.t + L(eg'VOk. VV;+I).6.t + L (Okg\7[". \7v;+I).6.t k=O k=O k=O System of Shallow Water Equations 19 N N N -L (1// g\7[k. \7V~+I).6.t + Eh L (Oti 'VOk, \7v~+I).6.1 + L (O~'Fk, \7v~+I).6.1 k=O k=O k=O N N N "'(k k+l)A "'(k k+l)A E"'('I'7k'l'7k+l)A + ~ P2' V2 ut + To ~ Pl' V2 ut + h. ~ vPI' v 112 ut k=O k=O k=O u u All of these terms are treated analogously to those in (29) with the exception of 82.53 u u and P17 - Pl9 which we detail now. From (26) and (28), 2 2U 1 52 = -To (Otb'l/.,l. v~) .6.t ::; f IIv~ II + [{ .6.t (h - ) + .6.t:?) , 2 2U 1 2 53 -Eh (Otb\7'/pI, \7v~).6.t::; f lI'Vv~11 + A.6.t (h - ) + .6.1 ). u u Observe that the treatment of P17 - Pl9 differs slightly from the treatment of related terms in (28) as follows: N T+~t 2 N PI7=~P2·1I2U "'( k k+l)Aut < f'-(A)4\ ut ior II'<.,tttt11 dS+ f'-'"\~e rk~tll V2k+lll:? utA k=O a k=O +K(.6.t)311(tttll~oo((0,~t);C2(flJ) ; N T+Ll.t 2 N Pl9 = Eh L (\7Pt. \7v~+1 ).6.t ::; K(.6.t)21 11\7~tt11 ds + K L erk~t 11\7,/!~+1112.6.t. k=O k=O 4.0.5. Bounding the CME-Error Equation. Using w = XHI as the test function in (22) followed by summation in time, yields N 1 2 (30) ~ Ilx + 11 + ~~ IIOtbXHll12 (~t)2 + ~~ II/h/ x~'+1112 ~t N +Eh (,sl + ~2 ) L II\7XHI W .6.t k=O ::;:* tIIOtbXHII12(.6.t)3+ E~2 tIIOtb\7xHII12(.6.t)3 k=O k=O k Hl k Hl - ~ ( (~: .\7X ) , X ).6.t - ~ ((V'x ) ~:' X ).6.t N( k) N k - L, (\71/} ork) (~)2' Xk+l .6.t - ~ L (k X .feX , XHI ).6.t k=O k=O 20 Chippada. Dawson. Mart.inez. Wheeler N N - L(IIkgV1f,k,Xk+l).6.t - L(1f,kFk,Xk+l).6.t k=O k=O N N - L ((lCk - K:k)(vk+~). Xk+l).6.t + L (H~:g\70k. Xk+l).6.t k=O k=O N N + L (Okg\7fk. Xk+l).6.t - L (1/>kg\7fA'. :l+I).6.t k=O k=O N N l +Eh L (,81 \7 cj>k+l+ ,82\7 cj>k.\7Xk+ ).6.t + L((}~'Fk, Xk+J).6.t k=O k=O N N N l l + L (q~. Xk+l ).6.t + ~L (K:kd~:, Xk+ ).6.t + Eh,81 L (sk, \7Xk+ ).6.t k=O k=O k=ll =51 + 52 + (AI + ... + A5) + (PI +. ' .+ P17) . Again, all the terms on the right-hand side are treated analogously to the terms in (29) with the exception of the following terms. In estimating 52, we use the inverse assumption to obtain ~ E N 1 2 ,52 = 1L IIOtb\7xk+ 11 (.6.t)3 ::; k=O And the truncation terms are estimated as follows N Pl5x = """(L.J ql'k X k+l) ut1\ k=O x 1 N Pl6 = '2 L(K:kd~,Xk+l).6.t < k=O System of Shallow Wat.er Equat.ions 21 