Author's personal copy - Harvard Universitymsantill/Mauricio_Santillana... · Author's personal copy A local discontinuous Galerkin method for a doubly nonlinear diffusion equation

Author's personal copy

A local discontinuous Galerkin method for a doubly nonlinear diffusionequation arising in shallow water modeling

Mauricio Santillana *, Clint DawsonInstitute for Computational Engineering and Sciences, University of Texas at Austin, United States

a r t i c l e i n f o

Article history:Received 14 May 2009Received in revised form 16 November 2009Accepted 18 November 2009Available online 26 November 2009

Keywords:Discontinuous GalerkinNonlinear diffusionDoubly nonlinearShallow water equationsDiffusive wave approximation

a b s t r a c t

In this paper, we study a local discontinuous Galerkin (LDG) method to approximate solutions of a doublynonlinear diffusion equation, known in the literature as the diffusive wave approximation of the shallowwater equations (DSW). This equation arises in shallow water flow models when special assumptions areused to simplify the shallow water equations and contains as particular cases: the Porous Medium equa-tion and the parabolic p-Laplacian. Continuous in time a priori error estimates are established betweenthe approximate solutions obtained using the proposed LDG method and weak solutions to the DSWequation under physically consistent assumptions. The results of numerical experiments in 2D are pre-sented to verify the numerical accuracy of the method, and to show the qualitative properties of waterflow captured by the DSW equation, when used as a model to simulate an idealized dam break problemwith vegetation.

! 2009 Elsevier B.V. All rights reserved.

1. Introduction

In this paper, we study a numerical scheme based on the localdiscontinuous Galerkin (LDG) method as a means to approximatesolutions to a doubly nonlinear diffusion equation, known in theliterature as the diffusive wave approximation of the shallowwater equations (DSW). This equation arises in shallow water flowmodels when special assumptions are used to simplify the shallowwater equations (SWE), and it gives rise to the following initial/boundary-value problem (IBVP):

@u@t !r " #u!z$a

jruj1!cru

! "% f on X& #0; T';

u % u0 on X& ft % 0g;#u!z$a

jruj1!cru

! "" n % BN on @X \ CN & #0; T';

u % BD on @X \ CD & #0; T';

8>>>>><

>>>>>:

#1$

where X is an open, bounded subset of R2, CN and CD are subsets of@X 2 C1 such that @X % CN ( CD:f : X& #0; T' ! R;u0 : X ! R; BN :@X \ CN & #0; T' ! R, and BD : @X \ CD & #0; T' ! R are given,z : X ! R( is a positive time independent function, n is the out-ward normal to CN; 0 < c 6 1;1 < a < 2 and u : X& #0; T' ! R isthe unknown. Here, j " j : Rd ! R refers to the Euclidean norm inRd (d % 1;2, in our work).

The DSW equation has been successfully applied as a model tosimulate overland flow and shallow water flow in vegetated areas,where water flow is driven mainly by gravitational forces anddominated by shear stresses. See for example [29,20,33,17,18]and [25]. In these water flow regimes, the solution u#x; t$ of theIBVP (1) represents the time evolution of the water height with re-spect to a given datum. The time independent function z#x$ repre-sents the bathymetry or topography over which the water flows, the– frequently time dependent – function f represents sources andsinks (e.g. rainfall or infiltration) and the boundary conditions, BN

and BD, simulate lateral inflow/outflow and the presence of speci-fied water elevation, respectively. A detailed mathematical formu-lation and derivation of the IBVP (1), in the context of shallowwater modeling can be found in [2].

The use of a single equation to describe the time evolution ofwater flow in lieu of the full shallow water system of equations be-comes advantageous, both from the conceptual and computationalpoints of view. Indeed, numerically solving the DSW equation isconsiderably cheaper than numerically solving the SWE [20]. How-ever, determining convergence and stability of numerical schemesto approximate solutions of the DSW equation is not a simple task[25]. Difficulties to analyze numerical schemes aimed atapproximating solutions of the DSW equation arise from the factthat – to the best of our knowledge – existence, uniqueness, andregularity of solutions to the DSW equation for general non-zerobathymetries, z#x$, have not been studied. Note that the DSWequation contains as particular cases two complicated nonlineardiffusion equations: the Porous Medium equation (PME), whenz % 0 and c % 1, and the p-Laplacian for 1 < p < 2, when a % 0

0045-7825/$ - see front matter ! 2009 Elsevier B.V. All rights reserved.doi:10.1016/j.cma.2009.11.016

* Corresponding author. Present address: Harvard University Center for theEnvironment. Tel.: +1 512 698 1564.

E-mail address: [email protected] (M. Santillana).

Comput. Methods Appl. Mech. Engrg. 199 (2010) 1424–1436

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and p ! c" 1, this case is not considered in our work, recall thatwe consider only 1 < a < 2.

The motivation for the present work emanates from our previ-ous work contained in [2] and [25]. Particularly from [25], wherewe studied numerically some qualitative properties of solutionsto the DSW for a collection of non-zero bathymetries z#x$ in 1D,using the continuous Galerkin method. Our findings indicated thatcharacteristics such as: the existence of compactly supported solu-tions, as well as the finite speed of propagation of disturbances(found analytically for solutions for the DSW equation, for flatbathymetries in 1D in [16] #z ! 0$), persisted for non-zero bathym-etries. The property of finite speed of propagation – as opposed tothe infinite speed of propagation in the heat equation, for example– can be understood as a consequence of the advection–diffusionnature of certain types of nonlinear diffusion equations such asthe PME (see [27]) and the DSW equation, and gives rise the pres-ence of free boundaries (locations where the solution goes fromu ! 0 to u > 0) and oftentimes traveling sharp fronts. Discontinu-ous Galerkin methods are well known to be able to capture sharpfronts in solutions to hyperbolic systems – as well as to be locallymass conservative – thus, making them suitable methods to solveour problem. Furthermore, the LDG approach outlined here fitsinto an overall discontinuous Galerkin framework being developedby our group for the approximation of shallow water systems [21].

The work presented in this paper is organized in the followingway. In Sections 1.1–1.4 and 1.5, we introduce the DSW equation,and present a brief introduction to DG methods, the notation usedin such methods, and all the preliminary information needed tocarefully set up and study our particular LDG method. The numer-ical method is constructed in Section 2, and the details of the con-tinuous in time error analysis are presented in Section 2.2. InSection 3 the results of 2D numerical experiments are shown.

