A Wetting and Drying Treatment for the Runge-Kutta

A Wetting and Drying Treatment for the

Runge-Kutta Discontinuous Galerkin Solution

to the Shallow Water Equations

Shintaro Bunya a , Ethan J. Kubatko b , Joannes J. Westerink c

and Clint Dawson d

aThe University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo, 113-8656, Japan

bThe Ohio State University, 417D Hitchcock Hall, United Sates

cUniversity of Notre Dame, Notre Dame, IN 46556, United States

dThe University of Texas at Austin, Austin, TX 78712, United Sates

Abstract

This paper proposes a wetting and drying treatment for the piecewise linear Runge-Kutta discontinuous Galerkin approximation to the shallow water equations. Themethod takes a fixed-mesh approach as opposed to mesh-adaptation techniques andapplies a post-processing operator to ensure the positivity of the mean water depthwithin each finite element. In addition, special treatments are applied in the numer-ical flux computation to prevent an instability due to negative mass for an arbitrarytime step. The proposed wetting and drying treatment is verified through compar-isons with five problems with exact solutions and convergence rates are examined.The combination of the proposed wetting and drying treatment and a TVB slopelimiter is also tested.

Key words: Shallow water equations, wetting and drying, moving boundary,discontinuous Galerkin method, RKDG method

1 Introduction

The shallow water equations (SWE) describe flows for processes such as tides,storm surge, and flows within ocean basins, on shelves, in bays, through in-lets, in rivers, and on coastal floodplains. Due to the wide ranges of spatialscales and complex geometries that are present in many applications, unstruc-tured grid methods are an attractive solution strategy. Within the frameworkof Finite Element (FE) solutions, the standard continuous Galerkin (CG)method has been widely used (e.g. [1–4]) and, more recently, the discontinuous

Preprint submitted to Elsevier 6 August 2008

Galerkin (DG) method has been applied [5–11]. The DG method has provento be very effective in solving the SWE and accurately represents long wavepropagation as well as advection processes within a unified framework.

Many problems of interest involve wetting and drying zones which occur, forexample, in inter-tidal flats and/or in riverine and coastal floodplains. In thewetting and drying zone, water inundates or recedes as it is driven by tides,wind forces, and/or storm surge. The difficulty in numerically modeling dryareas relates to the obvious fact that there is no water in these areas, i.e. thewater column height is zero, and the SWE are only defined for wet regions.This implies that we need to deal with moving boundary problems for theSWE. Moving boundary problems for the shallow water equations are oftencalled wetting and drying problems. In this paper we present a wetting anddrying method devised for the Runge-Kutta discontinuous Galerkin (RKDG)solution to the SWE.

Existing wetting and drying treatments for CG based finite element solutionsare categorized into three types [12]. The first type, mesh adaptation tech-niques, uses a deforming domain and mesh. In this case, the domain and thefinite element mesh are deformed as water surface elevations change so that theboundary of the mesh matches the edge of a water body. The mesh adaptationtechniques have advantages in accuracy as they can precisely track the locationof wetting and drying fronts. However, they tend to be computationally ex-pensive and encounter difficulties in handling complex shaped boundaries. Asreal world problems tend to have intricate topographic contours and boundaryshapes, a fixed mesh is more suited to geophysical and engineering problemsof interest. Also, it seems that methods based on mesh adaptation techniquesdo not always yield more accurate solutions than methods using fixed meshes.The second type, mesh reduction techniques, discretely eliminate elementsand nodes when the water depth falls below a specified minimum. The elim-inated elements and nodes are restored when the water depth rises above aspecified maximum. Because mesh reduction techniques involve discrete andsudden elimination of elements/nodes, they may cause oscillations as well asloss of mass and momentum. The last type, thin layer techniques, also usesa fixed mesh. These techniques maintain a thin layer of water in nominallydry elements. In this way, dry elements are conceptually modeled by reducingflow without eliminating elements or nodes. An advantage of the thin layer el-ement approach is that mass can be conserved because elements maintain fullconnectivity within the continuity or conservation of water volume equation.Furthermore, water in nominally dry elements is kept in these elements untilthey are restored as fully wet. On the other hand, conservation of momentumis not guaranteed. The thin water layer technique cannot place grid pointsprecisely on shorelines since they use fixed meshes. The exact shoreline maybe located in the middle of an element. In such elements, an erroneous gradientis maintained in water surface as part of the bottom may be higher than the

