Numerical study of Petrov-Galerkin formulations for the shallow water wave equations

Numerical Study of Petrov-Galerkin Formulations for the Shallow Water Wave Equations

Carlos A. A. Carbonel H.

Universidade Santa Úrsula. Rua Jornalista Orlando Dantas 59. Botafogo. 22230-010 Rio de Janeiro. RJ. Brazil

Augusto Cesar Galeão

Abimael Dourado Loula

Laboratório Nacional de Computação Científica.. Av. Getúlio Vargas 233. 25651-070. Petrópolis. RJ. Brazil

The behavior of Petrov-Galerkin formulations for shallow water wave equations is evaluated numerically considering typical one-dimensional propagation problems. The formulations considered here use stabilizing operators to improve classical Galerkin approaches. Their advantages and disadvantages are pointed out according to the intrinsic time scale (free parameter) which has a particular importance in this kind of problem. The influence of the Courant number and the performance of the formulation in dealing with spurious oscillations are adressed.

Keywords: Shallow water waves, Finite elements, Petrov-Galerkin formulations.

Introduction

The shallow water wave theory applies when the relative depth is very small. Thus, the vertical acceleration can be neglected and the path curvature is small. Therefore the vertical component of the motion does not influence the pressure distribution, which is assumed to be hydrostatic and the celerity of the waves is an increasing function of the water depth h such as .

As it is well known, for this type of problem, finite difference models which range from explicit to fully implicit (Hansen, 1956; Leendertse ,1967; Abbot and Ionescu, 1967; Casulli and Cheng, 1992) were proposed in the last 40 years. Similarly, several models using finite element methods for shallow water wave equations have been published, most of them based in (semi) discrete Galerkin formulations (Grotkop, 1973; Taylor and Davies, 1975; Sunderman, 1977; Kawahara, Shohei and Shunsuke, 1978; Praagman 1979; Kinnmark, 1986; Bova and Carey, 1996).

Finite element methods based on classical Galerkin formulations have shown some misbehaviors in time-dependent propagation problems. It is only possible to obtain numerical schemes that retain the high accuracy of the element-based spatial discretization for small values of the time step Dt. For quite modest values of Dt, the accuracy and phase-propagation properties of the Galerkin formulation are lost and the stability range is reduced in comparison with finite difference schemes (Donea and Quartapelle, 1992).

These disadvantages have motivated in the last years, the development of alternatively finite element formulations such as Taylor-Galerkin, least-squares Galerkin and Petrov-Galerkin methods to improve the time-accuracy approximations of evolution problems, particularly for advection-dominated problems.

Finite element formulations based on the SUPG and similar operators proposed initially for advection-difusion and compressible flow problems (Brooks and Hughes, 1982; Hughes and Tezduyar, 1984; Johnson 1987; Hughes and Mallet, 1986; Hughes, 1987; Shakib, 1988; Galeão and Do Carmo, 1988; Almeida and Galeão, 1996) are well known and valid alternatively. Mostly of these are implicit formulations, but it is possible to have explicit formulations (Hughes and Tezduyar, 1984). It is necessary to remark that, the extension of these formulations to the Navier-Stokes equations is non-trivial as well as to Shallow Water Wave equations, which describe the dynamics of nearly horizontal free surface flows. In this case, the dynamics is basically dominated by gravitational waves, however the non-linear contributions, mainly represented by the advective terms, are needed to describe cases of wave steeping, and wave reaction to arbitrary topography and boundaries.

Some Petrov-Galerkin formulations have been proposed using a symmetric form of the shallow water wave equations (Bova and Carey, 1995; Saleri, 1995; Carbonel, Galeão and Loula, 1995; Ribeiro, Galeão and Landau, 1996). Existing contributions related to shallow water waves failed to provide a picture of accuracy/stability dependence on the dimensionless Courant number. Their advantages and disadvantages can be distorted, because are related to the size of the time step, which is not an adequate referential parameter for a hyperbolic problem. Additionally, the choice of the intrinsic time scale, defined from theoretical approximations, does not give a global picture of the influence of this free parameter.

