Advances in Water Resources 31 (2008) 962–974

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Advances in Water Resources

journal homepage: www.elsevier .com/ locate/advwatres

A weighted surface-depth gradient method for the numerical integrationof the 2D shallow water equations with topography

F. Aureli a, A. Maranzoni b, P. Mignosa a, C. Ziveri a,*

a DICATeA, Università di Parma, V.le G.P. Usberti 181/A, 43100 Parma, Italyb DICATA, Università di Brescia, Via Branze 43, 25123 Brescia, Italy

a r t i c l e i n f o

Article history:Received 15 October 2007Received in revised form 29 February 2008Accepted 17 March 2008Available online 4 April 2008

Keywords:Shallow water equationsFinite volume schemesSurface gradient methodIrregular topographyC-propertyWet/dry frontsMass conservation

0309-1708/$ - see front matter � 2008 Elsevier Ltd. Adoi:10.1016/j.advwatres.2008.03.005

* Corresponding author. Tel.: +39 0521 905953; faxE-mail addresses: francesca.aureli@unipr.it (F. Au

unibs.it (A. Maranzoni), paolo.mignosa@unipr.it (P. Munipr.it (C. Ziveri).

a b s t r a c t

A finite volume MUSCL scheme for the numerical integration of 2D shallow water equations is presented.In the framework of the SLIC scheme, the proposed weighted surface-depth gradient method (WSDGM)computes intercell water depths through a weighted average of DGM and SGM reconstructions, in whichthe weight function depends on the local Froude number. This combination makes the scheme capable ofperforming a robust tracking of wet/dry fronts and, together with an unsplit centered discretization of thebed slope source term, of maintaining the static condition on non-flat topographies (C-property). A cor-rection of the numerical fluxes in the computational cells with water depth smaller than a fixed toleranceenables a drastic reduction of the mass error in the presence of wetting and drying fronts. The effective-ness and robustness of the proposed scheme are assessed by comparing numerical results with analyticaland reference solutions of a set of test cases. Moreover, to show the capability of the numerical model onfield-scale applications, the results of a dam-break scenario are presented.

� 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Two-dimensional shallow water equations (SWE) are currentlyaccepted to mathematically describe a wide variety of free surfaceflows under the effect of gravity, such as dam-break waves, prop-agation of flood waves in rivers, flood plain inundations, etc. Be-cause the analytical solutions available in the literature concernonly a few simple situations [24,27,29,25,22,18], the SWE need tobe numerically solved in order to deal with more generalapplications.

The irregular topography of the regions subject to flooding canstrongly affect the flow dynamics, giving rise to the formation ofhydraulic jumps, shocks and reflections; for this reason, in thenumerical simulation of such phenomena an efficient treatmentof the bed slope source term is necessary to obtain accurate resultsboth in the case of steady and unsteady flows.

In engineering applications the necessity to handle wetting anddrying moving boundaries is a challenge that researchers are alsotackling [4,5,13,10,6,7]. It is well known that small water depthsnear wet/dry interfaces can lead to numerical instabilities. Thesimple procedure of drying cells in which water depth is smaller

ll rights reserved.

: +39 0521 905924.reli), andrea.maranzoni@ing.ignosa), chiara.ziveri@nemo.

than a fixed tolerance [31] avoids these unphysical oscillations,but it is not completely satisfying, because it induces a mass errorthat grows into not acceptable values in field-scale applications[11,7,23]. Another numerical difficulty is related to partially wetcells in which pressure and gravity forces are not exactly balancedin static conditions [15,10].

In recent years a large amount of research has dealt with theapplication of finite volume Godunov-type methods for the numer-ical solution of SWE with source terms [30,8,14,9,33,16,19,34,26].In the fractional step method [31,20] the complete equations aresplit into a homogeneous problem and an ODE system. Althoughsimple, the procedure performs poorly if applied to SWE withgeometric source terms, especially near the steady state [31,20].A more complex approach, which gives more satisfactory results,concerns the upwinding of the source terms, in a manner similarto that adopted for the construction of numerical fluxes for solvinghomogeneous conservation laws. Bermúdez and Vázquez-Cendón[8] applied this treatment to first order Roe’s scheme and definedthe fundamental notion of C-property, that is the capability ofreplicating the exact solution for the stationary flow problem. Thisupwinding approach was extended to a wide range of problems byGarcía-Navarro and Vázquez-Cendón [14], Bermúdez et al. [9] andVázquez-Cendón [33]. Later Hubbard and García-Navarro [16] gen-eralized this technique to a finite volume high order TVD version ofRoe’s scheme and to arbitrary polygonal meshes. In [19] LeVequeproposed the quasi-steady state wave propagation algorithm inwhich, avoiding any splitting, the source term is incorporated into

F. Aureli et al. / Advances in Water Resources 31 (2008) 962–974 963

the wave propagation algorithm; a Riemann problem is introducedat the center of each cell whose flux difference exactly cancels thesource term. According to the author, this method is effective whenthe solution is near the steady state, but it presents difficulties inthe case of transcritical steady flows with shocks. In Zhou et al.[34] the surface gradient method was introduced. In order to evalu-ate more accurate numerical fluxes in the presence of a non-flattopography, the intercell water depth was computed startingfrom the MUSCL reconstruction of the water surface level. Togetherwith a centered discretization of the bottom slope source term,the scheme is capable of maintaining the static condition on aCartesian grid. This approach is very attractive for its simplicity,but, near wet/dry fronts on uneven topographies, the SGM recon-struction can lead to very small water depths (even negative) thatneed to be modified in order to avoid unphysical results. A morerobust behavior in the front tracking can be obtained with theconventional DGM (depth gradient method) approach which evalu-ates intercell water depths starting from the extrapolation of thesame conserved variable; on the other hand, this method doesnot maintain the static condition if a centered discretization isused for the bed slope source term. In the recent work of Valianiand Begnudelli [32] the bed slope source term is expressed asthe divergence of a suitable matrix related to the static force dueto the bottom slope; moreover, a SGM variable reconstruction isperformed to compute water depth at the cell boundaries. Thistechnique is simple and allows the preservation of the conditionof quiescent water on a totally wet domain on even though the gridis irregular.