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height of the exact shorelines and the water surface in such part is artificiallylifted above the bottom. This artificial gradient in water surface and the grav-ity generate ficticious flows. Removing or preventing these unphysical flowswithout alternating physical ones is a difficult task inherent in the thin waterlayer approach. Momentum conservation is often violated in this process. Theviolation of momentum conservation is only acceptable if the error is verifiedto be sufficiently small.

DG methods have emerged as an attractive solution to the SWE. The DGmethod is a FE method in that the formulation is based on a weak weightedresidual statement of the governing equations and the solution is locally inter-polated with bases of any order within each element. On the other hand, DGmethods and the finite volume (FV) methods are similar in that they both endup with the element as a control volume. In fact, both DG and FV solutionscan locally conserve quantities in each element. A similar approach to thinlayer techniques is commonly used in FV schemes to deal with dry cells. Forexample, Bradford and Sanders [13] used a tolerance in their two-dimensionalFV scheme. In their model, updated velocities are computed in a wet cell onlyif the water depth is greater than the tolerance.

Several wetting and drying treatments for the DG SWE solutions have beenreported. Bokhove [14] used a mesh adaptation technique to track wetting anddrying boundaries. As previously mentioned, mesh adaptation techniques areusually more accurate, but also their algorithms tend to be more complex. Aswe plan to couple the hydrodynamics model with transport and other models,it is desired to keep our algorithm less complex. We therefore prefer wettingand drying treatments that use fixed meshes. The work of Ern et al. [15] is theclosest to our approach at present. They use a slope modification technique toensure the positivity of water mass. Their method, however, adds mass whenthe positivity of mass is violated. The method presented in this article doesnot violate the positivity of mass without adding mass, taking advantage ofthe fact that each element is considered as a control volume in the DG spatialdiscretization.

In this paper, we propose a wetting and drying treatment that uses fixedmeshes for DG SWE solutions. Our wetting and drying treatment adopts thethin-water-layer technique so that it can achieve a good balance between ac-curacy and computational cost. Our treatment monitors the water depth ofeach dry/partially wet element and controls it by modifying water surface el-evations at the end of each Runge-Kutta time marching stage. This surfaceelevation modification redistributes the water mass within an element so thata positive water depth is maintained. The sum of mass within an element isunchanged through the modification. This ensures local mass conservation.Also, the surface elevation modification is local in each element, i.e. it is donebased on the surface elevation and velocity of the element; it does not use

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the states of the neighboring elements. This is beneficial in terms of paral-lel computational efficiency because the locality reduces the communicationbetween sub-domains. The proposed treatment also involves special handlingof numerical fluxes in order to ensure the positivity of water mass in eachelement with an arbitrary time step.

This paper is organized as follows. In section 2, our discretization of the SWEbased on the Runge-Kutta Discontinuous Galerkin (RKDG) method is sum-marized. In section 3, our new wetting and drying treatment is described indetail. In section 4, the robustness and accuracy of the proposed method areverified for five different wetting and drying problems, and, in section 5, con-clusions are presented.

