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Efficient implementation ofnon-oscillatory schemes for thecomputation of free-surface flowsMarinko Nujić aa Institute of Hydrosciences , Federal Armed Forces UniversityMunich , Werner-Heisenberg- Weg 39, D-H5577, Neubiberg, FRGPublished online: 13 Jan 2010.

To cite this article: Marinko Nujić (1995) Efficient implementation of non-oscillatory schemesfor the computation of free-surface flows, Journal of Hydraulic Research, 33:1, 101-111, DOI:10.1080/00221689509498687

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Efficient implementation of non-oscillatory schemes for the computation of free-surface flows Développement de schémas numériques performants non oscillatoires pour les écoulements a surface libre

MAR1NKO NUJIC Research assistant, Institute of Hydrosciences, Federal Armed Forces University Munich, Werner-Heisenberg- Weg 39, D-H5577 Neubiberg, FRC

ABSTRACT Two high-resolution numerical schemes based on Lax-Friedrichs numerical flux and ENO type of extra­polation are presented for the computation of free-surface flows. The performances of these schemes are similar to other high-resolution (TVD and ENO) schemes. However, the ease of implementation makes the new method very attractive for practical applications. The first numerical scheme is afterwards modified to account for the presence of the bottom slope terms. Several applications are presented.

RÉSUMÉ Deux schémas numériques performants bases sur Ie calcul des flux selon 1'algorithme de Lax-Friedrichs et sur 1'extrapolation du type ENO (essentiellement non oscillatoire) sont présentés dans leur application aux écoulements a surface libre. Leurs performances sont analogues a celles des autres schémas a haute résoluton (TVD et ENO). Cependant, la facilité d'utilisation de cette nouvelle methode la rend tres attractive dans les applications pratiques. Le premier schema numérique a été modifié dans une seconde étape pour prendre en compte la presence des termes dus a un fond en pente. Plusieurs applications sont presentees.

1 Introduction

The high-resolution numerical schemes TVD (total variation diminishing) and ENO (essentially non-oscillatory) have been recently applied by several authors for the computation of free-surface flows [1,2,3]. In the cited references, approximate Riemann solvers based on the flux-difference splitting of Roe [4] and a flux-vector splitting of van Leer [5] have been utilized as a building block. Although very accurate (shock capturing within one or two grid points), these methods require sig­nificant computational and programming time because of the field-by-field decomposition. Recently, non-oscillatory numerical schemes based on the Lax-Friedrichs solver were proposed [6,7]. Excellent results in aerodynamics were obtained by using Lax-Friedrichs solver in a combi­nation with discontinuous Galerkin finite element method, for the numerical solution of 2D Euler/ Navier-Stokes equations [8]. The main advantages of such an approach are its simplicity and ease of implementation. We also found this approach very attractive, for it is possible to account for the presence of the bottom slope term in a very convenient way.

Revision received September 26, 1994. Open for discussion till June 30, 1995.

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In chapter 2, the numerical schemes for the solution of ID hyperbolic systems of the form

" , + ƒ , = 0 (1.1)

are presented, where u represents a vector of unknowns and/is a flux function. This system is char­acterized by the following Jacobi matrix A, whose eigenvalues A., are all real and the columns of the matrix T represent eigenvectors.

du TAT A (1.2)

Only second-order accurate approximations are presented for the numerical solution of equation (1.1). However, higher-order generalizations are straightforward.

2 Numerical schemes

The approximate solution u of eq. (1.1), on a computational grid x; = jAx, t = nAt, can be written in the following predictor-corrector form:

" ( f" f" ï uj ~ ^ . ( J j + t / 2 - Jj-i/i)

0.5 ' AA J + 1 / 2 *J-[n'

(2.1)

Here/y+|/2 represents a numerical flux through the cell facey' + 1/2 between grid points y' + 1 andy'. The behaviour of the approximate solution depends very much on the particular form of this flux function. Two possible choices for the numerical flux are discussed in the next two sections. Just to mention that the two-stage Runge-Kutta (RK) method (2.1) belongs to a family of RK methods which preserve TVD property under certain CFL restrictions. More about it can be found in reference [6].