In this paper it is analyzed the numerical performance of some time dependent Petrov-Galerkin formulations in space and time, adopting continuous linear interpolation in the space discretization, combined with continuous and discontinuous interpolations in time. The role of the choice of the intrinsic time scale free parameter t in the solution of one dimensional transient shallow water wave problems, is discussed. This paper is organized in the following form: Initially, we present the governing equation of the hydrodynamic problem. Later, we describe the finite element methods based on variational Petrov-Galerkin formulations and the importance of the intrinsic time scale. Finally, we present the main results of some numerical experiments: First, we study the influence of the intrinsic time scale on the Petrov-Galerkin solutions for typical wave problems, a smoothed wave and a wave front propagating along a one-dimensional channel. Then, we examine the accuracy of the Petrov-Galerkin solutions in relation to the Courant number, the ability to suppress node-to-node oscillations and the importance of the spatial order of the interpolation function.

It is necessary to remark that, our simple wave problems (smoothed wave and a wave front propagation) are strictly according to the shallow water wave theory avoiding the "long wave paradox", an error encountered in the treatment of the nonlinear theory, when the relative depth is not small. This paradox could be described as follow : As the celerity is function of the depth water, the wave elements carrying more energy (greater depth water) have the tendency to catch up with the first wave elements (lower depth water) ahead of the wave. Consequently, a vertical wall of water results. A phenomenon that may occur physically, but much later that predicted by the shallow water theory. Additionally, when the relative depth is not small the pressure distribution is no longer hydrostatic.

Governing Equations

The shallow water wave equations may be derived by depth-integration of the incompressible Navier-Stokes equations, considering a hydrostatic pressure distribution and constant density (Stoker, 1957). In an one-dimensional domain W =[0, L] with boundary G (x=0 and x=L) and time interval [0,T], the governing equations including bottom friction and wind stress at the surface are mathematically described as follows.

(1)

(2)

In the above expressions H=h+h is the total water depth. h(x,t) is the water surface elevation and h(x) is the water depth, both measured from the undisturbed water surface; u is the velocity; l is the surface friction coefficient; W is the wind velocity and r is the friction coefficient.

To solve the equations (1) and (2), the following boundary conditions are applied.

u = 0 (3a)

at the closed boundary ( x = L), and at the open side ( x = 0), the water elevation is defined as

h = h^*(t) (3b)

where h^*(z) is a prescribed function in time. For the initial conditions, an appropriate initial state is assumed in the domain W and at the boundary G:

u = u_o, h = h_o at t = 0 (3c)

where u_o and h_o represent the initial velocity component and elevation respectively.

Finite Element Methods

To present the finite element methods, it is advantageous to rewrite the governing equations for the shallow water wave problem in the following form

(4)

(5)

where u is the velocity, d = 2 and c= is the celerity. The equation system (4)-(5) written in matrix form reads

(6)

where

(7)

Petrov-Galerkin formulations

We now construct now space-time Petrov-Galerkin finite element models for the above problem. To this end we introduce a space-time finite element partition ph, Dt, in which the time interval is partitioned into subintervals

I_n = t_n+1 – t_n = Dt, t_n Î(0,T] (8)

where tn, tn+1 belong to an ordered partition of time levels 0 = t₀ < .t_n< t_n+1.< t_F = T an the space domain W is partitioned in N sub-domains W_e with boundary Ge. The space-time integration domain is the slab S_n = W ´ I_n with boundary G_n = G_e ´ I_n such that the slab is composed by N elements = W_e ´ I_n

Let the finite element space of piecewise polynomial in space and time on the slab Sn. Defining

(9a)

(9b)

We say that the general space-time Petrov-Galerkin approximate solution for the shallow water wave problem (6) is the vector V^hÎ

, which satisfies

(10)

where

(11)

is the residual vector associated with (6) and

(12)

is a space-time operator. The first integral appearing in Eq. (10) is the space-time Galerkin residual. The second integral represents the added Petrov-Galerkin contribution, in which contains the stabilizing parameters (Hughes and Mallet, 1986). The third integral is a jumping term which enforces weakly the initial condition in S_n.

In particular, we are dealing with subcritical flows in shallow water waves (u << c), a diagonal matrix is adopted (Ribeiro, Galeão and Landau, 1996), where t is the intrinsic time scale free parameter. Using these assumptions, Eq. (10) can be rewritten as

(13)

where

(14)

is the Petrov-Galerkin weighting function.