2 Governing equations and discontinuous Galerkin method

2.1 Governing equations

The two-dimensional SWE consist of the depth-averaged continuity equationand x and y momentum equations as written here in conservative form:

∂ζ

∂t+

∂

∂x(Hu) +

∂

∂x(Hv) = 0, (1)

∂

∂t(uH) +

∂

∂x

(

Hu2 +1

2g(

H2 − h2)

)

+∂

∂y(Huv) = gζ

∂h

∂x− τuH + Bx, (2)

∂

∂t(vH) +

∂

∂x(Huv) +

∂

∂y

(

Hv2 +1

2g(

H2 − h2)

)

= gζ∂h

∂y− τvH + By, (3)

where ζ is the elevation of the free surface measured from a reference levelpositive upward, h is the bathymetric depth measured from the same referencelevel, but positive downward, H = ζ+h is the total height of the water column(water depth)), g is the acceleration of gravity, u and v are the depth-averagedvelocities in the x and y directions respectively, τ is the bottom friction factor,and Bx and By represent additional forces such as viscous stresses, Coriolisforce, tidal potential forces, and wind surface stresses. The bottom frictionterms, Bx and By are not considered in this article.

We define a variable vector as follows:

w ≡ [w1, w2, w3]T ≡ [ζ, p, q]T , (4)

where p ≡ uH and q ≡ vH . The SWE can then be written in the following

4

divergence form:

∂wi

∂t+ ∇ · Fi (w) = si (w) i ∈ 1, 2, 3, (5)

where Fi is the i-th row of the flux function matrix whose columns are the fluxfunction vectors in the x and y directions denoted by fx and fy respectively:

F = [fx, fy] =

uH vH

Hu2 + 12g (H2 − h2) Huv

Huv Hv2 + 12g (H2 − h2)

, (6)

and finally si is the i-th component of the vector s of source/sink terms whichis given by:

s =

[

0, gζ∂h

∂x, gζ

∂h

∂y

]T

. (7)

2.2 Discretization

We first introduce the notation that will be used. Given a domain Ω ⊂ R2,

which has been triangulated into a set of non-overlapping, but not necessarilyconforming elements, let ΩK define the domain of an element K and ∂ΩK

denote the boundary of the element. An inner product taken over ΩK will bedenoted by (·, ·)ΩK

, and an inner product taken over ∂ΩK will be denoted by〈·, ·〉∂ΩK

. The outward unit normal vector of ∂ΩK will be denoted by n.

We approximate w by wh, the components of which belong to the space ofpiecewise smooth functions that are differentiable over an element, but whichallow discontinuities between elements. We denote this space of functions byVh. The SWE are put into a discrete weak form by replacing w by wh ,multiplying each equation by a test function v ∈ Vh , integrating over eachelement, and integrating the divergence term by parts:

(

∂

∂t(wh)i, v

)

ΩK

− (∇v,Fi)ΩK+ 〈Fi · n, v〉∂ΩK

= (si, v)ΩK, (8)

where (wh)i is the i-th component of wh. Due to the fact that discontinuitiesare permitted along ∂ΩK , the flux F, which may be dual-valued along ∂ΩK ,is replaced in the boundary integral by a single-valued numerical flux denotedby F. Making this substitution the discrete weak formulation of the problemis now given by:

(

∂

∂t(wh)i, v

)

ΩK

− (∇v,Fi)ΩK+⟨

Fi · n, v⟩

∂ΩK

= (si, v)ΩK, (9)

5

where Fi is the i-th row of F.

To complete the spatial discretization, we use the local Lax-Friedrich flux tocompute the numerical flux F (see, for example, [17]), Gaussian quadrature toevaluate integrals, and Dubiner’s orthogonal triangular basis functions [18].The second-order Runge-Kutta scheme is used for the time discretization.Cockburn and Shu’s slope limiter [17] is applied. Refer to [11,19] for moredetails on our hydrodynamic model.