2.1 Numerical scheme I

This is a simplified version of the ENO scheme proposed in [6]. It uses LF scheme as a building block and it operates directly on fluxes instead of cell avarages. The numerical flux is splitted up into two parts

ƒ j + 1/2 _ J .;+ 1/2 + / j+ I (2.2)

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and each of the fluxes ƒ +j+U2 andf~j+in is approximated up to the required order of accuracy using

the ENO moving stencil idea. Second-order accurate numerical fluxes are defined as

(2.3) / ;+ ] / 2 = / ; + o.58/;

/;+,/2 = /;+1-o.5o/;+,

where

/ ; = 0 .5( / ; + an , ) , /T = 0 . 5 ( / , - a « , . ) .

Here a represents some positive coefficient, and it is required that

a > max A,,, (2.4) j j=l,...,N.

This condition on a may be relaxed, especially in the case of smooth flows, as will be seen in chap­ters 3 and 5. It is also possible to define the coefficient a locally, by taking maximum in eq. (2.4), only over points defined by the stencil for that particular cell-face [8]. An analysis concerning the coefficient a, and its influence on the numerical solution, is going to be presented in the authors second paper.

The quantities 5/+; and 5/~;+, are defined as

8/* = minmod (ƒ*+ , - f), f] -ƒ*_ , ) ,

5/;+, = minmod(/;+ ,-/; , /;+ 2-/;+ 1),

where the minmod function is defined as

(2.5)

minmod (a, b) a if \a\ < \b\ and ab > 0 b if \b\ < \a\ and ab>0 (2.6) Oif ab<0.

2.2 Numerical scheme II

The basic numerical scheme is described in reference [2], and it reads

fj+xn = 0.5 [ƒ , + ƒ , - | A ; + 1 / 2 | ( « , - « , ) ] (2.7)

where fR = f(uK) andƒ, = f(u,).

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The matrix Aj + m represents Roe's average matrix for the cell-face j + 1/2 and it satisfies Afj+U2 = 4/+1/2 AUj+1/2' The intermediate states uL and uK are obtained by using MUSCL type of extrapolation

u, = u, + 0.58M, L ' ' (2.8)

«« = K;+ i-0.58u; + 1

The quantities 6uj and 8i#y+1 are defined in an analogous way as 8/+, and 8/ ; + , , equation (2.5), using minmod function (limitcr). The modification to this scheme consists in replacing Roe's approximate solver by a simpler Lax-Friedrichs solver, so that the expression (2.7) now becomes

fi.w2 = 0.5[fR + fL-a(uR-uL)l (2.9)

where a represents the positive coefficient defined by (2.4). The modified scheme remains second-order accurate and field-by-field decomposition has been avoided. However, the performances of the new scheme are very similar to the original one as will be seen in the next chapter.

3 ID Numerical experiments

The performances of the new schemes will be first examined on an idealized 1D dam-break prob­lem. Similar numerical experiments on this problem, using high-resolution schemes, can be found in [1,3,9]. For the 1D shallow water equations, the vector of unknowns u and the flux function ƒ are defined as

u = h~

uh_ , ƒ =

uh

_uh + 0.5gh (3.1)

where h represents the water depth, u is the flow velocity and g is the gravity acceleration.

The dam is initially located at x = 0.5. The water depth ratio upstream/downstream is hl/hO = 100. A grid of 100 points has been used for the numerical simulation. Fig.l compares different numerical solutions with the analytical one (solid line). One can see that performances of the new schemes are very similar to the other high-resolution schemes which use field-by-field decomposition. Especially numerical scheme II produces very good results compared to the ID version of the MUSCL scheme from reference [2]. The value a = 0.4 | X,max | for the scheme II seems to be optimal for this problem. Lower values for a lead to under and overshots while greater values smear the solution. When the higher-order approximations are used, the difference between numerical solutions based on the approximate Riemann and Lax-Friedrichs solvers becomes smaller as indicated in [6], Therefore, in this case it is even more desirable to apply methods outlined in chapter 2.