Three stabilized formulation will be studied; (i) The streamline Petrov-Galerkin method (SUPG), (ii) The space time Petrov-Galerkin method (STPG) and (iii) The space-time (discontinuous in time) Petrov-galerkin formulation (STDPG). Linear interpolation in time will be considered combined with linear and quadratic interpolations in space

Streamline upwind Petrov-Galerkin formulation (SUPG)

In this case, finite element approximation is introduced only in the space domain, considering V^h at the levels t_n and t_n+1 ( respectively) with interpolation functions in the space. Defining

(15)

(16)

with

, q =(t- t_n )/Dt (17)

the SUPG formulation reads

(18)

The SUPG formulation equation (18) in fact represents the semi-discrete Petrov-Galerkin approximation and the following choices of the parameter q will be considered:

q = ½ Trapezoidal rule or Crank Nicholson

q = 2/3 Classical Galerkin

q = 1 Full implicit (Backward Euler scheme).

Space-time Petrov-Galerkin formulation (STPG).

In this case, in the Eq. 10, for each time step, the initial condition is strongly enforced as the last computed solution at the end of the previous time-step. As a result the jumping term disappear and the STPG formulation reads

(19)

The STPG method corresponds to a Petrov-Galerkin approximation with continuous in time interpolation

Space-time (discontinuous) Petrov-Galerkin formulation (STDPG).

This formulation corresponds to the complete variational form defined in Eq. 10, leading to

(20)

The STDPG approach can be seen as a space-time discontinuous Galerkin procedure with the addition of a stabilizing term.

The above formulations are variationally consistent, in the sense that if V^hàV, the stabilization terms vanish.. For the SUPG and STPG methods, the solution in its time step Dt =t_n+1-t_n is arranged only for nn unknowns in the level t+Dt, whereas for the STDPG method, the solution is arranged for 2nn unknowns associated to the levels t and t+Dt.

The intrinsic time scale t

The intrinsic time scale is a free parameter and acts to normalize the magnitude of the test function perturbation. The selection of t is an open problem because some estimations in relation to the convergence and error estimates, provide design conditions but are insufficient to determine a unique definition, particularly with time dependent equations in which the diffusive terms are not dominant. Except for the specification of t the Petrov-Galerkin formulations are complete. We should say that, the choice of t is somewhat dependent on the problem to be solved

For equation systems the definition of t is complicated. Shakib (1988) proposes a more general theoretical form valid for system equations. He applied to the time dependent diffusion advection equation but introducing a correction to get the exact solution for steady cases. For the determination of t , it is necessary the solution of an eigenproblem. In the case of one dimensional systems (Bova and Carey, 1995) with unknown vector F and matrices A_o , A₁ of the form

(21)

a possible definition of t is

(22)

where l denotes the element length. This definition could be extended for two dimensional problems. Particularly for one dimensional shallow water waves when u<<c is possible to get some definitions of t (Ribeiro, Galeão and Landau, 1996),

, a =1/Cr (23)

where t_sd, is the free parameter for a semi discrete formulation and t_st is the free parameter for the space-time formulation. As mentioned before, the definition of t is an open problem, and there are no information about the influence of t in the solution of transient problems and on how the choice of t (experimental and proposed theoretical approximations) is useful for the simulation of typical shallow water wave propagation focusing the correct description of the variables in space and time. Here, we evaluate the influence of the free parameter t of Petrov-Galerkin formulations in the solution of one dimensional transient shallow water wave problems.

Numerical Experiments

The performance of the finite element formulations are evaluated by considering some simple cases of water wave propagation in a channel of sub-critical flow regime. The channel of 10m depth is composed by linear elements of uniform length Dx=100m. As a reference we define

(24)

which is the time step corresponding to the Cr=1 (Cr is the Courant number). In the experiments considered in this paper, Dt_Cr = 10sec. The first case deals with the propagation of a smoothed wave. The second case deals with the propagation of a discontinuity represented by a water front. In the two cases the description of the perturbations are in accordance with the shallow water wave theory and the vector B of Eq. 6 is set equal to zero.