3 Wetting and drying treatment

Our wetting and drying treatment is based on the concept of the thin waterlayer technique, which is explained in Section 1. The most basic requirementto the wetting and drying treatment for this approach is to keep the watercolumn depth greater than zero so that the SWE can have a stable numericalsolution over the domain. A wetting (or drying) element is represented byreduced flows; a completely dry element/node is represented by still water ofa small depth. To realize this concept, we first propose an operator to keep thewater depth within an element positive pointwise for the condition that themean water depth of the element is positive. The positivity of the mean waterdepth is ensured by the treatments for numerical flux computation, which willbe introduced in Sections 3.3 and 3.4

In the discussion below, we assume triangular elements and linear approxima-tions of surface elevation ζ and discharges p (= uH) and q (= vH) . Aftereach stage of the Runge-Kutta method, the computed surface elevation ζ , anddischarges p and q are examined and modified as necessary. As is common infinite element terminology, we refer to the vertices of the triangle as nodes. LetHK denote the average water depth of element K, and ζi, Hi, pi and qi denotethe approximated surface elevation, water depth, discharge in the x-direction,and discharge in the y-direction, at node i ∈ 1, 2, 3, respectively.

3.1 Positive-Depth operator

Here we propose an operator to keep the water depth positive. We name theoperator the Positive-Depth (PD) operator and represent it by MΠh. Theoperator MΠh may be used in conjunction with a TVB limiter ΛΠh in thek-stage Runge-Kutta time-marching algorithm as follows:

• Set w0h = MΠhΛΠhPVh

(w0);

6

• For n = 0, . . . , N − 1 compute wn+1h as follows:

(1) For i = 1, . . . , k + 1 compute the intermediate functions:

w(i)h = MΠhΛΠh

i−1∑

l=0

αilw(l)h + βil∆tLh

(

w(l)h

)

(2) set wn+1h = wk+1

h

where PVhis a projection to the function space Vh and αil and βil are the

constants of the Runge-Kutta method. The purpose of the operator MΠh is toprevent non-positive water depths. To define the operator MΠh, we introducea threshold H0 to detect nodes that are about to dry.

We construct the operator MΠh on piecewise linear solution functions for theSWE in such a way that the following properties are satisfied:

(1) Accuracy: if Hh ≥ H0, ∀(x, y) ∈ ΩK ,

MΠhwh = wh, ∀(x, y) ∈ ΩK .

(2) Conservation of mass: for every element K, we have

∫

ΩK

MΠhHhdΩ =∫

ΩK

HhdΩ.

(3) Water surface modification: the slope of water surface is modified in sucha way that

MΠhHh ≥ H0, ∀(x, y) ∈ ΩK

if this is possible without violating property (2). If not possible, then thewater depth is set to the mean value over the subdomain HK :

MΠhHh = HK , ∀(x, y) ∈ ΩK .

(4) Discharge limiting : discharge vanishes at a corner node if the water depthat the corner node is less than H0. The removed discharge is distributedto other nodes to recover the momentum conservation unless the waterdepth is less than H0 at all the corner nodes.

Note that these properties do not prohibit violation of momentum conserva-tion while the second property requires mass conservation. We allowed loss ofmomentum to ensure that discharge at the wetting-and-drying front vanishes.To satisfy the properties listed above, it is clear that we have to introducetwo different operators: an operator for the water depth, which is denoted byMΠd

h, and the other operator for the discharge, which is denoted by MΠfh.

In general, we can write a piecewise linear function uh with the values at thecorner nodes ui and the linear basis functions φi that take a value of 1 at node

7

i and 0 at the other two nodes as follows:

uh(x, y) =3∑

i=1

uiφi(x, y) (x, y) ∈ ΩK .