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1 .4

1.2

1.0

0.8

0.6

0.4

0.2

on

-&—°

\*& l l l i 1 0.0 0.2 0.4 0.6 0.8 1.0

■ > :

o.o

Fig. 1. Idealized dam-break problem at t = 0.25 s. All three runs with CFL = 0.5. a) Numerical scheme I, a = 0.7 I A.mM | , b) Numerical scheme II, a = 0.4 | A.max | , c) MUSCL.

4 2D Dam-break problem

The next problem to be considered is a 2D dam-break problem described in [10]. The flood propa­gation is modeled using 2D shallow water equations which read

u, + fx + g,+ s = 0 (4.1)

where

u =

h

uh vh

ƒ

uh

u'h + 0.5gh' uvh

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s =

O gh{sfx-sbx) .gh(sfy-sby)

vh uvh

v h + 0.5gh' (4.2)

Here h represents the water depth, u the velocity component in x direction, v the velocity compo­nent in y direction and g the gravity acceleration. Bottom slopes in x and y directions are defined as

dz -=r- Sh ~dy

(4.3)

where z represents the bottom height. Friction slopes are taken care of by Manning's formula

2 n. 2 n u4u +v

2 n—2 n v4u +v (4.4)

in which n = Manning's roughness coefficient.

Spatial discretization of equation (4.1) is the same as previously described, separately applied to the fluxes ƒ and g. Treatment of the source terms is the same as in reference [10]. For more details on breach geometry, number of grid points used, and boundary and initial conditions see [10]. Fig. 2 compares experimentally determined water front at different times with the computational ones. A very good agreement is obvious and results are very much improved compared with those from reference [10]. Also, the calculated hydrographs from the fig. 3, fit much better to the experi­mental ones.

1000

-1000

-2000

-3000

EXPERIMENT SCHEME 1 MUSCL

2000 3000 X fmm]

4000

Fig. 2. Comparison between calculated and experimentally determined water fronts at different times, for 2D dam-break problem [10].

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0 1 2 3 4

EXPERIMENT SCHEME 1 MUSCL

T I M E [ S ]

Fig. 3. Calculated and measured hydrographs for 2D dam-break problem [10], at positions 4,7,19 and 22.

It should be mentioned that no water film in front of the breach was necessary (dry bed) and numer­ical solution was stable and free of oscillations. In reference [10], a ID analytical Ritter solution has been used at the dam site as initial condition. This was also unnecessary in the present case.

5 Variable bottom topography

If we try to apply high-resolution methods based on flux-difference or flux-vector splitting to the case of shallow water flow with variable bottom topography, very poor results are usually obtained. Figure 4 shows a 1D steady-state flow in a channel with variable bottom height. The flow has been obtained by solving the ID shallow water equation (5.1) with source term, and by using a ID version of MUSCL scheme. The complete equation reads

«, + ƒ , + s = 0

where u and /a re previously defined by equation (3.1).

(5.1)

b) 0.8 0.7 0.6 0.5 0.4 0.3 0.2

500 1000 X [m]

1500

-»%4M' -V—<fb*>—>

500 1000 1500 X [m]

Fig. 4. Steady-state flow in a channel (MUSCL). a) Water elevation, b) Discharge q.

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The source term s is defined through the equations (4.2)-(4.4), where only the components in x direction are of concern. Bottom slope term can be discretised by using central differences for example, or performing upwinding of the source term [1,11]. Note that upwinding of the source term does not help very much, because the real problem lies somewhere else as will be seen later on. A good insight into the behaviour of numerical error gives the discharge q. Namely, because of the steady-state flow conditions the discharge should be constant over the domain of interest and equal to q = 0.5 [m-Vs]. The numerical solution from the figure 4b), shows, however, large deviations from this exact value. This large numerical error is caused by two reasons:

- numerical diffusion - because of the non-smooth solution (water depth h), - numerical incompatibility - between bottom slope term and the term 0.5 gh2, from the flux

function/.