Wave Propagation in a Channel

The objective is to test the proposed finite-element Petrov-Galerkin formulations to the case of propagation of a smoothed wave described by a quadratic sine function in a frictionless channel of constant depth, evaluating the influence of t after 60 time steps (600 sec), when the Courant number is equal to 1/2. Figure 1 shows the SUPG solutions. They have a similar pattern in relation to the influence of t which is minimal. Notice that the SUPG solution with t =0 correspond to the classical semi-discrete Galerkin approximation (SDG). The solution in each case practically does not change when the values of t increases. It is noted that only in the case of q =1/2 (Crank-Nicholson scheme), the amplitude is maintained without algorithmic damping.

With the other choices, q =2/3 and q =1, the amplitude errors are too large to be considered in practical cases. With q =2/3 the amplitude decreases around 17% (Fig. 1b) and with q =1 the amplitude decreases around 36% (Fig. 1c). The worst solutions obtained with q =1 corresponds to the backward Euler semi-discrete finite element procedure.

In Fig. 2, the solutions for the continuous and discontinuous space time formulations are shown. The solution patterns are very different. In the case of the STPG formulation (see Fig. 2a), for na increasing t, the amplitude increases up to a limit value, capturing the real amplitude of 0.1m when t ³ 10Dt_Cr (increase of 20% in relation to STPG formulation with t =0 which is equivalent to the semi-discrete Galerkin approximation with q =2/3, see Fig.1b ).

In the case of the STDPG formulation (see Fig.2b), for increasing t , algorithmic damping is introduced which is acceptable only when the free parameter t <Dt_Cr.

Water front propagation in a channel

In this experiment, the influence of the free parameter t after 50 time steps is evaluated for each Petrov-Galerkin formulation. The Courant number is fixed equal to 1.

Figure 3 shows the solutions for the SUPG formulation. When q =1/2, the presence of residual oscillations in the upflow side of the water front (obtained for t = 0) decreases when the t increases, simultaneously with the reduction of the spreading of the leading edge of the front (see Fig.3a). A maximum value of t (t =1000Dt_Cr. =10000sec) gives a perfect description of the front but with a peak in the upflow side of around 10% of the front amplitude. When q = 2/3, the solutions are smoothed and the net influence of t is negligible (see Fig. 3b). When q =1, the influence of the t is non-perceptible (Fig. 3c).

Figure 4 shows the solutions for the continuous and discontinuous space time formulations. In the case of the continuous STPG formulation (Fig. 4a) the smoothed solution obtained when t =0 is improved for a increasing t up to values equivalent to DtCr For great values the solution exhibit asharp front, a reduction of the spreading of the leading edge of the water front and the generation of a residual oscillation in the upflow side of the front. In the case of the STDPG formulation (see Fig. 4b), the presence of residual oscillations in the upflow side of the front(obtained for t = 0) decreases when t increases. For t >Dt_Cr (>10sec), excessive algorithmic damping is introduced.

Preliminary Remarks.

From the results obtained in the two previous sections it is noted that for the SUPG and the STDPG formulation the net effect of an increasing t is the increase of algorithmic damping . In the STPG formulation the effect of increasing t is to reduce the excessive damping introduced by the Galerkin formulation helping in the description of discontinuities. The solutions using the SUPG formulations when q is equal to 2/3 and 1 are very poor, the damping introduced is unacceptable for real applications

In real applications, it could be present either smoothed and sharp water wave perturbations, which could be generated by the excitations of natural modes of the hydrodynamic system. Therefore, the choice of t must to be useful for the two types of possibilities. Using the STPG formulation, it could be possible to study problems in which smoothed waves are present, reaching a good approximation when t ³ 10Dt_Cr (³ 100sec) whereas for sharp fronts the better solutions are possible only for values of t < Dt_Cr,. For the SUPG formulation with q = 1/2, any value of t may be apparently good for smoothed waves but in the case of sharp fronts high values are necessary for a better approximation.

For the experiments of smoothed wave and front propagation, the theoretical values according to Eq. (23), are t_sd = 5 sec (equivalent to t = 0.5Dt_Cr), t_st equal to 2.24 sec for the wave propagation case and 3.54 sec for the water front propagation case (equivalent to t = Ö5/10 Dt_Cr, and Ö2/4 Dt_Cr respectively). But these values do not help to obtain an acceptable solution. Our opinion is that this type of theoretical approximation needs an improvement to preserve the physical and numerical mechanisms involved in the practical solution process .

Accuracy dependence on the dimensionless Courant number

The dependence on the dimensionless Courant number, which is the ratio of the physical velocity to the numerical velocity, is important due the natural implicit form of the Petrov-Galerkin formulations. Theoretically, it is possible to employ a time step many times larger than the corresponding to CFL condition.