We denote by Hh and Hi the water depth updated by the operator MΠdh and

its nodal values, respectively. Then, we have

Hh ≡ MΠdhHh =

3∑

i=1

Hiφi. (10)

We now determine Hi to complete the definition of the operator MΠdh:

(1) If Hi ≥ H0, ∀i ∈ 1, 2, 3, then

Hi = Hi ∀i ∈ 1, 2, 3 (11)

(2) If HK ≤ H0, then

Hi = HK ∀i ∈ 1, 2, 3 (12)

(3) Otherwise, we first find the order of nodal indeces ni ∈ 1, 2, 3 thatsatisfies the following inequalities:

Hn1 ≤ Hn2 ≤ Hn3, (13)

then, determine Hn1, Hn2 and Hn3 in the following sequence:(a) Set Hn1:

Hn1 = H0. (14)

(b) Set Hn2:

Hn2 = max(

H0, Hn2 −(

Hn1 − Hn1

)

/2)

. (15)

(c) Set Hn3:

Hn3 = Hn3 −(

Hn1 − Hn1

)

−(

Hn2 − Hn2

)

. (16)

With the operator MΠdh, the water depth in element K is kept the same if

the water depth is greater than or equal to H0 at all the corner nodes. If themean water depth is less than or equal to H0, the updated depth, Hh, is setto be HK ; i.e. the slope of the surface elevation is changed to be parallel tothe bottom. Otherwise, the slope of the depth Hh is limited in such a waythat Hh ≥ H0 for all (x, y) ∈ ΩK by redistributing water mass locally in anelement.

It is clear that the operator MΠdh satisfies the first, second and third properties.

8

We next define a discharge operator MΠfh. Discharge in the x-direction ph can

be written with the values at the corner nodes, pi as follows:

ph =3∑

i=1

piφi. (17)

We define Θi, npos and ∆p as follows:

Θi = Θ(

Hi − H0

)

, (18)

npos =3∑

i=1

Θi, (19)

∆p =3∑

i=1

pi (1 − Θi) , (20)

where Θ(·) is the unit step function defined as

Θ(a) =

0 if a ≤ 0,

1 if a > 0,(21)

Θi is the flag that tells whether node i is wet or dry based on the udpatedwater depth, npos is the number of wet nodes and ∆p is the sum of discharge

in x-direction at dry nodes. The operator MΠfh is now defined as follows:

ph ≡ MΠfhph =

3∑

i=1

piφi (22)

where

pi = Θi · (pi + ∆p/npos) (23)

The operator MΠfh satisfies the first and fourth properties. It violates the

momentum conservation only when Hi ≤ H0 for all i ∈ 1, 2, 3.

The discharge in y-direction, q, is limited by the same operator MΠfh:

qh ≡ MΠfhqh. (24)

3.2 TVB slope limiter

In RKDG methods, a slope limiter may need to be applied to remove high-frequency oscillations to avoid instability. Thus we use Cockburn and Shu’sTVB limiter [17]. The details of their slope limiter are not repeated here. Wewrite only the definition of the modified minmod function as the parameter M

9

will appear when we solve test problems with the slope limiter. The modifiedminmod function, m(·), is defined as follows:

m(a1, . . . , am) =

a1 if |a1| ≤ M∆x2

m(a1, . . . , am), otherwise, (25)

where m(·) is the minmod function and ai’s are arbitrary numbers.

In applying this slope limiter, we observed that the combination of the slopelimiter and our wetting and drying treatment sometimes conflicted leading toan instability. To remedy this conflict, we do not apply the TVB limiter onelements where the minimum water depth is less than a specified value. In thenumerical tests presented in this article, the slope limiter will not be appliedif the minimum water depth in an element is less than 0.01.

3.3 A stability condition for the thin-water-layer approach

The goal of this section is to provide a sufficient condition to ensure the pos-itivity of mean water depth in each element. We begin by considering thebalance between the existing mass in an element and the outgoing dischargethrough its boundary. The balance is, in fact, written as the mass conser-vation equation as in Eq. (9) for i = 1. We write it here again with somerearrangements:

(

∂Hh

∂t, 1

)

ΩK

= −⟨

F1 · n, 1⟩

∂ΩK

(26)

where test function v is replaced with 1 as we are now interested in the totalwater mass in the element and net mass exchange between neighboring ele-ments. Since we assume that our element is triangular and the interpolationis linear, Eq. (26) can be further rewritten with the mean water depth HK asfollows:

d

dt

(

HKAK

)

