^Laboratorio Nacional de Computaqao Cientifica, Rua Lauro Muller 455, Rio de Janeiro, RJ , Brasil

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1 A space-timefiniteelement formulation for the shallow water equations F.L.B. Ribeiro,' A.C. Galeao," L. Landau* "Programa de Engenharia Civil, COPPE / Universidade Federal do Rio de Janeiro, Caixa Postal 68506, Rio de Janeiro, RJ ^Laboratorio Nacional de Computaqao Cientifica, Rua Lauro Muller 455, Rio de Janeiro, RJ , Brasil Abstract This paper presents a space-time finite element formulation for problems governed by the shallow water equations. A constant time-discontinuous approximation is adopted, while linear three node triangles are used for the spatial discretization. The streamline upwind Petrov-Galerkin (SUPG) method is applied in its equivalent variational form to fit the time discretization. Also, the correspondent semi-discrete SUPG version is established, and some numerical results are presented in order to compare the performance of these methods. 1 Introduction As it is well known, the use of the classical Galerkin method to approximate convection-dominated phenomena leads to numerical solutions contaminated by spurious oscillations that are spread over the entire computational domain. In the context of weighted residual methods, a remarkable improvement on the numerical solution of such problems was provided by the consistent variational SUPG method proposed in by Brooks and Hughesfl]. Since then, many SUPG based methods have been used for multi variable systems of equations.. In Sharkib[2] and Almeida and Galeao[3], space-time Petrov-Galerkin (STPG) formulations were derived for the compressible Euler and Navier-Stokes equations, showing their inherent control of derivatives along the characteristics. This fact turns out to be an improvement over the classical semidiscrete SUPG formulation, where only streamline derivatives are controlled. The discontinuity capturing approach proposed in those references lead to

2 404 Computer Techniques in Environmental Studies stable and accurate methods to solve all details of sharp layers and/or shock discontinuities. If we realize that the mathematical structure of the above mentioned equations is identical to those that govern the shallow water problem, it can be immediately concluded that for such problems, P-G weighted residual methods will also perform well. This was done in Bova and Carey[4] and Saleri[5], where semi-discrete P-G (SDPG) finite elements were employed, and in Carbonel, Galeao and Loula[6], where a space-time P-G formulation was derived, using linear interpolation for both spatial and temporal discretization. In this paper, the continuous linear interpolation is retained for the spatial discretization, but constant time-discontinuous interpolation is adopted. For this choice of interpolation, the weighting function gives no contribution to the terms involving time derivatives. Even so, the resulting discrete space-time Galerkin (STG) equations coincide with the semi-discrete Galerkin approximation with an Euler backward difference scheme to approximate the first time derivatives. Nevertheless, this will not occur with the P-G weighting residual methods. The numerical examples that will be presented later will show that, under these circumstances, the STPG formulation will give more accurate results than the SDPG model. 2 Problem statement Let (x, y) G Q e W* define a set of points on an horizontal plane and let z e [- A, 77] denote the vertical direction, where h(x> y) represents the water depth and rj(x,y,z) is the water surface elevation, both measured from the undisturbed water surface. We start from the 3-D incompressible Navier-Stokes equations, after turbulent time-averaging, integrating these equations along the z direction using depth-averaged horizontal velocities. Under the simplifying assumption of a hydrostatic pressure distribution (negligible vertical acceleration), we arrive at the shallow water equations: 1 * - l KA + ^(qw vt' + ^g u fo+ //(*/,*,+f^) (la) -^Ww In these equations, H = A» 77 is the total water depth and u is the depthaveraged velocity, with components w and v in x and y directions respectively. The gravitational acceleration is given by g and/is the Coriolis parameter. The wind velocity is w, with components v/ and w/ a and C are, respectively, the surface and Chezy friction coefficients, and // is the eddy viscosity.

3 Computer Techniques in Environmental Studies 405 Multiplying the third equation by g and observing that, - cclc} O\ - K^;>, W where c = (gh), and considering similar expressions for (g//),* and (gff),y, we obtain the shallow water equations in the velocity-celerity variables (see Saleri [5] ) which, in matrix form, can be written as: U,,+A.VU = F ; A.VU = A*VU (3 a) (3b) u 0 c 0 w 0 c 0, /i c o" c (3c) C (3d) Once an initial state Ua(x, y) is specified at / = 0 and appropriate boundary conditions are prescribed, the system of equations above can be solved to give the unknown column vector U. To obtain the space-time description of (3a-d) we introduce the variable change: s = (I)/, where (1) has units of velocity. Then if we define: where I is the (3x3) identity matrix, we will say that the space-time solution of the original problem (3a-d) is the (3x1) column vector U that satisfies the transformed equation: (4a) in (4b)