In this paper a weighted surface-depth gradient method(WSDGM) is proposed with the aim of combining the good capabil-ities of SGM and DGM approaches and avoiding their drawbacks. Inthe framework of the SLIC scheme [31], the proposed algorithmcomputes water depths at the cell boundaries through a weightedaverage of the extrapolated values deriving from DGM and SGMreconstructions: in this way the scheme is capable of maintainingthe static condition on non-flat topographies and performing a ro-bust tracking of wet/dry fronts.

In cells in which the computed water depth is lower than asmall threshold, a flux correction is applied in order to drasticallyreduce the mass error.

The effectiveness and robustness of the proposed scheme onnon-horizontal beds and in the presence of wet/dry fronts ischecked on a set of test cases for which an analytical solution isavailable or a highly accurate numerical solution can be derived.

Finally, the simulation of a real case study concerning the hypo-thetical collapse of the dam on the Parma river (Northern Italy)confirms the applicability of WSDGM to field-scale problems onirregular topographies with wetting and drying.

2. Numerical model

Under the usual De St. Venant hypotheses of small bottomslopes and small streamline curvatures, the integral form of SWEreads [31]:

o

ot

ZA

UdAþZ

CH � ndC ¼

ZA

S0 þ Sfð ÞdA: ð1Þ

In (1) A is the area of the integration element, C the element bound-ary, n the outward unit vector normal to C, U the vector of the con-served variables and H = (F,G) the tensor of fluxes in the x and ydirections:

U ¼h

uh

vh

264

375; F ¼

uh

u2hþ 12 gh2

uvh

264

375; G ¼

vh

uvhv2hþ 1

2 gh2

264

375; ð2Þ

where h is the flow depth, u and v are the velocity components inthe x and y directions and g is the acceleration due to gravity. Thebed and friction slope source terms S0 and Sf are expressed accord-ing to the following relations:

S0 ¼ 0;�ghozox

;�ghozoy

� �T

;

Sf ¼ 0;�ghn2

f uffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2 þ v2p

h4=3 ;�ghn2

f vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2 þ v2p

h4=3

" #T

; ð3Þ

in which z is the bed level with respect to a horizontal reference andnf is the roughness coefficient of the Manning equation.

Among the numerical techniques for the integration of (1), thefinite volume method allows the computation of the variation DUover the time interval Dt as a function of the numerical fluxes ex-changed through the boundaries and of the forcing source terms. Ifapplied to a grid composed of polygonal elements with N sides, thevariation DU takes the form [30,20]:

DU ¼ �DtA

XN

k¼1

ðfk � nkÞDlk þDtA

ZAðS0 þ SfÞdA; ð4Þ

where nk is the outward unit vector normal to the k-side of the cellof length Dlk. If the grid is composed of Cartesian cells of areaDx � Dy and the first order Godunov splitting of the source termsis applied [30], the explicit updating algorithm for the discretizedvariable Ui,j becomes:

U�i;j ¼ Uni;j � Dt

Dx Fiþ12;j� Fi�1

2;j

� �� Dt

Dy Gi;jþ12� Gi;j�1

2

� �Unþ1

i;j ¼ U�i;j þ Dt S0 U�i;j� �

þ Sf U�i;j� �h i

:

8><>: ð5Þ

In (5) the superscript n denotes the current time level, while Fi±1/2,j

and Gi,j±1/2 are the numerical fluxes exchanged through the bound-aries between neighboring cells in the x and y directionsrespectively.

The numerical fluxes can be computed using several methodol-ogies. In previous works [2,3] the writers adopted the well-knownSlope LImiter Centred scheme (SLIC) [31] belonging to the MUSCLfamily and to the depth gradient method (DGM) class. The schemeconsists of the following steps.

(a1) A set of boundary extrapolated variables is evaluated at theinterfaces of each computational cell to achieve second orderof accuracy in space. For direction x it reads:

� �

UL

iþ12;j¼ Un

i;j þ12

Uþi�1

2;jUn

i;j � Uni�1;j ;

URiþ1

2;j¼ Un

iþ1;j �12

U�iþ32;j

Uniþ2;j � Un

iþ1;j

� �:

ð6Þ

In (6), the limiter diagonal matrices U [28] enforce a TVDconstraint on the slope of the boundary extrapolated vari-ables in order to avoid spurious oscillations. These limitersare a function of the ratios r± of the consecutive variations:

Uþi�1

2;j¼ U rþ

i�12;j

� �; U�iþ3

2;j¼ U r�iþ3

2;j

� �; ð7Þ

with vectors r± defined as:

rþi�1

2;j¼

Uniþ1;j � Un

i;j

Uni;j � Un

i�1;j; r�iþ3

2;j¼

Uniþ1;j � Un

i;j

Uniþ2;j � Un

iþ1;j: ð8Þ

Among several slope limiter functions proposed in the litera-ture [20], one of the following may be adopted:

964 F. Aureli et al. / Advances in Water Resources 31 (2008) 962–974

VanLeer Limiter :U rð Þ ¼rþ rj j1þr if r > 0

0 if r 6 0;

(

VanAlbada Limiter :UðrÞ ¼r2þr1þr2 if r > 0

0 if r 6 0;

(

Minmod Limiter :UðrÞ ¼minðr;1Þ if r > 0

0 if r 6 0;

(

Superbee Limiter :UðrÞ ¼max 0;minð2r;1Þ;minðr;2Þ½ �:

ð9Þ

Since the first component of (6) is reconstructed according tothe DGM technique, it is labeled as:

hL;DGMiþ1

2;j¼ hn

i;j þ12

Uþi�1

2;jhn

i;j � hni�1;j

� �;

hR;DGMiþ1

2;j¼ hn

iþ1;j �12

U�iþ32;j

hniþ2;j � hn

iþ1;j

� �:

ð10Þ

(b1) The boundary extrapolated variables are evolved over Dt/2in order to achieve second order of accuracy in time:

ULiþ1

2;j¼ UL

iþ12;j� Dt

2DxF UL

iþ12;j

� �� F UR

i�12;j

� �h i� Dt

2DyG UL

i;jþ12

� �� G UR

i;j�12

� �h i;

URiþ1

2;j¼ UR

iþ12;j� Dt

2DxF UL

iþ32;j

� �� F UR

iþ12;j

� �h i� Dt

2DyG UL

iþ1;jþ12

� �� G UR

iþ1;j�12

� �h i:

ð11Þ

(c1) The numerical intercell fluxes are evaluated according to theFirst ORder CEntred (FORCE) method [31]:

Fforceiþ1

2;j¼ Fforce

iþ12;j

ULiþ1

2;j;UR

iþ12;j

� �¼ 1

2FLF

iþ12;j

ULiþ1

2;j;UR

iþ12;j

� �þ FRI

iþ12;j

ULiþ1

2;j;UR

iþ12;j

� �h i;

Gforcei;jþ1

2¼ Gforce

i;jþ12

ULi;jþ1

2;UR

i;jþ12

� �¼ 1

2GLF

i;jþ12

ULi;jþ1

2;UR

i;jþ12

� �þ GRI

i;jþ12

ULi;jþ1

2;UR

i;jþ12

� �h i;

ð12Þ

which performs an arithmetic average between the first or-der centered flux of Lax–Friederichs [31]:

FLFiþ1

2;jUL

iþ12;j;UR

iþ12;j

� �¼ 1

2Fiþ1

2;jUL

iþ12;j

� �þ Fiþ1

2;jUR

iþ12;j

� �h iþ 1

2DxDt

ULiþ1

2;j� UR

iþ12;j

� �;

GLFi;jþ1

2UL

i;jþ12;UR

i;jþ12

� �¼ 1

2Gi;jþ1

2UL

i;jþ12

� �þ Gi;jþ1

2UR

i;jþ12

� �h iþ 1

2DxDt

ULi;jþ1

2� UR

i;jþ12

� �ð13Þ

and the second order centered flux of Richtmeyer [31]:

FRIiþ1

2;jUL

iþ12;j;UR

iþ12;j

� �¼ F URI

iþ12;j

� �;

URIiþ1

2;j¼ 1

2UL

iþ12;jþ UR

iþ12;j

� �þ 1

2DxDt

F ULiþ1

2;j

� �� F UR

iþ12;j

� �h i;

GRIi;jþ1

2UL

i;jþ12;UR

i;jþ12

� �¼ G URI

i;jþ12

� �;

URIi;jþ1

2¼ 1

2UL

i;jþ12þ UR

i;jþ12

� �þ 1

2DxDt

G ULi;jþ1

2

� �� G UR

i;jþ12

� �h i:

ð14Þ

The illustrated scheme is robust and stable when high water le-vel gradients occur together with small water depth gradients, asshown in the case of a supercritical steady flow over a bump, pre-sented in Section 3.1. Introducing a threshold value h� to avoidinstabilities in the presence of very small depths [31], the schemeis also capable of tracking the motion of wetting and drying fronts.

However, if a centered discretization is used for the bed slopeterm, this approach can not replicate the exact solution of thestatic flow problem gðx; y; tÞ ¼ hðx; y; tÞ þ zðx; yÞ ¼ g;uðx; y; tÞ ¼0; vðx; y; tÞ ¼ 0, i.e. does not satisfy the exact C-property [8].

This C-property can be instead satisfied on a Cartesian grid if theSurface Gradient Method is applied in the data reconstruction step[34].

If this method is incorporated into the SLIC scheme, numericalfluxes are computed by the following steps.

(a2) The evaluation of water depths at cell interfaces is obtainedfrom the reconstruction of the intercell water level g = h + z[34]:

gLiþ1

2;j¼ gn

i;j þ12

Uþi�1

2;jgn

i;j � gni�1;j

� �hL;SGM

iþ12;j¼ gL

iþ12;j� ziþ1

2;j;

gRiþ1

2;j¼ gn

iþ1;j �12

U�iþ32;j

gniþ2;j � gn

iþ1;j

� �hR;SGM

iþ12;j¼ gR

iþ12;j� ziþ1

2;j;

ð15Þ

where zi+1/2,j = (zi,j + zi+1,j)/2 represents an estimation of theintercell bed elevation if a linear variation of the bed profileis assumed.For the conserved variables uh and vh the same reconstruc-tion of Step (a1) is performed.

(b2) In order to satisfy the C-property when SGM is adopted, theevolution of the conserved variables over Dt/2 must includethe contribution due to gravity:

ULiþ1

2;j¼ UL

iþ12;j� Dt

2DxF UL

iþ12;j

� �� F UR

i�12;j

� �h i� Dt

2DyG UL

i;jþ12

� �� G UR

i;j�12

� �h iþ Dt

2S0i;j;

URiþ1

2;j¼ UR

iþ12;j� Dt

2DxF UL

iþ32;j

� �� F UR

iþ12;j

� �h i� Dt

2DyG UL

iþ1;jþ12

� �� G UR

iþ1;j�12

� �h iþ Dt

2S0iþ1;j

ð16Þ

where the bed slope source term is discretized with a cen-tered scheme [34]:

2 3 S0i;j ¼

0

�ghL

iþ12;jþhR

i�12;j

2

ziþ1

2;j�z

i�12;j

Dx

�ghL

i;jþ12þhR

i;j�12

2

zi;jþ1

2�z

i;j�12

Dy

6666477775: ð17Þ

(c2) Numerical fluxes Fi�12;j

, Gi;j�12

are evaluated via the FORCEmethod described at point (c1).

After the computation of numerical fluxes, the solution is up-dated replacing (5) by the following algorithm:

U�i;j ¼ Uni;j � Dt

Dx Fiþ12;j� Fi�1

2;j

� �� Dt

Dy Gi;jþ12� Gi;j�1

2

� �þ Dt S0i;j

Unþ1i;j ¼ U�i;j þ Dt Sf U�i;j

� �;

8><>: ð18Þ

in which S0i;j is discretized with the centered approximation (17) asa function of the evolved variables ðhL;R

i�12;j;hL;R

i;j�12Þ. In (18) the unsplit-

ting of the bed slope source term guarantees the exact conservationof the static condition on a completely wet irregular topography.

The SGM is equivalent to the DGM in the case of a flat bottom,and entails an equivalent computational effort [34]. SGM recon-structions are preferable for those applications in which smallwater level gradients occur in the presence of high water depth gra-dients, as, for example, in the case of a subcritical steady flow over abump, as reported in Section 3.1. The application of SGM to field-scale cases often leads to unsatisfactory results in the treatment

F. Aureli et al. / Advances in Water Resources 31 (2008) 962–974 965

of the wetting and drying fronts, especially on highly irregulartopographies. In fact, near wet/dry interfaces on a non-flat bottom,the SGM reconstruction can give rise to very small water depths(even negative) and, as a consequence, to unphysical results.