= −⟨

F1 · n, 1⟩

∂ΩK

(27)

where AK is the area of element K. Eq. (27) is then discretized in time usingthe s-stage Runge-Kutta method as follows:

M(i)K ≡ H

(i)K AK

=i−1∑

l=0

αilH(l)K AK − ∆tβil

⟨

F(l)1 · n, 1

⟩

∂ΩK

, αil ≥ 0, i = 1, . . . , s

=i−1∑

l=0

αil

(

H(l)K AK − ∆t

βil

αil

⟨

F(l)1 · n, 1

⟩

∂ΩK

)

, (28)

10

where superscript (l) indicates quantities in the l-th Runge-Kutta stage. Thesufficient condition to ensure the positivity of the mean water depth of elementK is

M(i)K > 0 ∀i ∈ 1, . . . , s. (29)

We define M(i)K,l as

M(i)K,l ≡ H

(l)K AK − ∆t

βil

αil

⟨

F(l)1 · n, 1

⟩

∂ΩK

. (30)

Eqs. (28) and (30) lead to

M(i)K =

i−1∑

l=0

αilM(i)K,l. (31)

Since∑l−1

l=0 αil = 1 and αil ≥ 0, ∀i, for a general Runge-Kutta method, M(i)K is a

convex combination of M(i)K,l. Also, the combinations of (i, l) where αil = 0 can

be ignored as βil = 0 if αil = 0 [20]. Therefore, Eq. (29) is satisfied sufficientlyif

M(i)K,l > 0 ∀l ∈ 0, . . . , i − 1|αil 6= 0. (32)

Eqs. (30) and (32) give the following restriction on ∆t to satisfy Eq. (29)sufficiently:

∆t <αil

βil

H(l)K AK

⟨

F(l)1 · n, 1

⟩

∂ΩK

if⟨

F(l)1 · n, 1

⟩

∂ΩK

> 0

∀l ∈ 0, . . . , i − 1|αil 6= 0. (33)

Eq. (29) is satisfied with any ∆t if⟨

F(l)1 · n, 1

⟩

∂ΩK

≤ 0. We choose nonnegative

βil for the sake of efficiency [20] and αil are also nonnegative. Thus, if the mean

water depths of previous Runge-Kutta stages, H(l)K , is positive, then the right

hand side of Eq. (33) is positive. Therefore one can always find a positive ∆t

that ensures the positivity of mass integrated over an element; i.e. M(i)K > 0.

Furthermore, once the positivity of the mean water depth is ensured, thePD operator will ensure the pointwise positivity of water depth by the watersurface modification.

One can set a sufficiently small ∆t to satisfy Eq. (33). However, it is noteasy to estimate the right hand side of Eq. (33). It is rather more practical toconsider the condition of Eq. (33) in a wetting and drying treatment so that thepositivity of the mean water depth is satisfied with any given ∆t. Therefore,we consider manipulating the numerical fluxes between neighboring elementsin such a way that mass exchange is prohibited once the violation of Eq. (33)

is detected. To detect this violation, the flux F(l)1 must be computed once, and

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if the violation is detected with the computed flux, then a new flux must berecomputed so that it satisfies F

(l)1 · n = 0.

We compute the numerical flux that prohibits mass transfer in the form of thereflection flux. The reflection flux is computed by an approximated Riemannsolver with win ≡ (ζin, pin, qin) and w

refin ≡ (ζref

in , prefin , qref

in ). The symbol win

is the vector of variables on the elemental boundary from the interior of theelement. The symbol w

refin represents the reflected variable vector defined as

follows:ζrefin = ζin, (34)

prefin

qrefin

· n = −

pin

qin

· n, (35)

prefin

qrefin

· t =

pin

qin

· t (36)

where t is the unit tangential vector to the elemental boundary ∂ΩK . Thereflection numerical flux can be written as follows:

F = F(

win,wrefin

)

. (37)

The violation of Eq. (33) is detected on each edge of a triangular element andthe reflection numerical flux is used only on the edge.