The first reason can be eliminated by taking the water elevation H = h + z as independent variable in the continuity equation, instead of the water depth h. This is because, on the irregular topogra­phy, the water elevation H is normally much smoother then the water depth h. The second problem can be solved in two different ways:

a) Extract the term 0.5gh2 from the flux function/ differentiate and combine it with the bottom-slope term. In this way one gets

ƒ uh

_u~h_ >s =

0 gh{sfx-sHx)

where sHr = -̂ r— dH 3A

(5.2)

b) Extract the term 0.5gh2 from the flux function/ equation (3.1), and discretize it so that it is "compatible" with the bottom-slope term.

The second approach was adopted here, because it retains automatically the proposed numerical schemes conservative. Let us assume that the water was initially at rest inside of some closed area with variable bottom topography. Then, without influence from outside, the water should stay at rest. Now, the term "compatible" means that the same requirement should be fulfilled by our discrete model. In other words, numerical scheme should not introduce any artificial source terms in the numerical solution. The shallow water equations can be brought into the following form, appropriate for the numerical solution by the proposed methods

ui + ƒ> + Px+ s = 0 (5.3)

where

H' uh_

0

0.5 gh\

ƒ =

s =

' uh'

_u h. 0

Jgh( V-. - Sbx) -

(5.4)

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The numerical flux ƒ (convection), is discretized by using the non-oscillatory method described in section 2.1. The coefficient a is defined in the same manner as in (2.4)

a max (|w| + j

Jgh), 7 = 1 , N. (5.5)

Such an approach arises from a fact that convection is the main source of troubles in the numerical treatment of hyperbolic equations. A number of numerical experiments confirm that the numerical solution remains free of oscillations as expected, when using the incomplete flux function from equation (5.4). The numerical flux/? is discretized by using standard central differences

Pj^/2 = 0.5(pj+1+Pj). (5.6)

The bottom-slope is also discretized by central differences, so that the following approximation for the bottom-slope and for they'th grid point is valid

(sb). = 0.5s(ft..t, + V i ) (Z j+ i ,) 2Ax

(5.7)

The use of the average h = 0.5(/z, + , + hs _,) instead of the value hj in equation (5.7), make it possible to satisfy compatibility condition stated previously. By using the modified scheme outlined above, the numerical error decreased by an order of magni­tude as can be seen from fig. 5. The proposed scheme remained conservative, as already mentioned, so that mixed regimes of flow can be simulated without difficulties (fig. 5). All these things together make the proposed scheme adequate for the numerical simulation of flooding.

Y 4

b) 0.8 0.7 0.6 0.5 0.4 0.3 0.2

500 1000 X [m]

1500

500 1000 X [m]

1500

Fig. 5. Steady-state flow in a channel. Calculated by modified scheme I, with CFL = 0.95 and a = 0.1 I A,lmix |. a) Water elevation, b) Discharge q.

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An application to real flooding problems, using the numerical scheme outlined above, is given in reference [12]. As a conclusion to this section, one can say that the variable bottom topography can, if not handled correctly, cause a lot of numerical difficulties. This is subject not only to high-resolution methods, but to many other finite-difference and finite-element methods. Some authors perform smoothing of topographical data prior to its use [13], wile the others use very fine discretisation in the region of rapid changes in bottom topography. A natural way to overcome this problem is, however, through the satisfaction of compatibility condition stated previously.