In order to overcome the limitations of SGM and DGM recon-structions and, at the same time, retain their good capabilities, aweighted surface-depth gradient method was implemented accord-ing to the following computational steps.

(a3) In the data reconstruction step, the water depth at cell inter-faces is estimated through a weighted average of the bound-ary extrapolated values deriving from MUSCL DGM and SGMextrapolations:

hLiþ1

2;j¼ #i;jh

L;DGMiþ1

2;jþ 1� #i;j� �

hL;SGMiþ1

2;j;

hRiþ1

2;j¼ #iþ1;jh

R;DGMiþ1

2;jþ 1� #iþ1;j� �

hR;SGMiþ1

2;j;

ð19Þ

in which #i,j is a weighting parameter.The form of #i,j should allow a smooth transition between

a fully SGM extrapolation where water is at rest (Fr = 0) andan essentially DGM extrapolation at wet/dry moving fronts.Both previous requirements can be satisfied by adoptingthe Froude number as the control parameter.

Although any function with the features mentioned abovecan be used, WSDGM adopts a simple trigonometric expres-sion for the weighting function:

#i;j ¼12 1� cos pFri;j

Frlim

� �h i0 6 Fri;j 6 Frlim

1 Fri;j > Frlim;

(ð20Þ

where Frlim is an upper limit beyond which a pure DGMreconstruction is performed.Since a shock wave cannot be adjacent to a region of dry bed[31], water depths must tend gradually to zero and Fr to infi-nite at wet/dry interfaces; however, in a finite volume frame-work, Fr values remain finite also at the shoreline: the choiceof Frlim needs to guarantee an essentially DGM-type behaviornear wet/dry moving fronts.In order to rigorously satisfy the C-property and to preventthe well-known instabilities [4,5,13,6] that arise at the shore-line when a fixed mesh is adopted, a special treatment isperformed at wet-dry interfaces. If the water surface in the(i, j)-cell is lower than the bed elevation of the adjacent drycell, the intercell bed elevation is set at the level of theextrapolated water surface and, as a consequence, the recon-structed water depth is zero [15,1]. The centered estimate ofthe intercell bed elevation is restored when the shoreline nolonger involves the cell.

(b3) The evolution of the extrapolated variables includes the forc-ing effect of the bed slope source term as in Step (b2).

(c3) Numerical fluxes are computed according to Step (c1).

The solution is updated applying algorithm (18). In problemsinvolving friction the source term Sf is discretized applying asecond order semi-implicit trapezoidal method [12].

The occurrence of very small water depths in numerical simula-tions can lead to instabilities, such as negative water depths andunphysical velocities; moreover, in problems with friction, thestructure of the Manning equation is such that, when water depthtends to zero, the bed resistance tends to infinite. In order to avoidthese difficulties, computed waterdepths lower than a fixed toler-ance are usually set at zero. This common procedure leads to a lossof mass when the updated water depth is positive but smaller thanh�, and to a gain when it is negative. In real case studies in whichfronts can be very uneven due to the strong bottom irregularities,the mass error can grow into unacceptable values.

In order to drastically reduce this mass error, when thewater depth h�i;j deriving from the first step of (18) is lower thanh�, in WSDGM a flux correction is performed to obtain h�i;j ¼ 0.Defining:

Ci;j ¼h�i;jDxDy

Dt c1 � F1i�1

2;jþ c2 � F1

iþ12;j

� �Dyþ c3 � G1

i;j�12þ c4 � G1

i;jþ12

� �Dx

h i ;ð21Þ

where F1 and G1 are the first component of the numerical fluxes in xand y directions and ck (k = 1, 2, 3, 4) are integer coefficients equal to0 or 1, three different cases can be distinguished:

– ð0 6 h�i;j 6 h�Þ \ ðhni;j ¼ 0Þ, as occurs, for example, at the wetting

front, when the cell, initially dry, is wetted with a water depthsmaller than h�. If F1 � n < 0 or, similarly, G1 � n < 0 (flow enteringthe (i, j)-cell), the correspondent coefficient ck is set at 1, other-wise at 0. If ck = 1 all the components of the numerical flux vectorare reduced by the same factor ai,j = (1 � Ci,j), otherwise the fluxvector remains unchanged;

– ð0 6 h�i;j 6 h�Þ \ ðhni;j P h�Þ, as occurs, for example, at the drying

front, when the cell, initially wet, is not completely dried. IfF1 � n > 0 or, similarly, G1 � n > 0 (flow leaving the (i, j)-cell), thecorrespondent coefficient ck is set at 1, otherwise at 0. If ck = 1all the components of the numerical flux vector are increasedby the same factor ai,j = (1 + Ci,j), otherwise the flux vectorremains unchanged;

– ðh�i;j 6 0Þ, as occurs, for example, at a drying front, when the cellis overdraft. If F1 � n > 0 or, similarly, G1 � n > 0 (flow leaving the(i, j)-cell), the correspondent coefficient ck is set at 1, otherwiseat 0. If ck = 1 all the components of the numerical flux vectorare reduced by the same factor ai,j = (1 + Ci,j), otherwise the fluxvector remains unchanged.

After the flux correction, the first step of (18) is recomputed;since this procedure can lead to h�i;j < h� in the adjacent cells, it isiteratively performed: in this way the mass error is drastically re-duced, without significantly increasing the computational effort,because the cells involved by the algorithm are always a small per-centage of the total number of elements at wet/dry interfaces.

In order to compute numerical fluxes at the cell sides placed onthe boundary of the computational domain, boundary conditionsfor the conserved variables h, uh, vh and the bottom elevation zare imposed on ghost cells adjacent to the domain [20], accordingto the different character of the flow.

The proposed WSDGM is globally second order accurate inspace thanks to the MUSCL variable reconstruction [31], but firstorder accurate in time due to the splitting of the friction sourceterm (Eq. (18)) [20]; for frictionless problems, the second orderof accuracy is achieved also in time.

The time step Dt is controlled by means of the 2D CFL stability cri-terion 0 < Crx + Cry 6 1 [30], where Crx and Cry are the Courant num-bers in x and y directions respectively. This assumption leads to:

Dt ¼ Cr maxui;j

þ ffiffiffiffiffiffiffiffiffighi;j

qDx

þvi;j

þ ffiffiffiffiffiffiffiffiffighi;j

qDy

0@

1A

24

35�1

; ð22Þ

where 0 < Cr 6 1.