1.1. The DSW equation

For completeness in our presentation, we mention some of thekey characteristics of the DSW equation that make it an interestingproblem to be studied, as well as the context in which we intend toapproach our convergence analysis.

The DSW is a doubly nonlinear diffusion equation, since theproduct of two nonlinearities involving u and ru, namely#u% z$a and ru=jruj1%c, appear inside the divergence term. Also,when written in the form

@u@t

%r & #a#u;ru$ru$ ! f #2$

with the diffusion coefficient a given by

a#u;ru$ ! #u% z$a

jruj1%c; #3$

one immediately notices that the nonlinearity involving the gradi-ent of u inside the divergence, ru=jruj1%c, is at best c-Hölder con-tinuous w.r.t. ru, since it scales as jrujc and 0 < c 6 1. As aconsequence, the familiar coercivity and continuity conditions

lkuk2V 6 #a#u$ru;ru$ and #a#u$ru;rw$6 MkukVkwkV for u;w 2 V ; #4$

commonly assumed in the numerical analysis of nonlinear diffusionequations (see [28,15] or [26]) will not hold.1 This fact motivates theneed for further assumptions or properties on the type of solutionsto be approximated if one is to produce a meaningful numericalmethod. To this end, we follow the strategy we presented in [25],for the convergence of the continuous Galerkin method, to restrict

our analysis to the approximation of solutions satisfying physicallyconsistent properties based on shallow water modeling theory. Eventhough the DSW is a degenerate diffusion equation, we will assumethat (1) the solution u does not vanish (i.e. u > !, for a small ! > 0),which corresponds to a wet condition throughout the domain, and(2) that the gradient of u is bounded. The latter assumption is consis-tent with the derivation of the DSW from the SWE, for general andsmooth bathymetries. With these assumptions, we proved in [25]that the continuous Galerkin (CG) method converges to the (as-sumed to be unique and regular) solution of the DSW equation, forfinite elements of polynomial order k with order O#hkc$ (here h rep-resents the diameter of the spatial triangulation). We also foundthat, in theory, we had to use polynomials of degree k > 4 in orderto ensure the boundedness needed on the discrete solution for theproof to succeed. In our numerical experiments in [25], however,we found that we could achieve convergence for our method evenfor solutions of the DSW that vanished in large regions of the do-main, as well as for solutions with unbounded gradient. Moreover,we found that for nondegenerate solutions we could achieve – opti-mal – convergence rates O#h2$ with piecewise linear elements. Theseresults show the gap between our conservative theoretical conver-gence analysis and the actual numerically achievable convergencerates. We do not address this gap in this work. Instead, we focuson extending the results obtained for the CG method to the LDGmethod.

1.2. DG methods

The LDG method is one of many discontinuous Galerkin (DG)methods. These methods are characterized by the fact that conti-nuity across elements is not enforced in the linear space wherethe basis functions live, and thus, the approximate solutions pro-duced are discontinuous or ‘‘broken”. This major difference withthe continuous Galerkin finite element method gives rise to veryinteresting properties that characterize all DG methods. Thesecan be summarized as follows: (1) They can easily handle variousshapes in different elements across the domain, as well as localspaces of different types (orders). This is the case since continuityis not enforced strongly across elements. (2) The previous propertymakes these methods suitable to handle structured and unstruc-tured meshes in domains with general geometries. (3) Their highdegree of locality makes them highly parallelizable. (4) They areelement-wise conservative (This statement is meaningful whenmodeling nonlinear conservation laws). (5) They are ideally suitedfor hp-refinement (or hp-adaptivity). A good reference that offers areview on the development of discontinuous Galerkin methods isthe book by Cockburn et al. [13].

The LDG method was introduced by Cockburn and Shu in [14]as an extension, to general convection–diffusion problems, fromthe numerical techniques introduced by Bassi and Rebay in [4] tosolve the compressible Navier–Stokes equations. One of the basicideas in the LDG method is to rewrite, say the parabolic equationat hand, as a degenerate first order system of equations, and solvefor u andru#! q$ as independent unknowns. Even though this strat-egy is also utilized in methods based on a mixed formulation, in theLDGmethod one further discretizes the resulting first order systemusing particular DG techniques. It is particular to the LDG methodstudied in this work that the approximation to u, and the approx-imation to each of the components of q belong to the same approx-imation spaces. This choice makes the coding of the methodsimpler than the standard mixed methods. Also, the so-callednumerical flux, bU , (introduced to properly define the values of thesolution u and the fluxes across all element boundaries) does notdepend on q, making it possible for the local variable q to be solvedin terms of u. The particular numerical fluxes, bU and bQ , used in ourmethod are introduced in Section 2. Examples of other consistent1 In (4), #&; &$ represents the appropriate duality pairing.

M. Santillana, C. Dawson / Comput. Methods Appl. Mech. Engrg. 199 (2010) 1424–1436 1425


numerical fluxes, in the context of elliptic problems, can be foundin [9].

Works addressing the properties of the LDG method in the con-text of convection–diffusion problems include for example:[14,12,10] and [1]. The applicability of the LDG method has beenexplored for example, for elliptic problems in [3], and in [19]; fornonlinear diffusion problems in [7], and in [22]; for a class of non-linear problems in fluid mechanics in [8]; for Richard’s equation(another nonlinear parabolic equation) in [23]; for nonlinear sec-ond-order elliptic and hyperbolic systems in [24]; for nonlinearconvection–diffusion and KdV equations in [30]; and for PDE’s withhigher order derivatives in [31] and [32].

1.3. Regularized problem

In [25] we used a strategy that consisted of constructing a reg-ularized numerical scheme to approximate the possibly degeneratediffusion coefficient a!u;ru" in (3) with nondegenerate diffusioncoefficients a! in (1), such that 0 < ! 6 a!!u" and with the propertythat a!u" # lim!!0a!!u", for a small parameter !. We will use a sim-ilar strategy in our study.