In Fig. 5 it is shown the solution at 600 sec in the case of a wave propagating in a channel (quadratic sine function) , for the SUPG formulation (q =1/2) and for the space-time formulations when the Courant number change. SUPG solutions when q =2/3 and q =1 are not presented due the excessive damping when the Courant number increases. The SUPG solutions with q =1/2 and t =Dt_Cr=10sec showed in Fig. 5a, are affected by a phase shift, residual oscillations and the amplitude decreases around 5%, when Cr=2. The STPG formulation leads to a reduction of amplitude ( around 15% when Cr=2) and produces a strong phase shift, when the Courant number increases (see Fig. 5b). The STDPG formulation gives very interesting solutions without a phase delay and minimal residual oscillations, decreasing the amplitude around to 5% for Cr=2 (see Fig. 5c).

Ability to suppress node-to-node oscillations

The ability of the three variational formulations, to suppress spurious 2Dx oscillations is investigated numerically. Initial conditions for sea water elevation will be of the 2Dx type, that is varying in sign from node-to-node in a channel of 200 uniform elements. The water elevation of the spurious oscillation h_s ranges from –0.1m to 0.1m covering 75% of the channel. The experiments are performed for Cr = 1/2

Figure 6 shows the behavior of the water level evaluated at the middle point of the channel as a function of the number of time steps. The curves plotted in Fig. 6a represent the responses of the Petrov-Galerkin formulations when t = 0, equivalent to their correspondent Galerkin formulations. As can be seen during the first 30 time steps the Galerkin solutions have a tendency to conserve the initial amplitude without damping.

On the other hand the Petrov-Galerkin solutions presented in Fig. 6b exhibit a efficient damping suppressing the spurious oscillations in a time interval little to 7 time steps, that is, h® 0 when t à ¥ These simple experiments have confirmed that the Petrov-Galerkin formulations are a family of stable methods for the solution of shallow water wave problems.

Increasing the order of the spatial interpolation function

Here, it is repeated the experiments described in the wave and front propagation experiments for the STDPG formulation using a quadratic polynomial function in space. The distance between node to node is conserved and the results are presented in Fig. 7. For the water wave propagation case (Fig.7a), and an increasing t, the damping is weaker compared with the solution using linear interpolation function (Fig. 2b). For the water front experiment (see Fig. 7b), it is observed that the oscillations in the upflow side of the front were practically suppressed and the accuracy is improved for different values of t, compared to the solution shown in Fig. 4b

Conclusions

The experiments demonstrate that finite elements methods based on the present Petrov-Galerkin formulations could be used to solve shallow water wave problems, when an adequate intrinsic time scale t is employed. The existing theoretical ways to estimate this parameter do not gives the best t for general applications (smoothed waves and sharp fronts), needing improvements to represent the physical and numerical mechanisms involved in the practical cases. Additionally in real application problems, smoothed and sharp water wave perturbations could be present, therefore the correct evaluation of t (theoretical or empirical choice) is an important point addressed in the future by the theoretical investigators.

It is confirmed that for the SUPG semi-discrete method and the space–time discontinuous formulation (STDPG), the net effect of increasing t is to increase damping. The associate Galerkin continuous in time method leads to an over-damped scheme in such a way that, the effect of increasing t in the corresponding STPG formulation is to reduce the excessive damping introduced by the continuous in time Galerkin formulation and partially capturing the real solution. The SUPG and STPG formulations are affected by a phase delay when Cr >1. The STDPG formulation gives results with acceptable performance for an increasing Courant number.

Among the formulations considered in this paper, the SUPG formulation with q =1 is not useful for practical unsteady application due the excessive damping.

It is necessary to study ways to improve these Petrov-Galerkin formulations to obtain solutions without phase delay and minimal amplitude damping, particularly for situations when the Cr ³ 1 exploring in this form the natural implicit solution, because most of the present formulations, are practically (due damping effects and phase delay) limited to the CFL condition (Cr=1).

Manuscript received: September 1999. Technical Editor: Átila P. S. Freire.

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Publication Dates

• Publication in this collection
  15 Dec 2000
• Date of issue
  2000

History

• Received
  Sept 1999