3. Test cases with reference solution

In this section the robustness and effectiveness of the proposedscheme are assessed applying WSDGM to non-flat topography ref-erence test cases without friction. The dynamics of wetting and dry-ing and the occurrence of complex structures in the flow field, suchas moving shocks and reflections, mean that the tests are severe.

966 F. Aureli et al. / Advances in Water Resources 31 (2008) 962–974

Among the possible options proposed in the literature, the VanLeer limiter function was adopted.

3.1. 1D steady flows over a steep bump

These tests concern 1D steady flows in a [�10 m 6 x 6 + 10 m]frictionless channel. The bottom profile, characterized by the pres-ence of a steep bump, is described by the following equation:

zðxÞ ¼ bc 1� x2

4

� ��2 m 6 x 6 2 m

0 elsewhere;

(ð23Þ

where the height of the bump bc is set at 0.8 [21] instead of theusual value 0.2 (e.g. [34,32,7]), in order to make the test more se-vere. The computational domain is discretized with 0.1 m gridspacing.

According to the conditions imposed at the boundaries (Table1), the water can be at rest (Test A, with initial conditiong(x) = 1 m) or the flow can be transcritical with (Test B) or without(Test C) the occurrence of a hydraulic jump, supercritical (Test D)or subcritical (Test E).

The five cases were simulated by means of WSDGM with differ-ent values of Frlim ranging from Frlim = 0 (pure DGM reconstruction)to Frlim ?1 (pure SGM reconstruction).

Table 2 compares the L1 errors in h and q as a function of Frlim,which are defined as:

L1ðhÞ ¼1N

XN

i¼1

hi;num � hi;ref

hi;ref

;

L1ðqÞ ¼1N

XN

i¼1

qi;num � qref

qref

; ð24Þ

where N is the total number of computational cells, hi,ref is the ref-erence solution derived from the energy equation (together withthe momentum principle if a hydraulic jump is present) and qref isthe constant value of the specific discharge.

Even though in Test E a pure SGM reconstruction (Frlim ?1)gives a slightly better result, on average the lowest L1 errors are ob-tained for values of Frlim in the interval [1,2]. Since in this range thenorms are almost the same, the value Frlim = 2 is chosen to give in-sight into the results. Fig. 1 compares, in the subdomain[�3 m 6 x 6 3 m], reference water level and unit discharge profileswith those computed assuming Frlim = 0, Frlim = 2 and Frlim ?1.

As shown by numerical profiles and L1 norms for Test A, thepure DGM reconstruction (Frlim = 0) does not satisfy the C-property.

Table 1Boundary conditions and Fr range for 1D steady flows over a bump

Test Upstream boundary conditions Downstream boundaryconditions

Fr range

A q = 0.0 m2/s h = 1.0 m 0B q = 0.4 m2/s h = 0.75 m 0.10–4.00C q = 0.4 m2/s Transmissive 0.10–5.03D q = 1.5 m2/s, h = 0.25 m Transmissive 2.31–3.83E q = 1.0 m2/s h = 1.70 m 0.14–0.41

Table 2L1 error norms for h and q as function of Frlim

Frlim Test A Test B Test C

L1(h) L1(h) L1(q) L1(h)

0 2.0E�3 3.2E�3 3.9E�3 4.4E�40.2 0.0E+0 3.2E�3 3.7E�3 4.4E�41 0.0E+0 3.1E�3 3.5E�3 4.3E�42 0.0E+0 3.2E�3 3.5E�3 6.0E�45 0.0E+0 3.3E�3 3.7E�3 6.5E�420 0.0E+0 3.9E�3 4.8E�3 6.2E�3?1 0.0E+0 4.0E�3 5.3E�3 1.1E�2

In the subcritical regions of Tests B, C, E, the results obtainedadopting Frlim = 0 show a dip in the water elevation upstreamand downstream of the bump, where the bottom profile is notsmooth.

If the flow is supercritical downstream from the bump (Tests Cand D), the numerical scheme with Frlim ?1 badly converges to-ward the steady solution, as shown by L1 norms and oscillations inwater elevation and unit discharge profiles. It was verified thattests analogous to B, C and D, characterized by boundary condi-tions giving rise to a thinner nappe over the last part of the bump,cannot be carried out if a pure SGM reconstruction is performed(e.g. Test B with q = 0.25 m2/s and h = 0.60 m).

In Test B a deviation of the computed discharges from the refer-ence solution occurs close to the hydraulic jump; a similar behav-ior can be found in [34,32] even though bc = 0.2 m. This deviation isthe main source of error and gives rise to comparable L1 norms forall the examined values of Frlim.

When Frlim = 0 and the flow is completely subcritical (Tests Aand E) the bottom discontinuities induce deviations in q at thebeginning and end of the bump; the same behavior occurs in TestD for Frlim ?1.

On the whole, the results obtained for these test cases provethat WSDGM with Frlim = 2 performs better than the schemesbased on pure DGM or SGM reconstructions. For this reason, theparameter Frlim was set at 2 also in the following tests.

In order to numerically prove that WSDGM is second orderaccurate, a convergence analysis on the grid size was performedwith reference to Test C. Four different grid spacings were used,starting from Dx = 0.2 m, and successively halving the value to ob-tain the remaining grids. Fig. 2 shows that the L1 norms in h, q andenergy per unit mass E vanish like OðDx2Þwhen the grid is progres-sively refined.

3.2. Dam-break in a frictionless sloping channel

In this subsection, the rapidly varying 1D flow induced by adam-break in a frictionless sloping channel is investigated(Fig. 3). Among the tests for which an analytical solution of wet/dry fronts is available [24,29], this is one of the most stronglyinfluenced by the threshold value h�, because the water bodylengthens indefinitely, becoming thinner and thinner. For this rea-son, the dam-break in a sloping channel was chosen as the refer-ence test for the evaluation of the threshold effects on the fronttracking.