4.0.6. Bounding the SUlll of the Error Equations. We observe that the right-hand-side of (28) can be bounded by N 2 N 2 N (31) [ L e-rk~t Ilv,k 11 .6.t + [ L e-rk~t II\7v,k 11 .6.t + [ L II\7xk+1112 .6.t A,=O k=O k=O N k 2 2 k +[ Ilv~112+ [11\7v~112 + K L (ilo 11 + II\7ok 11 + II0ti \70 In .6.t k=O N 2 N ., 2U l 2 +K L IlcfJkll .6.t + K L II\7cfJkW .6.t +!{ (h - ) + .6.t ) k=O k=O , 4 ') , 2 ') + /\ (.6.t) II~ttttlIC2«(0,(N +1 )~t);L2(O)) + !i (.6.t) II~tt11;:2«(O.(N +l)~t );t.:"(O)) +!{(.6.t)211\7~ttll~2((O.(N+1)~t)L2(O) + K(.6.t)311~tttll~OO((0~t)L2(O)) N N N .., l ..} ? +Kl Lerk~t Ilv~+IW.6.t + !{L Ilxk+ W.6.t + K L Ile-rk~tv,kW.6.t k=O k=O k=O 2 2 +K ~N IIf;k e-rj~tV,j.6.t .6.t + !{ tlit e-l'j~t\71j,j.6.t .6.t, k=O ) =0 The right-hand-side of (29) can be bounded by N 2 N 2 N (32) [ L e-rk~t II¢k 11 .6.t + [ L e-,.k~t II\7v,k 11 .6.t + [ L II\7xk+1112 .6.t k=O k=O k=O N N 2 2 2 + J{ L (II ok 11 + II\7 Ok 11 + IlOti \7 ok 112) .6.t + K L II cfJkII .6.t k=O k=O N 2 N k 1 2 2 +KLII\7cfJ ll .6.t+KLllxk+ 112 .6.t+K (h (i-l)+.6.t ) .6.t'J k=O k=O , 4 2 • ., 2 +/\ (.6.t) II~ttttIIL:!«O.(N+l)M).c2(O)) + !i(.6.t)-II(ttIIC2«O.(N+l)~t);.c2(O)) +K(.6.t)211\7(ttll~2«(0,(N+l)~t)P(O)) + K(tlt)311~tttll~oo«o.~t);L2(O)) N N +K2 L erk~t Ilv~+1112 .6.t + K3 L erk~t II\7v~+1112 .6.t. k=O k=O Finally, the right-hand-side of (30) can be bounded by 2 2 (33) C·* h ;h~Ehl'h ) t IIOtbXk+1112 (.6.t)3 + [t e-rMt II,pk 11 .6.t k=O k=O N N rMt 2 2 +[ L e- 11\7¢k 11 .6.t + [ L II\7xk+111 .6.t k=O k=O 22 Chippada, Dawson. Mart.inez, Wheeler Choose r such that, and Eh e _ !{3 2: 1'3 - 1'-e2 o. And, observe that 2 2 2 k e-rk~t 111/,k11 + Ilxk+111 + IILe-e7'j~t1/,j.6.t j=O 1-{1(11) 2 k 2 > Ile-rk~t¢kI12 + Ilxk+11l + IILe-erj~t1/).6.t j=O Now, sum (28)-(30), using bounds (31)-(33) and the choice for l' above to obtain N 2 (34) ~e-r(N+l)~t II¢NI12 + (To + ~eeO) Le-rk~t II¢kI1 .6.t k=O N 2 N 2 + ( To: 1) L e-rk~t Il0tb¢k 11 (.6.t)" + L e-rk~t Il0tb¢k 11 .6.t k=O k=O ... J 'or (a) N N 2 2 +Eh L e-rk~t 11\7¢k 11 .6.t + ~h L e-rk6.t Ilatb \7¢k 11 (.6.t)2 k=O k=O y (b) +'''*llv~W+,*II\7v~112+ ~llv~1I2+~h lI\7v~1I2 ~ "-..--' (e) (d) N N rk + 7'1 L erk~t lI\7v~+1 W .6.t + 1'2 L e 6.t Ilv~+11l2 .6.t k=O k=O ... .J ... .. 