We present the nondegenerate problem that we will approxi-mate numerically along with some properties and results that willbe used in the analysis carried out in the next sections. We beginby introducing the nondegenerate version of the IBVP (1), obtainedby replacing the function !s$ z"a with a sequence of bounded Lips-chitz functions fb!!s"g, with the properties that (i) fb!!s"g con-verges uniformly to !s$ z"a as !! 0, and (ii) for small ! > 0 thefollowing holds b!!s" P ! for all t 2 %0; T&. To this end, the bathym-etry z!x" will be assumed to be a smooth and bounded time inde-pendent function defined in X. The nondegenerate IBVP is given by

@u@t $r ' b!!u" ru

jruj1$c

! "# f on X( !0; T&;

u # u0 on X( ft # 0g;b!!u"

jruj1$cru

! "' n # BN on @X \ CN ( !0; T&;

u # BD on @X \ CD ( !0; T&:

8>>>>><

>>>>>:

!5"

In the next section we develop a numerical scheme to approximatethis nondegenerate problem as explained in Section 1.1. The factthat solutions to the nondegenerate problem (5) are close to the ori-ginal solution to problem (1) as !! 0 will be understood as in [2]for z # 0, and will be assumed for the general case z – 0.

Remark 1.1. For intuition purposes one could choose for examplethe following sequence b!!u" # !u$ z"a ) !:

1.4. Previous results

For completeness, we present some essential results needed inthe subsequent sections. For proofs of the next two lemmas seeSection 1.5 in [25] and the references therein.

Lemma 1.1. Let u1 and u2 be non negative L1!X" functions, then foraP 1

jua1 $ ua2j 6 a max!ku1kL1!X"; ku2kL1!X""! "a$1

ju1 $ u2j: !6"

Lemma 1.2 (Coercivity and continuity). Let g1 and g2 be boundedvector valued functions in Rn!n P 1", then the following estimateshold true

cA0jg1 $ g2j2 6 g1

jg1j1$c $

g2

jg2j1$c

!!g1 $ g2" !7"

and

g1

jg1j1$c $

g2

jg2j1$c

#####

##### 6 A0jg1 $ g2j 62c jg1 $ g2j

c; !8"

where

A0 :#Z 1

0jkg1 ) !1$ k"g2"j

c$1dk:

1.5. DG notation

Let fThg denote a family of regular finite element partitions ofX such that no individual element Xe crosses @X. For the erroranalysis described below, we will assume that Th is a locally qua-si-uniform finite element mesh. Let he denote the element diame-ter with h being the maximal element diameter. We will alsoassume each element Xe is Lipschitz and affinely equivalent toone of several reference elements [6]. Let Pk!Xe" denote the spaceof (possibly) discontinuous piecewise polynomials of degree atmost k; k P 1, defined on Xe, and let

M # fv : vjXe2 Pk!Xe"g:

We will assume that Pk!Xe" is chosen such that the usual space ofcontinuous, piecewise polynomials of order k defined on the trian-gulation Th are contained in M.

We will denote by ei the set of all interior element faces, with eDthe set of all element faces along the Dirichlet boundary CD, and eNthe set of all element faces along the Newmann boundary CN . Notethat if e is an interior face in the finite element mesh, then e hastwo elements adjacent to it, we will denote them by X$

e and X)e .

Also, if v andw are smooth real valued and vector valued functions,respectively, defined on these elements, we will denote their traceson e, from the interior of the element X$

e , as v$ and w$; and fromthe exterior of the element X)

e , as v) and w). We will denote by n$

the outward normal vector to the element X$e at e and by n) the

outward normal vector to the element X)e at e. The previous defini-

tion implies naturally that n) # $n$. We will define the averagef'g and the jump s ' t on the face e as:

fvg # !v$ ) v)"2

; fwg # !w$ )w)"2

; !9"

svt # v$n$ ) v)n); swt # w$ ' n$ )w) ' n): !10"

We will also denote by !'; '"E, the usual L2 inner product over a d-dimensional domain E, and by h'; 'i@E, the !d$ 1"-dimensional inte-gral over the surface @E. To simplify notation, we will omit thedependence on the domain and denote with !'; '" the integrals overthe whole domain E # X; !'; '"X.

Throughout the paper, Cwill be a generic positive constant withdifferent values and the explicit dependence with respect toparameters will be written inside parenthesis.

We refer the reader to Chapter 4 in [6] and [11] for proofs of thefollowing lemmas.

Lemma 1.3 (Interpolation error). Let u 2 Hk)1!X", then there existsan ‘‘interpolant” u 2 M which satisfies

ku$ ukHs!X" 6 Chk)1$skukHk)1!X":

Lemma 1.4 (Inverse inequalities). Let v 2 M then, there exists aconstant K0 independent of h and v such that

kvkL1!X" 6 K0h$1kvkL2!X" and krvkL1!X" 6 K0h

$1krvkL2!X":

The following trace theorem is well known. See Chapter 4 in [6]:

Theorem 1.1 (Trace inequality). Suppose that region R has aLipschitz boundary. Then there exists a constant C # C!R" such thatfor v 2 H1!R",

1426 M. Santillana, C. Dawson / Comput. Methods Appl. Mech. Engrg. 199 (2010) 1424–1436


kvkL2!@R" 6 C kvk1=2L2!R"

kvk1=2H1!R"

:

By the trace inequality and inverse inequality, for any v 2 M

kvkL2!@Xe" 6 C!Xe;K0"h#12

e kvkL2!Xe": !11"

2. The local discontinuous Galerkin method

In this section, we study the approximation properties ofnumerical solutions to the DSW equation, through the regularizedinitial/boundary-value problem (5), obtained using the LDG meth-od. In order to formulate the LDG method it is appropriate to re-write the nonlinear degenerate parabolic IBVP (1) as adegenerate first order system of equations where u;ru, anda!u;ru"ru, are now considered as independent unknowns:

ut #r $ q % f on X& !0; T';~q % ru on X& !0; T';q % a!u; ~q"~q on X& !0; T';

8><

>:!12"

where

a!u; ~q" % !u# z"a

j~qj1#c; !13"

with initial and boundary conditions given as before by

u % u0 on X& f0g;q $ n % BN on @X \ CN & !0; T';u % BD on @X \ CD & !0; T';

8><

>:!14"

where @X % C % CN ( CD. Moreover, assuming u;q and ~q aresmooth enough, we multiply each equation in (12) by test functionsw 2 M;v 2 !M"d and ~v 2 !M"d respectively (where d is the spatialdimension), and integrate by parts over an element Xe to obtainthe local weak form of (12):

!ut ;w"Xe( !q;rw"Xe

# hq $ ne;wi@Xe% !f ;w"Xe

8w 2 M;

!~q; ~v"Xe( !u;r $ ~v"Xe

# hu;v $ nei@Xe% 0 8~v 2 !M"d;

!q;v"Xe# !a!u; ~q"~q;v"Xe

% 0 8v 2 !M"d;

8>><

>>:

!15"

where ne represents the outward normal vector to the faces of theelement Xe. The discontinuous Galerkin method consists of findingapproximations !U;Q ; ~Q" to the solution !u;q; ~q" of (15), whereU 2 M and Q ; ~Q 2 Md, satisfying for all t 2 )0; T'

!Ut ;w"Xe( !Q ;rw"Xe

# h bQ $ ne;wi@Xe% !f ;w"Xe

8w 2 M;

! ~Q ; ~v"Xe( !U;r $ ~v"Xe

# hbU ;v $ nei@Xe% 0; 8~v 2 !M"d;

!Q ;v"Xe# !a!U; ~Q" ~Q ;v"Xe

% 0 8v 2 !M"d;

8>><

>>:

!16"

for every element Xe in the domain X.By construction, the approximants !U;Q ; ~Q " may be discontinu-

ous across element boundaries. As a consequence, at a given face ethe functions !U;Q ; ~Q "may bemulti-valued. This is why the numer-ical fluxes bQ and bU are introduced in (16). This issue is clearly ex-plained in the context of elliptic problems in [9] and in [3], and inthe context of nonlinear diffusion problems in [7].

For the LDG method that we will analyze and implement, thenumerical fluxes are chosen in the following simple way:

bU %fUg if e 2 ei;BD if e 2 CD;

U if e 2 CN;

8><

>:!17"

and

bQ %fQg# rsUt if e 2 ei;Q # r!Un# BDn" if e 2 CD;

BN if e 2 CN:

8><

>:!18"

Note that the numerical flux bU does not depend on Q . Thismakes it possible for the local variable Q to be solved in terms ofU by using the second and third equations of (16). This is a partic-ular property that distinguishes the LDG method (hence the name‘‘local”). The penalty parameter r appearing in the definition of thenumerical fluxes will be chosen in order to enhance the stabilityand thus, the accuracy of the method.

Remark 2.1. The fluxes defined in (17) and (18) are both consistentand conservative as defined in [3] and [9].

The resulting LDG formulation is obtained in two steps. First, bysumming over all elements Xe to find

!Ut ;w" ( !Q ;rw" # h bQ ; swtiei # h bQ $ n;wi@X % !f ;w" 8w 2M;

! ~Q ; ~v" ( !U;r $ ~v" # hbU ; svtiei # hbU ;v $ ni@X % 0 8~v 2 !M"d;

!a!U; ~Q" ~Q ;v" # !Q ;v" % 0 8v 2 !M"d;

8>><

>>:

!19"

where we have denoted with !$; $" %P

e!$; $"Xethe sum of all element

integrals. And second, by substituting the values of the numericalfluxes (17) and (18) in (19)

!Ut ;w" ( !Q ;rw" # hfQg; swtiei ( hrsUt; swtiei#hBN ;wiCN

# hQ $ n;wiCD( hr!U # BD";wiCD

% !f ;w" 8w 2 M;

! ~Q ; ~v" ( !U;r $ ~v" # hfUg; s~vtiei # hU; ~v $ niCN% hBD; ~v $ niCD

8~v 2 !M"d;

!a!U; ~Q " ~Q ;v" # !Q ;v" % 0 8v 2 !M"d;

8>>>><

>>>>:

!20"where, for simplicity, we have denoted with

h$; $iei :%X

e

h$; $i@XenC; h$; $iCN%

X

e

h$; $i@Xe\ CN; and

h$; $iCD%

X

e

h$; $i@Xe\CD

the sum of the boundary integrals in all interior element boundariesei, in all element boundaries along the Newman boundary CN , andin all element boundaries along the Dirichlet boundary CD, respec-tively. In order to enforce the initial condition we set

!U0;w" % !u0;w" 8w 2 M; t % 0: !21"

Note that using integration by parts for some of the terms in thesecond equation of (20), the following expression holds,

!U;r $ ~v" # hfUg; s~vtiei # hU; ~v $ niCN

% #!rU;v" ( hsUt; f~vgiei ( hU; ~v $ niCD: !22"

Based on the previous observation we will rewrite the LDG formu-lation for the IBVP (5) as,

!Ut ;w" ( !Q ;rw" # hfQg; swtiei ( hrsUt; swtiei#hBN ;wiCN

# hQ $ n;wiCD( hr!U # BD";wiCD

% !f ;w" 8w 2 M;

! ~Q ; ~v" # !rU; ~v" ( hsUt; f~vgiei ( hU; ~v $ niCD% hBD; ~v $ niCD

8~v 2 !M"d;

!a!U; ~Q " ~Q ;v" # !Q ;v" % 0 8v 2 !M"d

8>>>><

>>>>:

!23"

Remark 2.2. It is clear that any continuous classical solution ofproblem (12)–(14) will satisfy problem (23) since all termsinvolving jumps across elements s $ t, will be zero and all boundaryterms will satisfy strongly the boundary conditions.



Remark 2.3. As mentioned in Section 1.3, the diffusion coefficienta!u; ~q" in (13) will be approximated by the family of Lipschitz non-degenerate diffusion coefficients of the form

a!!u; ~q" #b!!u"j~qj1$c

; !24"

and we will denote with b!%" any member of the family fb!!%"g in thesubsequent analysis to simplify the notation. Furthermore, notethat any solution of the IBVP (5) will also be a solution (12)–(14)with the regularized diffusion coefficient (24).

Remark 2.4. It is not difficult to see that, for a given r > 0, the sys-tem of nonlinear ordinary differential equations arising from (23)will have at least one solution. Indeed, the fact that the right handside of this system is – at least – locally Hölder continuous withrespect to U and each component of eQ ensures existence of at leastone solution. See [25] for a more detailed argument.

2.1. Stability analysis

Even though the proof of Theorem 2.1 can be established as aCorollary of Theorem 2.2, we present it here for clarity. Indeed,many of the mathematical manipulations presented in the proofof Theorem 2.1 can be easily followed and will be used in the moreelaborate setting of the proof of Theorem 2.2.