Following Hunt [17] and introducing the set of dimensionlessvariables:

�x ¼ xh0; �t ¼ t

ffiffiffigpffiffiffiffiffih0

p S0;�h ¼ h

h0; �u ¼ uffiffiffiffiffiffiffiffi

gh0

p ;

�c ¼ cffiffiffiffiffiffiffiffigh0

p ¼ffiffiffi�h

p; ð25Þ

where h0 is the water depth behind the dam, S0 is the bed slope andc is the speed of small amplitude waves in still water, it is possible

Test D Test E

L1(q) L1(h) L1(q) L1(h) L1(q)

6.2E�4 2.0E�4 8.7E�5 3.2E�4 4.3E�44.6E�4 2.0E�4 8.7E�5 3.0E�4 4.1E�41.9E�4 2.0E�4 8.7E�5 6.3E�5 1.0E�41.6E�4 2.0E�4 8.7E�5 2.9E�5 4.0E�51.8E�4 1.1E�3 3.3E�4 1.9E�5 2.6E�55.4E�3 2.4E�3 9.3E�4 1.7E�5 3.2E�59.8E�3 2.6E�3 1.1E�3 1.7E�5 2.3E�5

Fig. 1. 1D steady flows over a steep bump: comparison between reference solutions and numerical results. In the shaded region Fr P 2.

F. Aureli et al. / Advances in Water Resources 31 (2008) 962–974 967

to rewrite the 1D De St. Venant equations in the dimensionlesscharacteristic form:

d�x�d�t¼ �u� �c

dd�t

�u �x�;�tð Þ � 2�c �x�;�tð Þ � �tð Þ ¼ 0:

8>><>>: ð26Þ

From (26) the dimensionless expression of the drying and wettingfront (�xdry, �xwet) can be derived [17]:

�xdry ¼12

�t � 2ð Þ2 � 1 �t P 2ð Þ; ð27Þ

�xwet ¼12

t þ 2� �2 � 2: ð28Þ

In the numerical simulation a [�10 m 6 x 6 150 m] channel ofslope S0 = 0.1 was discretized with a mesh of size 0.1 m. The damsection was placed at x = 0 and the maximum water depth h0

was set at 1m.

Fig. 4. Dam-break in a frictionless sloping channel: comparison between analyticaland numerical solutions of wetting and drying fronts.

Table 3L2 error norms of the drying front position as function of Dx

�h� Dx = 0.05 m Dx = 0.1 m Dx = 0.2 m

10�5 0.088 (behind) 0.284 (behind) 0.584 (behind)10�4 0.135 (ahead) 0.256 (ahead) 0.176 (ahead)

Fig. 2. 1D steady flows over a steep bump (Test C): L1(h), L1(q) and L1(E) as functionof Dx, computed with WSDGM and Frlim = 2.

Fig. 3. Definition sketch for the dam-break problem in a sloping channel.

968 F. Aureli et al. / Advances in Water Resources 31 (2008) 962–974

The comparison between analytical and numerical fronts isshown in Fig. 4. The wetting front is not particularly sensitive tothe value of the dimensionless threshold �h� ¼ h�=h0, but the dryingprocess is more so. In particular, the numerical drying fronts ob-tained imposing �h� ¼ 10�5 and �h� ¼ 10�4 are respectively behindand ahead the analytical solution. The drying front is obviouslyinfluenced by the grid dimension too. Table 3 compares the L2 errornorms of the drying front position as a function of Dx, which are de-fined as:

L2 �xdry� �

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1N

XN

i¼1

�xdry;i� �

num � �xdry;i� �

an

�2

vuut ; ð29Þ

with N the number of computational time steps in the dimension-less interval 2 6 t 6 5. Despite the halving and the doubling ofthe grid size, it is confirmed that the analytical drying front lies be-tween the numerical fronts obtained assuming �h� equal to 10�5 and10�4. Thus, it can be stated that a value of �h� lying in the interval[10�5,10�4] is suitable to achieve a satisfactory tracking of wet/dry interfaces.

3.3. 2D periodic motions in a parabolic basin

The capability of the proposed method of providing accurate re-sults in the presence of 2D wetting and drying moving boundarieson non-flat topographies was verified comparing numerical resultswith two exact solutions given by Thacker [29], which concern theoscillation of a water volume in a frictionless paraboloidic basin ofequation:

z ¼ z0 1� x2 þ y2

L2

� : ð30Þ

In (30) the depth function z is positive below the equilibrium level(Fig. 5a and Fig. 7a), z0 is the depth of the vertex of the paraboloidand L is the radius at z = 0.

In the following sections, two cases will be considered, corre-sponding to particular choices for initial values. In the first (Section3.3.1) the water body rotates in the basin, maintaining its free sur-face planar and the velocity field uniform, while in the second (Sec-tion 3.3.2) the water surface is a parabola of revolution whichexpands and contracts periodically.

3.3.1. Planar water surfaceIn this case the moving shoreline is a circle of radius L whose

center C describes a circle of radius n (Fig. 5a). The equations of thismotion are given by [29]:

gðx; y; tÞ ¼ 2n z0L

xL cos xt � y

L sin xt � n2L

�uðx; y; tÞ ¼ �nx sin xt; vðx; y; tÞ ¼ �nx cos xt;

(ð31Þ

where g is the surface elevation, positive above the equilibrium le-vel and x ¼

ffiffiffiffiffiffiffiffiffiffi2gz0

p=L is the frequency of the rotation around the

center of the basin. The magnitude of the velocity vectors is con-stant over time at the value |V| = nx, whereas the direction rotatesover time describing an angle a = 3p/2 � xt during a period(0 6 t 6 T). This test is extremely severe for numerical models sincea great number of cells is continuously wetted and dried.

The test was performed in a square domain[�1.6 m 6 (x,y) 6 + 1.6 m] with z0 = 0.05 m, L = 1 m and n = 0.5 m;the basin dimensions are such that the water never reaches theboundaries. The numerical simulation was carried out for fourperiods with Dx = Dy = 0.02 m and h� = 3 � 10�6 m. Since the waterbody remains compact and intersects sharply the bathymetry, thenumerical solution is not particularly sensitive to the threshold va-lue h�.

Fig. 5b shows a contour map of the computed results at t =(15/8)T, when a = �p/4: the shoreline is still circular, the surfacealmost perfectly planar and the velocity field nearly uniform.