'V ...... (e) (1) N N +r3 Lerk~t lI\7v~+1112 .6.t+~ IIxN+lW + ~L 1I0tbxk+1112 (.6.t)2 k=O k=O ... V -I (g) System of Shallow Water Equat.ions 23 N 2 N 1 2 +~ t; I IJfb/ (Xk+l)11 .6.t+Eh (PI + ~2) {; II\7xk+ 11 .6.t , ~ v (h) 2 2 2 ::;(f* h ;/~ EhP2 ) £= Il0tbXk+lll (.6.t)3 + E IIv? 11 + E II\7v? 11 k=O N 2 N 2 N 2 +E L e-I'kl>t II1jJk 11 .6.t + EL e-rk~t 11\71jJk11 .6.t + EL II\7Xk+111 .6.t k=O k=O l:=O N ~ N ~ +Kl L e7'k~t II\7v~+III- .6.t + [(2 L erMt Ilv;+III- .6.t k=O k=O N N k 2 k 2 k +[(3 L erk~t II\7v;+1112 .6.t + K L (11B 11 + II\7B 11 + II0ti \7B In .6.t k=O k=O l +K£= (11¢kW + Ilv¢kW) .6.t+[\'£= (11¢k+lW + 11\7¢k+ W).6.t k=O k=O +[{ (.6.t)4 II~tttt 11~2«O.( N+l )~t );C2(fl)) + [{(.6.t)2 II~tt11~2«(O,(N + I)~t);C!(flll - 2 ~'3 ') +1\ (.6.t) 11\7~tt II£:2((0,( N +1 )~t);C2( fl) I + l\ (.6./)' II(ttt11£:00((O.M);C2(fl)) +[-((.6.1)2 IIVtt 11~2(O,(N+l )~t);C2(fl)) + K (.6.t)2 11\7Vt 11~2((0,(N+l )~t);C2(fl)) N +[-( LAk.6.t + [-((h2(i-l) + .6.t2) + K (h2(i-l) + .6.t2) .6.t2 , k=O > O. Using the above choice for .6.t, hiding all terms multiplied by E, and observing that terms (a)-(h) are all non-negative, we can write (34) as follows N 2 (35) ~e-"(N+l)~t II1jJNI12 + (~ + ~eeO) Le-rk~t II1jJkI1 .6.t k=O N N 2 2 + L e-rk~t Il0tbl{,k 11 .6.t + ~h L e-rk~t 11\71jJk11 .6.t + ~* IIv? 11~'(fl) k=O k=O N N ) N +~ Ilx +1112 + 0'4 L IIOtbXk+1112 (.6.t? + ~h (PI + ~2 L IlvXk+1112 .6.t k=O k=O N k k l 2 ::; K Ile 1I~2((o.N~t)P(fl)) + K lI\7e 1I~2«(0,N~t)P(fl)) + K L II0ti \7e ' 11 .6.t k=O 2 +y ¢k , . + y \7¢k " II 112l2(0,(N+l)~t);C2(fl))" II 11 q(0,(N+1)~t);C2(fl)) +K(.6.t)4 II~tttt 11~2(0,(N +1 )~t);C2(fl)) + K(.6.t)2 II~tt11~2((O.(N+l )~t);C2(fl)) 24 Chippada. Dawson, Martinez. Wheeler +J((.6.t)211\7~ttll~2((0(N+!)~t);.c2(n)) + f{(.6.t)311~tttll~oo«o.~t)p(nJ) . ,2 2 ,') 2 +Ii. (.6.f) IIVttll.c2«O,(N+l)~t);.c2(n)) + l\(.6.t)-II\7Vtll.c2((0,(N+l)~tJ;.c2(n)) N 2 2 2 +Ii' L AA'.6.t + J{ (h (£-I) + .6.t ) + I( (h (l-I) + .6.t2) .6.t2 . k=O Recalling that apply the generalized discrete Gronwall's Inequality, to finally obtain 2 N 2 (36) ~e-r(N+l)~t 111/,NI1 + (~ + ~e90) Le-rk~t 111/,kI1 .6.t k=O N N rMt 2 + Le- IIOtb!/ikI12.6.t + ~h Le-rk~t 11\71/,k 11 .6.t + ~ IHII~[(n) k=O k=O N N N 1 2 1 2 +~ Ilx + 11 + 0"4 L II0tb Xk+1112 (.6.t)2 + ~h (,81 + ~2 ) L II\7xk+ 11 .6.t k=O k=O k 2 ::; J{ {IIBII;2«0'N~t);1{[(n)) + t,II0ti \7B 11 .6.t + 11 +( .6.t)4 II(tttt 11~2«0,( N +1)~t);.c2(n)) + (.6.t)2 II~tt 11~2(0,(N +1 )~t);1{1 (n)) 3 2 2 I 2 +( .6.t) II(tttllJ.:oo((0,~t);J.:2(fl)) + (.6.t) I Vtt 11.c2«o.