Theorem 2.1 (Stability). Let U and eQ be solutions of (23) and (21)with BN # 0, and BD # 0. Then

kU!t"k2L2!X" & k ~Qk1&cL1&c!0;T;L1&c!X"

&Z T

0kr1

2sUtk2L2!ei" & kr12Uk2L2!CD"

! "

6 C !; ku0k2L2!X"; kfk2L2!0;T;L2!X""

! ": !25"

Proof. Note that choosing w # U, ~v # Q , and v # eQ and adding allterms on the left hand side in (23) we obtain, after severalcancellations

12@

@tkU!t"k2L2!X" & hrsUt;sUtiei & hrU;UiCD

& a!U; eQ " eQ ; eQ! "

#!f ;U":

!26"

From the observation that

!k eQ k1&cL1&c!X"

6Z

Xb!U"j eQ j1&c # !a!U; eQ " eQ ; eQ ":

Eq. (26) leads to

12

@

@tkU!t"k2L2!X" & kr1

2sUtk2L2!ei" & kr12Uk2L2!CD"

& !k eQ k1&cL1&c!X"

6 !f ;U":

!27"

Furthermore, since

!f ;U" 6 12kU!t"k2L2!X" &

12kfk2L2!X":

Eq. (27) implies

12

@

@tkU!t"k2L2!X" & kr1

2sUtk2L2!ei" &12kr1

2Uk2L2!CD"& !k eQ k1&c

L1&c!X"

6 12kU!t"k2L2!X" &

12kfk2L2!X": !28"

Since the second, third, and fourth terms of the left hand side of theprevious equation are nonnegative we obtain

12

@

@tkU!t"k2L2!X" 6

12kU!t"k2L2!X" &

12kfk2L2!X"

which, by Gronwall’s Lemma, leads to

kU!t"k2L2!X" 6 C!kU0k2L2!X"; kfk2L2!0;T;L2!X""" for all t 2 '0; T(:

Integrating (28) in time from 0 to T, the following must also hold:

k eQ k1&cL1&c!0;T;L1&c!X"

6 C!!; kU0k2L2!X"; kfk2L2!0;T;L2!X""": !29"

Likewise for the second and third terms of (28). Finally, by choosingw # U0 in the first equation of (21) we obtain

!U0;U0" # !u0;U0" 612kU0k2L2!X" &

12ku0k2L2!X"

which implies

kU0k2L2!X" 6 ku0k2L2!X"

Thus, the result of the Theorem follows at once. h

2.2. Continuous in time a priori error analysis

In this section, we will study how close (possibly nonunique)solutions to the LDG approximation problem (23), U, are to the trueweak solution u of problem (12)–(14) with the regularized diffu-sion coefficient (24).

The analysis requires that comparison functions be carefullychosen.

We define u 2 M and q 2 Md to be L2 projections, given by

!u$ u; v" # 0 8v 2 M !30"

and

!q$ q;v" # 0 8v 2 Md: !31"

Furthermore, define ~q 2 Md by

!~q;v" # !ru;v" $ hsut; fvgiei $ hu$ BD;v % niCD8v 2 Md: !32"

We note that, defining the projection p~q 2 Md by

!p~q;v" # !ru;v" 8v 2 Md

we have

!~q$ p~q;v" # !r!u$ u";v" $ hsu$ ut; f~vgiei $ hu$ u;v % niCD:

!33"

Choosing v # ~q$ p~q and applying the trace theorem and (11) it iseasily seen that

k~q$ p~qkL2!X"

6 CX

e

kr!u$ u"kL2!Xe" & h$1=2e ku$ uk1=2

L2!Xe"ku$ uk1=2

H1!Xe"

h i: !34"

We make the following assumptions on the boundedness of thesolution and approximations, namely that there exist positive con-stants K1;K1;K2 and K2 such that

kukL1!0;T;L1!X"" 6 K1; !35"

kUkL1!0;T;L1!X"" 6 K1; !36"

k~qkL1!0;T;L1!X"" 6 K2; !37"

k ~QkL1!0;T;L1!X"" 6 K2: !38"

We will show in Lemmas 2.1 and 2.2 that the constants in ourerror estimate are independent of !K1 and K2 provided we choosethe maximum diameter of the mesh, h, small enough and the finiteelement space is of high enough order. We note that (37) holds ifru is bounded and ru$ ~q is small (or merely bounded).

Theorem 2.2. Let !u; ~q;q" be the solution of problem (12)–(14) andlet !U; ~Q ;Q" be the solution of problem (23). Let vu # u$ u; ~vq #



~q! ~q and vq " q! q. Furthermore assume (35)–(38) hold. Then for allt 2 #0; T$, there exists a constant C " C%!; c;K1;K1;K2;K2; T& such that

kU%t& ! u%t&k2L2%X& ' k ~Q ! ~qk2L2%0;T;;L2%X&&

'Z T

0kr1

2sU%t& ! u%t&tk2L2%ei& ' kr12 U%t& ! u%t&% &k2L2%CD&

! "

6 kvu%t&k2L2%X& ' k ~vqk

2L2%0;T;L2%X&& ' C kU%0& ! u%0&k2L2%X& ' kvuk

2L2%0;T;L2%X&&

! "

' CZ T

0kr!1

2fvqgk2L2%ei&

' kr12vuk

2L2%CD&

' kr!12vqk

2L2%CD&

' kr12fvugk

2L2%ei&

! "

' CZ T

0

Z

X~vq

## ##2c: %39&

Proof. Since the solution u of (12)–(14) satisfies the weak form(23), the following three equations hold:

%Ut ! ut ;w&'%Q ! q;rw&! hfQ ! qg;swtiei ' hrsU! ut;swtiei! h%Q ! q& (n;wiCD

' hr%U! u&;wiCD

"%ut ! ut ;w&'%q! q;rw&! hfq! qg;swtiei ' hrsu! ut;swtiei'

! h%q! q& (n;wiCD' hr%u! u&;wiCD

; %40&

% ~Q ! ~q; ~v& ! %r%U ! u&; ~v& ' hsU ! ut; f~vgiei' hU ! u; ~v ( niCD

" %~q! ~q; ~v& ! %r%u! u&; ~v&' hsu! ut; f~vgiei ' hBD ! u; ~v ( niCD

; %41&

and

b%U&~Q

j ~Q j1!c!

~q

j~qj1!c

!;v

!! %Q ! q;v&

" b%u&~q

j~qj1!c!

~q

j~qj1!c

!;v

!! %q! q;v&

! b%U& ! b%u&% &~q

j~qj1!c;v

!: %42&

Note that the first and second terms on the right hand side of (40)and the second term on the right hand side of (42) are zero by def-inition of u and q. Furthermore, by (32), the entire right hand side of(41) vanishes.

To simplify notation, let nu " U ! u; ~nq " ~Q ! ~q and nq " Q ! q.Now, choosing w " nu; ~v " nq, and v " ~nq, and adding Eqs. (40)–(42) we obtain, after multiple cancellations:

12

@

@tknu%t&k

2L2%X& ' hrsnut; snutiei ' hrnu; nuiCD

' b%U&~Q

j ~Q j1!c!