Fig. 6a shows the comparison between numerical and analyticalwater depth time series in the last two periods of simulation atpoints (1.0;0.0), (0.5;0.0), (0.4;0.0). The first point gets wet anddry during the periodic motion, the second gets dry only att ¼ ð0:5þ nÞT; ðn 2 NÞ, whereas the third remains wet all the time.

Fig. 5. (a) Definition sketch for Thacker test with planar water surface [29] and (b) contour map of the numerical results at t = (15/8)T.

Fig. 6. Thacker test with planar surface: (a) comparison between analytical and numerical results for water depth h and (b) L2 error norms for h, |V| and a.

F. Aureli et al. / Advances in Water Resources 31 (2008) 962–974 969

An overall quantitative information about the accuracy of thenumerical reconstruction is given by L2 error norms for waterdepth h, velocity magnitude |V| and direction a estimated asfollows:

L2 hð Þ ¼ 1�han tð Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPNweti¼1 hið Þnum � hið Þan

�2

Nwet

s;

L2 Vj jð Þ ¼ 1Vj jan

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPNweti¼1 Vj jið Þnum � Vj jan

�2

Nwet

s;

L2 að Þ ¼ 1p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPNweti¼1 aið Þnum � aan �2

Nwet

s;

ð32Þ

where Nwet is the total number of wet cells and �hanðtÞ is the averagewater depth, in this case not depending on time and equal to z0/2.The trend of the norms during the last two simulation periods isshown in Fig. 6b. The error on the velocity norm and on the direc-tion a are mainly due to the cells close to the shoreline: here waterdepths become very small and the derived variables u and v, ob-tained by dividing unit discharges by water depth, can assumeslightly incorrect values. At the end of four simulation periods,the volume error (relative to the volume at t = 0) is about5 � 10�7. This residual error is due to a few cells on the shorelinein which water depth remains less than h� after the iterative proce-dure; in these cells water depth is then set at zero. If a pure SGM

Fig. 7. (a) Definition sketch for Thacker test with curved water surface [29] and (b) slices of numerical results and analytical solution at selected times.

970 F. Aureli et al. / Advances in Water Resources 31 (2008) 962–974

reconstruction is performed, spurious oscillations caused by unreli-able extrapolations at wet-dry interfaces grow in time and the sim-ulation crashes.

3.3.2. Curved water surfaceThe second exact solution analyzed concerns the periodic mo-

tion of a circular paraboloidic water volume initially at rest andsubject to gravity (Fig. 7a). The equations of the motion are [29]:

g x; y; tð Þ ¼ z0

ffiffiffiffiffiffiffiffi1�A2p

1�A cos xt � 1� x2þy2

L21�A2

1�A cos xtð Þ2� 1

h i� �u x; y; tð Þ ¼ 1

1�A cos xt12 xxA sin xt� �

v x; y; tð Þ ¼ 11�A cos xt

12 xyA sin xt� �

;

8>>><>>>:

ð33Þ

Fig. 8. Thacker test with curved water surface: comparison between analytical and numewater depth h and velocity magnitude jVj.

where the frequency x and the parameter A are:

x ¼ffiffiffiffiffiffiffiffiffiffi8gz0

pL

; A ¼ z0 þ g0ð Þ2 � z0ð Þ2

z0 þ g0ð Þ2 þ z0ð Þ2: ð34Þ

According to the first of (33), the surface remains a parabola ofrevolution during the motion and the water body moves awayfrom the center in the first half of the period T and then convergestowards in the second half. Moreover, at times:

t� ¼ nT � 1x

arccos1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� A2

pA

!; n 2 N ð35Þ

rical results for (a) water depth h and (b) velocity component u. (c) L2 error norms for

F. Aureli et al. / Advances in Water Resources 31 (2008) 962–974 971

all the terms between curly brackets in (33) vanish, so the parabo-loidic water surface degenerates in a planar surface withg(x,y, t*) = 0.

Due to the large variation over time of the number of wet cells,this problem is extremely severe for numerical models, too. Thetest was performed in a square domain [�1.75 m 6(x,y) 6 + 1.75 m] with z0 = 0.05 m, L = 1 m and g0 = 0.10 m; withthe assumed parameters the radius of the shoreline at the maxi-mum expansion (t ¼ ð0:5þ nÞT;n 2 N) is Rmax ¼

ffiffiffi3p

m and theboundaries of the domain are never reached by the water body.The numerical simulation was carried out for four periods withDx = Dy = 0.02 m and h� = 5 � 10�6 m.

Fig. 7b shows slices along the x-axis of numerical results andanalytical solution at some selected times. A moderate underesti-mation of the water depth occurs near the center after four periods.

Fig. 9. Circular dam-break on a non-flat bottom: comparison between 2D WSDGM

Fig. 8a and b show the comparison between numerical and ana-lytical solution for water depth h and velocity component u atpoints (0.0;0.0), (0.5;0.0), (0.7;0.7). The first two points remainwet all the time, whereas the third gets wet and dry during theperiodic motion. Small local differences with the analytical solu-tion are shown more clearly in the insets, where all the computa-tional points are reported.

The L2 norm of h (Fig. 8c) was evaluated according to (32),where the average water depth �hanðtÞ is equal to (h(0,0, t) � z0)/2.Since the velocity field vanishes at t ¼ n=2T;n 2 N, L2(|V|) was nor-malized by the analytical velocity magnitude at the shoreline att = t*.

Although the mass error (with respect to the mass at t = 0)slightly increases over time for the same reason explained in Sec-tion 3.3.1, after four simulation periods the introduction of the flux

results and 1D radial reference solution for water surface level and velocity.

Fig. 11. Circular dam-break on a non-flat bottom: comparison between 1D radialreference solution and 2D numerical profiles of water surface level and velocity att = 0.6 s applying a pure DGM reconstruction.

972 F. Aureli et al. / Advances in Water Resources 31 (2008) 962–974

correction procedure yields to obtain a relative mass error of about6 � 10�8 without corrupting the tracking of wet/dry fronts; ifthe flux correction is not applied the error is about 0.01. Like inthe previous case, if pure SGM extrapolations are performed, thenumerical simulation crashes due to spurious oscillations close tothe shoreline.