4 406 Computer Techniques in Environmental Studies 3 Petrov-Galerkin finite element model In order to construct the space-time finite element subspace, let us consider partitions 0 = ^ < t\ <...( < * +, of 5R* and denote by / = ((, ; +,) the n* time interval. For each n the space-time integration domain is the "slab" S» = Q x /, with boundary F = DC/,,. If we define S* as the e'** element in, e = 1,... n, where (TVy is the total number of elements in 5%, then for n = 0, 1,2,... (/) the space-timefiniteelement partition H^ ^ is such that: ^,=^^';,%=0,,x/,,; Q=UQ,;Q,nQ.=0 g=l e=l ybrz#y (5a) (ii) the space-time finite element subspace consists of continuous piecewise polynomials on the slab $, and may be discontinuous in time across the time levels t^ that is: tf* EC"(S.); U" ^ep*(5;); f/" -=O (5b) where P^ is the set of polynomials of degree less than or equal to k. According to the above definitions, the variational STPG formulation for the problem (4a-b) reads: Find t/* e»* such that for n = 0, 1,2,... W. X J(T ^. V L/" ). A*d&dt + (6a) where, = A.VU* - F" = Uj+AW*).VU'' - F* (6c)

5 Remarks Computer Techniques in Environmental Studies 407 (1) If in (6a) the integrals are taken over Q and fig instead of & and 5^, respectively; the P-G weighting function (rz.vf)'') is replaced by (ra.vu**)', and in (5b) the finite element subspace is defined for all / (making the jumping term in (6a) to disappear); and finally, if we approximate the partial time-derivative by a time-differencing operator, we generate the semi-discrete SUPG method. (2) If in (6a) we do not consider the added SUPG contribution represented by the second term under the summation symbol, we reproduce the timediscontinuous Galerkin method. If in this case, for instance, we use constant time-discontinuous interpolation, the resulting space-time finite element method will be identical to the backward Euler semi-discretefiniteelement procedure. Since constant time interpolation is used, C/,J' = 0, and therefore it is clear that the jumping condition, represented by the third integral in (6a), is the term responsible for this equivalence. Nevertheless, the STPG method and the correspondent SDPG method will be different. (3) The definition of the matrix T will be also different in these two formulations. This point is focused in the next section. 4 Purely hyperbolic problems To simplify our analysis let us assume F = 0. Using the T matrix definition found in Sharkib[2], we have: for the space-time formulation, _ s ^ -1-1/2 tfvaf 1 (7a) where x<, = t : *i = x ; X2 = y ; & (k = 0,1,2) are the local coordinates of the parent element S^ ; and A\, AI are the Jacobian flux matrices defined in (3c). for the semi-discrete formulation, - -1/2 (7b)

6 408 Computer Techniques in Environmental Studies where / (/ = 1,2) are the local coordinates of the parent element i% If, once again, for the sake of simplicity, we consider unidimensional problems and 11*'^ partitions of equal (hexat) S* elements, (7a-b) simplify to: (8a) (8b) where, 4 4 (9a) (9b) are the eigenvalues and correspondent eigenvetors of (z With these definitions we have, )"^ and /4,^ respectively. *. 1 7^-^ C «-c u-c + - u + c H-Cl-lU U 4- C \ + \ U - C c (lob) Now let us introduce the non-dimensional factor a = -~-, or what is the same, CFL = =>a>\> where CFL denotes the well known Courant-Friedrichsa Levy number. Using this factor, the definition of T*t becomes:

7 Computer Techniques in Environmental Studies 409 A 28 (lla) (lib) =av (lie)»remarks (i) Notice the intrinsic dependence of r,, on the used time step A/, which is not accounted for in %</. In the limit, as At -» 0 ; T,, -» 0, in a consistent way. This does not occur with the semi-discrete formulation because r^ is independent on At. (ii) Although not realizable from the practical point of view, %,= r^ if and only if a = 0. Remind that a must be greater than one, in order to the CFL condition be attained. We restate this comment saying that r,, approaches %</ as the time step At becomes larger, or, in other words, when accuracy decreases. (in) If we assume that u (( c, then r,, and r«/ become almost diagonal matrices, and can be replaced by, I * ^ I i; r,u=- i Because a > 1, the above ratio between brackets is always less than one. Even in the most unfavorable situation, CFL = 1 = a, r>, introduces less dissipation than r,/. In order to get a deeper insight about the performance of the STPG and SDPG methods, some numerical experiments will be performed in the next section. For these examples, constant discontinuous time interpolation and piecewise linear continuous spatial interpolation will be adopted for the STPG method. For the correspondent SDPG formulation this same spatial discretization will be used, combined, with an implicit backward finite difference scheme for time discretization.