(N +1 )M);.c2(fl)) 2 2 +(~t)211\7Vtll~2«0,(N+l)Ll.t);J.:2(n)) + (h (i-l) + .6.t ) (1 + .6.t2)} for .6.t sufficiently small and J( = J( exp (IiT /( 0"5 - Ii.6.t)) where 0"5 = ~min{ 1, 'Y*, e-r~t}. Bounding above and below, we obtain the estimate on the affine errors. 2 (37) II¢N 11 + 11¢11;2((o.N ~t );1{1(n)) + 118tb¢k 11:2«0,N ~t)p(n)) + IlxN+lW + II\7xll;2«0,(N+l)~t)p(n)) k ::; J{ {IIBII;2«(0,N ~t);1{1 (n)) + 1I0ti \7B 11:2«0,N ~t )p(n)) + 11 Using the standard approximation results, we obtain (38) I1!/iN II + 11¢lli2«(O.N ~t);1{l(n)) + II0tb 1/,k Ili2(IJ,N M);.c2( n)) + IlxN+llI + 11\7xlli2«(0,(N+1J~t);J.:2(n)) ::; J{ {h(-l + .6.t} System of Shallow Wat.er Equat.ions 25 The result of the theorem now follows by an application of the triangle inequality to the projection error and to the affine error (38). D REMARK: It is clear that stability can be proven using arguments similar to those used in the proof of the error estimate. In fact, the stability estimate is as follows. THEOREM 4,2. Let 0 ::; So ::; f ::; Sl and f, Sl 2: 2. Let (e' .1/) be the solution to (6)-(8) and suppose that assumptions A2-A14 hold. Then.:3 a constant C = C(T, Sl, 1') such that II~N+lll + 11(lli2((0,N ~t);1tl (n») + Il0tb~k Ili2«O.N ~t);c(n)) + IlvN+lll + II\7vlli2«o.(N+1')~t);.c2(n) ::;C (1ICIIl1tl(n) + II~oll1tl(n) + IIOtb~011.c2(n) + IlvlJll1t1(n) + IITWslli2((O.N~t);.c2(n») + IIFlli2((0,N~t);.c2(n))) , 2 1 for.6.t::; min{Kl,K3}' and K3 = h (T*h2 + CEh{32r Using a slightly different analytical approach in proving Theorem 4.2, it can be shown that the results of this theorem hold for .6.t ::; Kl when {3l = {32= t. 26 Chippada. Dawson. Martinez. Wheeler REFERENCES [1] S. CHIPPADA, C, N, DAWSON, M. L. MARTINEZ, AND M. F, WHEELER. Finite element appro:l'- imations to the system of shal/ow water eq1J,ations. Part I: Continuous time a priori error estimates, Tech. Report TR95-35. Ric" Universit.y. Houst.on. TX. 1995. [2] L. C, COWSAR, T, F, DUPONT, AND M, F, WHEELER. A p"iori estimates for mixed finite element methods for the wa,'e equation. 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