3.4. Circular dam-break on a non-flat bottom

This test was developed by the authors themselves in order tostudy the flow consequent to the sudden collapse of an idealizedcircular dam placed on a non-horizontal bottom. A cylindricalwater volume of radius R0 = 10 m is initially placed in a circular do-main of radius R = 25 m centered in (x = 0,y = 0). The bottom profileis described by the following equation:

z rð Þ ¼ 0:5 1þ cos2p5

r� � �

; ð36Þ

where r ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2

pis the radius. As initial conditions it is assumed:

g r;0ð Þ ¼ 10 m 0 6 r 6 R0

h r;0ð Þ ¼ 0 m R0 < r 6 R

ur r;0ð Þ ¼ 0 m=s everywhere:

8><>: ð37Þ

At the circular boundary a reflective condition is imposed.Since the problem has a cylindrical symmetry, an inhomoge-

neous set of 1D differential equations can be derived along r [31]:

oUotþ oF Uð Þ

or¼ Sr Uð Þ þ S0 Uð Þ; ð38Þ

where

U ¼ h;urh½ �T; F Uð Þ ¼ hur; hu2r þ 1

2 gh2h iT

;

Sr Uð Þ ¼ � 1r hur;hu2

r

h iT; S0 Uð Þ ¼ �gh 0; oz rð Þ

or

h iT:

ð39Þ

In (39) ur = ur(r, t) is the radial velocity, Sr(U) is the source term in-duced by the metrics and S0(U) is the source term due to the non-flat bottom.

A reliable approximation of the exact solution was obtainednumerically solving the 1D problem (38) on a very fine mesh(Dr = 0.005 m). The reference solution was computed by means ofthe same WSDGM in which the unsplit centered discretizationof S0 allows the satisfaction of the C-property, while the splittingof Sr, discretized with a pointwise approach, completes the updat-ing of the conserved variables. In the 2D simulation the computa-

Fig. 10. Circular dam-break on a non-flat bottom: contour map of numerical watersurface level at t = 2 s.

tional domain was discretized through a square grid of 0.25 m; thewet/dry tolerance h� was set at 10�4 m.

Fig. 9 shows 2D numerical profiles of water level and velocityalong h = 0 (y = 0) and h ¼ p

4 (y = x) at some selected times com-pared to the 1D profiles computed on the finer mesh. At t = 0.6 sthe capability of WSDGM to preserve the static condition can beappreciated. At t = 2 s and t = 6 s the shock wave coming from the

Fig. 12. Real field case study: contour map of the area under investigation.

F. Aureli et al. / Advances in Water Resources 31 (2008) 962–974 973

boundary is moving toward the center, whereas at t = 8 s the sameshock has already passed through the focusing point at r = 0 and itis now expanding outwards.

The 2D numerical solution is able to reproduce the complexstructures induced by the non-flat bottom and by the reflectionagainst the lateral circular wall. The radial symmetry is well main-tained as can be appreciated also from Fig. 10, which shows thecontour map of the numerical water surface level at t = 2 s.

In Fig. 11 the numerical results obtained adopting Frlim = 0 con-firms that a pure DGM reconstruction can not maintain the static

Fig. 13. (a) Water stage with velocity vectors and (b) velocity

Fig. 14. Water stage contour maps at

condition of flow at rest if a centered discretization for the bedslope source term is applied.

Adopting a pure SGM reconstruction (Frlim ?1), the schemedoes not work at all due to bad extrapolations at wet/dryinterfaces.

4. Case study: collapse of the dam on Parma river

In order to test the applicability of WSDGM to real case studiescharacterized by an irregular topography, the flooding after the

magnitude contour maps at 2 minutes after the breaking.

different times after the breaking.

974 F. Aureli et al. / Advances in Water Resources 31 (2008) 962–974

hypothetical collapse of the barrage on Parma river (NorthernItaly) is presented. The dam, built in 2005 for flood control, is lo-cated about 10 km upstream from the city of Parma. The reservoirhas a storage capacity of about 12 � 106 m3 and the maximumwater depth with reference to the bottom of the stilling basin is16.4 m.

The 10 m DTM of the whole area of interest was reconstructedon a through an interpolation of data acquired from the availablecartography. Fig. 12 illustrates the contour map of the area to-gether with the different conditions imposed at the boundaries.The reflective condition in the North Eastern limit simulates thepresence of a road embankment; it was verified a posteriori thatthe condition to impose on the western and southern boundariesis not important, since the flooding does not reach these areas.At the initial time, the static condition corresponding to the maxi-mum retaining depth (105.6 m a.s.l.) was imposed in the reservoir.The Manning roughness coefficient was set at 0.03 s m�1/3 in theriver and 0.05 s m�1/3 elsewhere; the wet/dry tolerance h� wasset at 2 � 10�3 m. As it can be seen from the flow field (Fig. 13a)and from the contour map of the velocity magnitude (Fig. 13b) at2 minutes after the breaking, WSDGM is able to preserve the staticcondition in the region not yet reached by the rarefaction wave. InFig. 14 water stage contour maps resulting from the numericalmodeling at two selected times are shown as an example.

5. Conclusions

In this paper a finite volume MUSCL-Hancock scheme for thenumerical integration of 2D shallow water equations was pre-sented. In the framework of the SLIC scheme, the weightedsurface-depth gradient method (WSDGM) computes water depthat the cell boundaries through a weighted average, based on thelocal Froude number, of the extrapolated values deriving fromDGM and SGM reconstructions. In particular the scheme appliesa pure SGM reconstruction in static conditions when Fr = 0, and apure DGM reconstruction when Fr is greater than an upper limitFrlim, which was set at 2 after a numerical analysis. The WSDGMreconstruction enables the scheme to perform a robust trackingof wet/dry fronts and, together with an unsplit centered discretiza-tion of the bed slope source term, to maintain the static conditionon non-flat topographies (C-property).

A flux correction applied to the shoreline cells with water depthlower than a threshold value drastically reduces the mass errorwithout corrupting the wet/dry front tracking.

WSDGM was validated through its application to a set of severereference tests involving high bed slopes, 1D and 2D wet/dryfronts. The comparison between reference and numerical resultsproved that WSDGM provides more accurate solutions than thoseobtainable applying pure SGM or DGM reconstructions. Moreover,the case study concerning the flooding after the hypothetical col-lapse of the dam on Parma river confirmed the applicability ofWSDGM also for field-scale applications.

Acknowledgements

This work was supported by the Italian Ministry of the ScientificResearch (MIUR) within the frame of the PRIN project 2005–2006‘‘Dynamics of flooding over complex topography areas”.

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