3.4 Flux between dry elements

Besides the flux modification, to ensure the positivity of the water depth,flux between dry elements is also modified to prohibit water mass exchangebetween neighboring dry elements. The reasons why the mass/momentumexchange needs to be restricted are threefold; 1) to prevent unphysical os-cillations from propagating to dry regions, 2) to prevent dry elements fromlosing all their mass, and 3) to save computation time. The third reason mayneed some explanation. As long as neighboring elements are coupled through anumerical flux, all the elements including dry elements must be considered inthe computation. We can decouple neighboring dry elements by evaluating theflux between dry elements as the reflection flux. After decoupling neighboringdry elements, all the wet elements and only the dry elements neighboring awet element should be considered in the computation. The rest of the dryelements can be neglected to save computational time.

Here we define the wet-or-dry status of element K. Let ω(l)K denote the wet-

or-dry status; i.e. ω(l)K = 0 if element K is dry at the l-th Runge-Kutta inter-

12

mediate stage and ω(l)K = 1 otherwise. The wet-or-dry status is determined as

follows:

ω(i)K = ω

(i−1)K Θ

(

HK − H0

)

+(

1 − ω(i−1)K

)

Θ(

HK − H0

)

Θ (|ζK |H max − (H0 − |hK |min)) (38)

where |ζK|H max is the surface elevation at a point where the water depth isthe greatest over ΩK . The symbol |hK |min represents the highest bottom depthover ΩK . Note that the positive direction of the bottom depth is vertical down-ward. This wet-or-dry judgment is partially based on Bates and Hervouet’sclassification of dam-break-type and flood-type wetting/drying elements [21].

To prevent mass/momentum exchange between dry elements, we compute thereflection flux defined in Section 3.3 and use it as a numerical flux along theinter-element boundary that is sandwiched by dry elements. Thus, with thisreflection flux, interaction between neighboring dry elements no longer exists.

3.5 Cancellation of gravity terms

While an element is wetting or drying, till it gets fully wet or dry, an artificialgradient in surface elevation may be observed as the slope is limited by theslope of the bottom. This artificial slope of surface elevation induces unphysicalgravitational force according to the gravity terms in the momentum equations.Thus we cancel the gravity terms in dry elements, in which ω

(l)K = 0.

In our formulation presented in Eq. (9), the acceleration of gravity, g, appearsin Fi, Fi and si. In evaluating the domain integration terms that contain Fi

and si, we simply set g = 0 if element K is dry (i.e. ω(l)K = 0). On the other

hand, in obtaining a single-valued numerical flux Fi, it is incorrect to set g = 0since it will violate the balance between the boundary integration terms anddomain integration terms. In a wet element neighboring a dry element, inevaluating the domain integration terms, which contains Fi and si, g will beset to a prescribed nonzero value.

To achieve consistent evaluations of Fi with that of Fi and si, we allow dual-valued numerical fluxes on such inter-element boundaries that are sandwichedby wet and dry elements. One of the numerical flux vectors is computed inthe standard way with a prescribed nonzero g. This standard numerical flux isused to evaluate the boundary integrals of wet elements. The other flux is theone for dry elements. We compute it with the same Riemann solver, but withg = 0. Denoting the numerical flux computed with g = 0 by Fd

i , the numericalflux is redefined on the inter-element boundary of wet and dry elements as

13

follows:

F =

F1

F2

F3

on the wet side,

F =

F1

Fd2

Fd3

on the dry side.

Note that, only on the dry side, the momentum fluxes are replaced with thosecomputed with g = 0. Thus we violate the momentum conservation to cancelthe gravity terms. On the other hand, the same mass flux is used on bothsides. Therefore, mass conservation is intact.