~q

j~qj1!c

!

; ~nq

!

" !hfvqg; snutiei ' hrsvut; snutiei ! hnu; vq ( niCD

' hrvu; nuiCD' b%u&

~qj~qj1!c

!~q

j~qj1!c

!; ~nq

!

! b%U& ! b%u&% &~q

j~qj1!c; ~nq

!: %43&

Furthermore, from the result of Lemma 1.2 and providedb%U& P ! > 0

c!Aj ~nqj2L2%X& 6 b%U&

~Qj ~Q j1!c

!~q

j~qj1!c

!; ~q! ~q

!; %44&

where A :" inf %0;T&)X%A0& " 1=%k ~QkL1%0;T;L1%X&& ' k~qkL1%0;T;L1%X&&&&1!c.

Using (44), we can establish that

12@

@tknu%t&k

2L2%X& 'kr1

2snutk2L2%ei&

'kr12nuk

2L2%CD&

'c!Ak ~nqk2L2%X& 6

X6

i"1

Ti;

%45&

where Ti; i " 1; . . . ;6, are the terms arising from the right hand sideof (43). We now proceed to bound the terms T1 ! T6.

For T1, we multiply and divide by r12 to get

T1 " hr!12fvqg;r

12snutiei 6

!12kr1

2snutk2L2%ei&

' 12!1

kr!12fvqgk

2L2%ei&

:

%46&

For term T2 we have the following inequality

T2 " hrsvut; snutiei 612kr1

2svutk2L2%ei&

' 12kr1

2snutk2L2%ei&

: %47&

The terms T3 and T4 are handled identically to T1 and T2 with ei re-placed by CD.

As for terms T5 and T6, note that

T5 "Z

Xb%u&

~qj~qj1!c

!~q

j~qj1!c

!~nq 6

2c C1

Z

Xj ~vqj

cj ~nqj

6 2c C1

12!2

Z

X~vq

## ##2c ' !22k ~nqk

2L2%X&

$ %; %48&

where C1 " C%K1&, and

T6 " %b%U& ! b%u&&~q

j~qj1!c; ~nq

!6 C2

Z

Xju! Uk~qjcj ~nqj

6 C2k~qkcL1%X&ku! UkL2%X&k ~nqkL2%X&

6 C2k~qkcL1%X&12!3

%kvuk2L2%X& ' knuk

2L2%X&& '

!32k ~nqk

2L2%X&

$ %; %49&

where C2 " C%K1; !K1&. From (45)–(49) and choosing !1; !2 and !3small enough so that for ! and !* small positive numbers,0 < ! 6 c!A! 1

c C1!2 ' 12C2k~qkcL1%X&!3

! "and 0 < !* 6 1

2 %1! !1&, weobtain12

@

@tknu%t&k

2L2%X& ' !*kr1

2snutk2L2%ei&

' 12kr1

2nuk2L2%CD&

' !k ~nqk2L2%X&

6 C knuk2L2%X& ' kvuk

2L2%X& ' kr1=2svutk

2ei ' kr!1

2vq ( nk2L2%CD&

!

'kr!12fvqgk

2L2%ei&

' kr12vuk

2L2%CD&

'Z

X~vq

## ##2c%: %50&

Since the second, third, and fourth terms of the left hand side in theprevious inequality are nonnegative, we can use Gronwall’s Lemmato find that for all t 2 #0; T$,

knu%t&k2L2%X& 6 C knu%0&k

2L2%X& ' kvuk

2L2%0;T;L2%X&& '

Z T

0

Z

X~vq

## ##2c&

'Z T

0kr!1

2fvqgk2L2%ei&

' kr12vuk

2L2%CD&

!

'kr1=2svutk2ei ' kr!1

2vq ( nk2L2%CD&

"i: %51&

Likewise, integrating (50) again in time from 0 to T, we can establishthe boundedness of the three remaining terms of the left hand sideof (50):Z T

0kr1

2snutk2L2%ei&

' kr12nuk

2L2%CD&

! "and k ~nqk

2L2%0;T;L2%X&&: %52&

The result of the Theorem follows immediately from the trian-gle inequality. h

Corollary 2.1. If u%t&; ~q%t&;q%t& are sufficiently smooth for 0 < t 6 T,and the approximations U; ~Q ;Q are constructed with piecewise poly-nomials of degree at most k and satisfy the assumptions of Theorem2.2, then for all t 2 #0; T$ and r " O%1h&, there exists a constant



C ! C"!; c;X; T;K1;K1;K2;K2; kukL1"0;T;Hk#1"X$$; kqkL1"0;T;Hk"X$$$ "53$

such that

kU"t$%u"t$kL2"X$ # k ~Q"t$% ~q"t$kL2"0;T;L2"X$$

#Z T

0

1h

ksU"t$%u"t$tkL2"ei$ # k"U"t$%u"t$$kL2"CD$

! "6 Chkc: "54$

Proof. The corollary follows from Theorem 2.2 and the followingbounds.

kvu"t$k2L2"X$ 6 Ch2"k#1$kuk2Hk#1"X$; "55$

k ~vqk2L2"X$2 6 Ch2kkuk2Hk#1"X$; "56$

kvuk2L2"0;T;L2"X$$ 6 Ch2"k#1$

Z T

0kuk2Hk#1"X$: "57$

Using the trace inequality,

kr12svutk

2L2"ei$

# kr12vuk

2L2"CD$

6 Ch%1XekvukL2"Xe$kvukH1"Xe$

6 Ch2kkuk2Hk#1"X$: "58$

Note also that

Z

X~vq

## ##2c 6Z

X~vq

## ##2$ %2c

2

Xj j1%c ! k ~vqk2cL2"X$

Xj j1%c

6 Ch2kckuk2cHk#1"X$

"59$

and

kr%12fvqgk

2L2"ei$

# kr%12fvqgk

2L2"CD$

6 ChX

e

kq% qkL2"Xe$kq% qkH1"Xe$ 6 Ch2kkqkHk"X$: "60$

The result of the Corollary follows immediately. h

Lemma 2.1 (Boundedness of the approximation). Under theassumptions of Theorem 2.2, choosing c P 1=2, and provided h is suf-ficiently small and k P 3, if kukL1"0;T;L1"X$$ 6 K1, then kUkL1"0;T;L1"X$$ 6K1"1# !k1 $ for a small parameter !k1 .