8 410 Computer Techniques in Environmental Studies 5 Numerical results Our first example is the well known dam break problem, which consists of a wall separating two undisturbed water levels that is suddenly removed (Figure la). Friction effects are neglected and the spatial discretization is given by a 2x100 triangular elements mesh, as illustrated in Figure Ib. Figures 2-3 show the results for / = 2.50, respectively, comparing the solutions obtained with the Galerkin, the space-time and the semi-discrete formulations. For a time step Af = 0.10 (Figure 2), Galerkin solutions exhibit oscillations in the entire computational domain. This does not occur for both, the semi-discrete and the space-time P-G solutions, which accurately approximate the high gradients between the three horizontal water levels. For this time step, the semi-discrete solution is sharper than that obtained with the space-time formulation. The effect of reducing the time-step is shown in Figure 3, where the results corresponding to Af = 0.05 are plotted for / = For the Galerkin solutions, the oscillations grow up. For the semi-discrete solutions some localized oscillations appear near the sharpest layer, while a sharper solution without oscillations is obtained with the space-time method. g = 10 L= 100 (a) h= 1 L/2 4 L/2 L/ 100 I l^~~ JM L/ 100 Figure 1: Dam break problem.

9 Computer Techniques in Environmental Studies = 2.50 Galerkin Space-time Semi-discrete Figure 2: Solution for time / = 2.50, A/= = 2.50 Galerkin Space-time Semi-discrete Figure 3: Solution for time / = 2.50, A/ = 0.05.

10 412 Computer Techniques in Environmental Studies The second example, illustrated by Figure 4a, is the problem of a reflecting wave in a frictionless horizontal channel of length L = 5000, discretized with 2x10 elements, as shown in Figure 4b. The channel is open at the inflow boundary and closed at the opposed boundary. The system is subjected to a boundary condition at point A, raising the water level suddenly from the initial state of rest (H = 10) to H = 10.1, within one time step. The results can be seen in Figures 5-6. In these figures, the time-history responses for the water surface elevation at point B are depicted. For A/ = 10 (Figure 5), the curves of both, Galerkin and semi-discrete solutions, almost coincide. The STPG method presents an overdiffusive behavior for this time step, leading to a solution that progressively damps along time. However, a completely different behavior occurs when A/ = 1. For this time step, the STPG solution reaches the rectangular pulse form, while the Galerkin and SDPG solutions present some oscillations, as can be observed in Figure 6. (a) h= 10 L = 5000 (b) ~>-J zz^l JZz^J L^: Lc=rc2 Lf L/25 Figure 4: Reflecting wave in a frictionless channel.

11 Computer Techniques in Environmental Studies 413 At = 10 Galerkin -Space-time Semi-discrete J Figure 5: Solution at point B, with A/ = 10. Galerkin Space-time Semi-discrete o.oo Figure 6: Solution at point B, with A/ = I.

12 414 Computer Techniques in Environmental Studies 6 Conclusions In this paper, a STPGfiniteelement model was derived for problems governed by the shallow water equations. A piecewise linear continuous interpolation (in the space variables) was used, and piecewise constant discontinuous functions were adopted for time interpolation. In addition, the correspondent semidiscrete P-G model was also presented. The numerical results showed in this work indicate that the STPG method performs better than the SDPG formulation. Finally we should comment about the necessity of using a consistent time-space shock-capturing PG weighting function such that, in the limit, when Af-> 0, the approximated numerical characteristics approaches the true characteristics. This can not be actually provided by the SUPG method. The CAU generalized method proposed in Almeida and Galeao [3] for compressible flows, fulfill these requirements and, can therefore be successfully applied to shallow water problems. References 1. Brooks, A. N., Hughes, T. J., Streamline Upwind Petrov-Galerkin Formulation for Convection-Dominated Flows with Particular Emphasis on the Incompressible Navier-Stokes Equations, Compiit. Meth. Appl. Mech. Engrg, Vol. 32, pp , Shakib, F, Finite Element Analysis of the Compressible Euler and Navier- Stokes Equations, Ph.D. Thesis, Stanford University, Almeida, R. C, Galeao, A. C, An Adaptive Petrov-Galerkin Formulation for the Compressible Euler and Navier-Stokes Equations. Comput. Meth. Appl Mech. Engrg, Vol. 129, pp , Bova, S. W., Carey, G. F., An entropy Variable Formulation and Petrov- Galerkin Methods for the Shallow Water Equations, in: Finite Element Modeling of Environmental Problems-Surface and Subsurface Flow and Transport, ed. G Carey, John Wiley, London, England, Saleri, F., Some Stabilization Techniques in Computational Fluid Dynamics, Proceedings of the 9"' International Conference on Finite Elements in Fluids, Venezia, Carbonel, C, Galeao, A. C, Loula, A. D., A Two-dimensional Finite Element Model for Shallow Water Waves, Proceedings of the 73"' Brazilian Congress and 2"** Iberian American Congress of Mechanical Engineering, Belo Horizonte, 1995.