6 Conclusion

It has been shown that non-oscillatory, high-resolution methods, based on the Lax-Friedrichs solver, can be successfully applied for the numerical solution of free-surface flows. They are much easier to implement than methods which use field-by-field decomposition, and require less compu­tational time. However, the accuracy of the new schemes is similar to the other high-resolution schemes. Another advantage of this approach is that the other equations, like dispersion equation or turbu­lence model equations, can be treated in the same way. This is because the Lax-Friedrichs solver does not depend on the structure of the hyperbolic system, like the approximate Riemann solvers which use field-by-field decomposition do. Shallow water flows with variable bottom topography can be successfully treated with methods proposed in this article, because of the appropriate choice of the flow variables and satisfaction of certain compatibility relations. This is in general not easy to achieve for high-resolution methods based on the flux-difference and flux-vector splitting, present in the literature up to now. Therefore, this should be a useful contribution to the computation of free-surface flows with strong variations in bottom topography, using high-resolution shock-capturing schemes.

Notations

A ƒ 8 8 h H m ii

N

P s *b

sf t u II

V

Jacobi matrix flux function (in x direction) gravity acceleration flux function (in y direction) water depth water elevation number of unknowns (or number of equations) Manning coefficient total number of grid points flux function source term bottom slope friction slope time velocity component in x direction vector of unknowns velocity component in v direction

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z bottom height a coefficient for the Lax-Friedrichs flux A. eigenvalue

References

1. GLE1STER, P. (1988) "Approximate Riemann solutions of the shallow water equations", J. Hydr. Res., '26(3), 293-306.

2. ALCRUDO, F. and GARCIA-NAVARO, P. (1993) "A High-Resolution Godunov Type Scheme in Finite Vol­umes for the 2D Shallow-Water Equations", Int. j . numer. methods fluids. Vol. 16, pp. 489-505.

3. YANG J.Y., HSU, C.A., CHANG, S.H. (1993) "Computations of free surface flows, Part 1: One-dimensional dam-break flow", J. Hydr. Res., 31(1).

4. ROE, P.L. (1981) "Approximate Riemann Solvers, Parameter Vectors and Difference Schemes", J. Comput. Phys., 43, pp. 357-372.

5. VAN LEER, B. (1982) "Flux Vector Splitting of the Euler Equations", Lecture Notes in Physics, Vol. 170, pp. 507-512.

6. SHU, C.W. and OSHER, S. (1988) "Efficient Implementation of Essentially Non-oscillatory Shock-Capturing Schemes", J. Comput. Phys., 77, pp. 439-471.

7. NESSYAHU, H. and TADMOR, E. (1990) "Non-oscillatory Central Differencing for the Hyperbolic Conser­vation Laws", J. Comput. Phys., 87, pp. 408-463.

8. LIN, S.Y. and CHIN, Y.S. (1993) "Discontinuous Galerkin Finite Element Method for Euler and Navier-Stokes Equations", AIAA Journal, Vol. 31, No. 11, pp. 2016-2026.

9. ALCRUDO, F., GARCIA-NAVARO, P. and SAVIRON, J.M. (1992) "Flux Difference Splitting for ID Open Channel Flow Equations", Int. j . numer. methods fluids, Vol. 14, pp. 1009-1018.

10. BECHTELER, W., KULISCH, H. and NUJIC, M. (1992) "2D Dam-Break Flooding Waves - Comparison between Experimental and Calculated Results", 3rd Int. Conf. on Flood and Flood Management, Flor­ence, 24-26. Nov.

11. ROE, P. L. (1986) "Upwind Differencing Schemes, Hyperbolic Conservation Laws with Source Terms", 1st Int. Congress on Hyperbolic Problems, St. Etienne, (ed. C. CARASSA and D. SERRE).

12. BECHTELER, W., NUJIC, M. and OTTO. J.A. (1994) "An Analysis of Flood Propagation Using the Program Package FLOODSIM", Int. Conf. on Modelling of Flood Propagation over Initially Dry Areas, 29-30 June, Milan, Italy.

13. LICK, W.J. (1989) "Difference Equations from Differential Equations", Lecture Notes in Engineering; 41, Springer-Verlag, Berlin.

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