Proof. Clearly

kUkL1"0;T;L1"X$$ 6 kU % ukL1"0;T;L1"X$$ # kukL1"0;T;L1"X$$: "61$

From Corollary 2.1 and Lemma 1.4 we obtain

kUkL1"0;T;L1"X$$ 6 kU % ukL1"0;T;L1"X$$ # K1

6 Ch%1kU % ukL1"0;T;L2"X$$ # ku% ukL1"0;T;L1"X$$ # K1

6 C"h3c%1 # h3$ # K1

Thus, we can choose a sufficiently small h so that Ch1=2 6 !k1K1,which implies

kUkL1"0;T;L1"X$$ 6 K1"1# !k1 $ !

Lemma 2.2 (Boundedness of the gradient of the approxima-tion). Under the assumptions of Theorem 2.2, choosingc P 1=2# k=4 with 2 P k > 0, and provided h is sufficiently smalland k P 4, if krukL1"0;T;L1"X$$ 6 K3, then k ~QkL1"0;T;L1"X$$ 6 K3

"1# !k2 $ for a small parameter !k2 .

Proof. Returning to (41), for any 0 < t < T ,

" ~Q % ~q; ~v$ ! "r"U % u$; ~v$ % hsU % ut; f~vgiei % hU % u; ~v & niCD:

"62$

Setting ~v ! ~vq, and using trace and inverse inequalities

k ~vqk2L2"X$ ! "rnu; ~vq$ % hsnut; f ~vqgiei % hnu; ~vq & niCD

6 krnukL2"X$k ~vqkL2"X$ # knukL2"ei$k ~vqkL2"ei$ # knukL2"CD$k ~vqkL2"CD$

6 Ch%1knukL2"X$k ~vqkL2"X$:

Therefore,

k ~vqkL2"X$ 6 Chkc%1: "63$

Now following the argument used in the proof of Lemma 2.1, we seethat the result follows if k P 4 and h is small enough so thatCh4c%2 6 Chk 6 !k2K2. h

3. Numerical experiments: 2D

In this section, we investigate numerically, the order of accuracyof the proposed LDG method. We also present results of somenumerical experiments aimed at solving two ideal 2D problems: adam break event, and flow in a channel with vegetation resultingfrom a dam break event. The main motivation to show the latter re-sults is to provide the readerwith convincing evidence that theDSWequationcaptures thephysicsof theaforementioned ideal problems.In fact, the setting of the simulatedflow in a channelwith vegetationwas inspired by an actual experiment shown in [5].

The 2D LDG finite element formulation on unstructured trian-gular elements was coded in order to carry out the numericalexperiments. A second-order backward difference formula (BDF)time integrator was used to solve the problem forward in time. Pi-card iteration was used to linearize the resulting nonlinear system,and the conjugate gradient method was used to solve the resultinglinear systems.

Fig. 1. Mesh for the dam break simulation (left) without vegetation (right) withvegetation.

Table 1Convergence rates to approximate Barenblatt solutions for a ! 5=3 and c ! 1=2 usingt0 ! 2 and tf ! 2:1 and X ! '%2;2(.

dt h kU % ukL2"X$ Conv. rate

1/100 1/2 2:81) 10%3 –1/400 1/4 6:76) 10%4 2.061/800 1/8 1:58) 10%4 2.101/1600 1/16 3:68) 10%5 2.10



3.1. Numerical convergence

In order to verify the accuracy of the implemented 2D LDGscheme, we chose to reproduce an analytic Barenblatt solution tothe DSW equation for a flat bathymetry !z!x; y" # 0". The explicit

expression for such solution u!x; t" in 1D (spatially) is presentedin [16] and used in [25] to numerically investigate the convergencerates of a one dimensional CG scheme. We extended this analyticsolutions to 2D (spatially) by simply by setting u!x; y; t" # u!x; t"where

Fig. 2. Dam break simulation. Figures showing evolution of water depth (meters) at times = 0.0, 0.5, and 5 s. (left) 3D views, (right) 2D views.



u!x; t" # t$1

c!m%1"&C $ k!m; c"jUjc%1c '

cmc$1% ; !64"

where &s!x"'% denotes the positive part of s!x",m # 1% a=c, C is a po-sitive function related to the initial mass M, given by

M #Z 1

$1u!x; t"dx;

k!m; c" # mc$ 1m!c% 1"

1c!m% 1"

! "1c

; and U # x t$1

c!m%1":

Fig. 3. Dam break simulation. Figures showing evolution of water depth (meters) at times = 9.0, 54.0, and 70.0 s. (left) 3D views, (right) 2D views.



We used our 2D code to reproduce this solution on the domainX ! "#2;2$ % "#0:5; 0:5$, for the time interval t 2 "2;2:1$, fora ! 5=3 and c ! 1=2. We prescribed the appropriate Dirichletboundary conditions (U y; t' ! u t' and U&2; y; t' ! u&2; t')on the boundaries x ! #2 and x ! 2, and zero-Newmann boundaryconditions on the boundaries y ! #0:5 and y ! 0:5. We restricted

our error analysis to a numerical domain X such that u is nonde-generate (u > 0) everywhere for our simulation time, t 2 "ti; tf $.The results are shown in Table 1. Note that the function givenby (64) is Lipschitz continuous and compactly supported, in partic-ular, its gradient is bounded and continuous in our cylinderX% "t0; tf $.

Fig. 4. Dam break simulation with vegetation. Figures showing evolution of water depth (meters) at times = 0, 1.0, and 3.0 s. (left) 3D views, (right) 2D views.



Remark 3.1. Assuming that the BDF integrator gives rise to orderDt2 errors, where Dt is the time step, we chose to push the limits inour investigation to see if we could observe optimal convergence(h2 for piecewise linear elements), despite the fact that Corollary2.1 suggests convergence results of the type, ku!tn" # UnkL2!X" 6

C!u; tn"!Dt2 $ h1=2" for c % 1=2, when approximating nondegener-ate solutions u 2 H2!X", using piecewise linear basis functions(k % 1). The previous motivation lead us to chose the time stepmuch smaller than the grid diameter in our convergenceexperiments.

Fig. 5. Dam break simulation with vegetation. Figures showing evolution of water depth (meters) at times = 9.0, 11.0, and 13.0 s. (left) 3D views, (right) 2D views.



The convergence rates shown in Table 1 show that the errorestimates obtained in Corollary 2.1 for c ! 1=2 are very conserva-tive. Corollary 2.1, loosely speaking, suggests that the error de-creases as O"h1=2# for piecewise linear basis functions, yet inpractice, we observe O"h2# convergence. This is not necessarily sur-prising since removing the degeneracy in the IBVP (1) gives rise toa presumably well-behaved parabolic problem, where optimalconvergence rates – such as the ones observed in the numericalexperiments – could, in principle, be achieved.

3.2. The Dam break problem

In this section, we present the results of 2D simulations of theevolution of water depth profiles in an ideal dam break problem.This problem consists of simulating the water flow resulting fromremoving an ideal dam that keeps water on a confined area of thedomain. The set up is as follows, a channel was designed to connecttwo reservoirs, one completely filled with water (up hill) and theother completely empty (down hill). The channel is considered tobe dry at the beginning as well. When the ideal dam is removedfrom the upper reservoir, water is expected to flow down hill,flooding first the channel with a well defined front, and later flood-ing the lower reservoir; first with a well defined and radially sym-metric front, and later filling it gradually. This process is expectedto continue until all the water is transferred fully to the lowerreservoir.

The units used in this ideal setting were meters for the waterdepth and height, and seconds for the time. This numerical exper-iment was computed in a domain with a uniform friction coeffi-cient cf ! 1 (this value was chosen for simplicity and withoutany physical meaning) and with zero Newmann boundary condi-tions on @X. The mesh of the computational domain is shown inFig. 1 (left), the initial condition and water bed of this problemare presented at the top left of Fig. 2. The – wet condition – param-eter, introduced in Sections 1.1 and 1.3, was chosen to be ! ! :01 toprovide stability in the code. Recall that the typical depth in the do-main is O"1#. The mesh radius is of the order h $ 0:125 m (in a do-main with characteristic lengths of order L $ 6 m and W $ 3 m,respectively), and the time step was comparable in size, i.e.Dt ! 0:125 s. The experiment was run from t ! 0:0 s to t ! 70:0 s.3D and 2D views of the numerically simulated evolution of thewater depth are presented in Figs. 2 and 3.

As discussed before, the main features of the phenomenon arecaptured, these include: (1) The down-hill flow of water, (2) theappearance of a flooding wave with a well defined front propagat-ing in the direction of lowest potential energy points (lowestpoints in space), see Figs. 2 and 3, (3) the radial symmetry of thewater flow both, at the entrance of the channel (uphill) as wellas at the exit of the channel (down hill) throughout the event,(4) the radial symmetry in the flooding front when reaching thelowest reservoir, see upper views of Fig. 3, (this is a consequenceof the previous observation), and (5) the eventually gradual trans-fer of water from the upper part to the lowest one.

Some of the characteristics of the phenomenon that are not cap-tured are mostly related to two factors: the diffusive nature of theDSW equation, and the vertical integration utilized to derive it. Re-lated to the first factor, the physical interaction of the water flowwith the walls is not captured. For example, when water flows ina confined channel, ripples form as a consequence of momentumtransfers between the water and the walls (as well as friction).Also, when water frontally hits a wall (as it happens in the lowerviews of Fig. 3) water sloshes and forms reflecting waves. Thesefeatures are not present in the experiments we show. Anotherobvious characteristic not captured with the DSW equation as amodel, and related to the second factor, is the vertical velocity pro-file of the water flow.

3.3. The Dam Break problem with vegetation

In this section, we present the results of 2D simulations of theevolutionofwaterdepthprofiles in an ideal dambreakproblemwithvegetation in some regionsof thedomain. This problemwas inspiredby the experimental setting shown in [5]. The numerical implemen-tation was set up similarly to the one presented in Section 3.2. Themain difference consists of including three islands of vegetation indifferent locations of the domain. These vegetated regions, consid-ered to have the same vegetation density, modify the water flowlines in the experimental setting of [5]. It is observed, as intuitionwould suggest, thatwaterflowsmore rapidly away fromthem. Theirinclusion in the numerical simulations is done only by assigning ahigher value of the friction coefficient cf inside these areas. Through-out the domain cf ! 1 and in the vegetated regions cf ! 5. Thebathymetry remained the same as well as all the remaining compu-tational variablespresented in thedambreakproblem inSection3.2.Again, we chose to simulate thewater flow resulting from removingan ideal dam that keeps water on a confined (uphill) area of the do-main. The mesh for this problem and the location of the islands ofvegetation are shown in Fig. 1 (right). This experimentwas run fromt ! 0:0 to t ! 70:0 aswell. However, since themost relevant featuresof this event take place before t ! 20:0, only views for t 2 %0;20& arepresented. 3D and 2D views of the numerically simulated evolutionof the water depth are presented in Figs. 4 and 5.

Figs. 4 and 5 show very good agreement with the expected fea-tures of the phenomenon. In particular, they clearly display the factthat, as expected, water flows more rapidly away from the vege-tated areas. Also, the flooding front propagates throughout the do-main in a way that qualitatively captures the expected dynamics.Again, the limitations of the DSW equation as a model appear asdescribed in the previous section.

4. Conclusions

In this study, we prove that nondegenerate approximate solu-tions to the DSW equation, obtained using the LDG method, con-verge to true solutions of such equation, provided the truesolution is sufficiently smooth. We show that for discontinuous fi-nite elements of polynomial order k (with k P 4, see Corollary 2.1),one can ensure convergence O"hkc#. Numerical experiments in 2Dshow that the theoretical convergence rate obtained in Corollary2.1 – for a nondegenerate true solution u 2 C1"X; t# of the DSWequation, and c ! 1=2 – is conservative. Indeed, in this case we ob-serve numerical convergence rates O"h2# for piecewise linear finiteelements "k ! 1#.

We also present numerical experiments aimed at showing thequalitative characteristics of water flow captured by the DSWequation when used as a model to simulate an idealized dam breakproblem with vegetation. The numerical experiments show verygood agreement with the expected features of the phenomenon.

Acknowledgements

This work was supported in part by the National Science Foun-dation, Project Nos. DMS-0411413, and DMS-0620697, and Centrode Investigación en Geografía y Geomática,‘‘Ing. Jorge L. Tamayo”,A.C. The conclusion of this manuscript was possible due to the gen-erous Henson Environmental Fellowship awarded to Mauricio San-tillana at the Harvard University Center for the Environment.

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Author's personal copy - Harvard Universitymsantill/Mauricio_Santillana... · Author's personal copy A local discontinuous Galerkin method for a doubly nonlinear diffusion equation