4 Numerical Tests

In this section, we verify the wetting and drying method proposed in theprevious section. We consider still water at rest, the dam break problem, thedrying Riemann problem over a flat bottom, the Carrier-Greenspan solutionand a parabolic bowl problem. Convergence studies are conducted for the lastfour problems. The influence of slope limiting is investigated in some of thetest problems. The influence of the threshold H0 is examined for the Carrier-Greenspan problem. The behavior of the wetting and drying method in a twodimensional problem is tested in the parabolic bowl problem.

Convergence rates are computed by measuring the following L2 error norm:

L2(w) =1

A

[∫

Ω(wh − w)2dΩ

]1

2

(39)

where Ω is the entire computational domain, A is the area of the domain Ω,wh is the numerical solution of ζ , p or q, and w is the exact solution.

4.1 Still water at rest

We test our wetting and drying treatment with respect to whether it canproduce still water at rest. The problem definition and the finite elementmesh are depicted in Figure 1. The slip boundary condition with no normalflow is applied to the entire boundary. The CFL number is set to 0.1. Thesurface elevation is initially set to 0 on the nodes where the bottom is below

14

the datum level. The surface elevation on the other nodes are set to 10−5 mabove the bottom elevation. The initial discharges are set to zero. Thus thewater body is initially at rest.

In Figure 1, initially wet elements are shadowed in gray. The water front is notstraight since the mesh is unstructured. The wet-or-dry judgment introducedin Eq. (38) tends to estimate a smaller wet region. This is a desirable propertyas one should avoid inducing a fictitious flow generated by an artificial gradientin water surface near a water front. It approaches the actual water front asmesh resolution is increased.

Figure 2 shows the time histories of maximum of the magnitude of dischargesover the entire computational domain. In this figure, the effect of the gravitycancellation, which was introduced in Section 3.5 is examined. The upper partof the figure is produced from the solution computed with the gravity cancella-tion. The lower part of the figure is the case without the gravity cancellation.From the upper figure, it is observed that the water body practically staysstill if the gravity terms are canceled. From the comparison of the upper andlower figures, it is clear that the gravity cancellation removes the fictitiousgravity force. The maximum magnitude of discharges did not grow over thelevel shown in Figure 2 in longer simulations for either case.

Initial surface elevation

x

y

x

ζ15 m

100 m

100 m 50 m

ζ=0Bottom

Fig. 1. Schematic defining the still-water problem. Initially wet elements are shad-owed in gray.

15

0

2e-17

4e-17

6e-17

8e-17

1e-16

0 10 20 30 40 50 60

Max

imum

of D

isch

arge

(m

2 /s)

Time (sec)

Gravity Canceled in Dry Elements

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50 60

Max

imum

of D

isch

arge

(m

2 /s)

Time (sec)

Gravity Not Canceled

Fig. 2. Time history of the maximum magnitude of the discharge over the entirecomputational domain. Top: The gravity terms are canceled, Bottom: The gravityterms are not canceled.

16

4.2 Dam break on a dry bed

In this section, we solve the one-dimensional dam-break problem on a drybed using a 2D mesh. The numerical solution is compared with the analyticalsolution provided in [22]. Figure 3 shows the computational domain, the initialcondition, and one of the computational meshes used in the convergence studypresented later. The left side of the dam located at x = 0 is initially wet andhas still water with the depth of 10 m. The right side of the dam is an initiallydry region where the initial water depth is set to H0. The acceleration ofgravity is 10 m/s2. The CFL number is 0.1. Lateral slip and no normal flowboundary conditions are applied to all boundaries.

Bottom

10 m

H0x

ζ

Dam at x=0

-300 300 m0

x

y

100 m

Fig. 3. Schematic defining the dam break problem. The ∆x of the computationalmesh shown above is 5 m.

The convergence study is conducted with three sets of parameters shown inTable 1. In Case D1, the slope limiter is not applied. In Cases D2 and D3, theslope limiter is applied with different values of the TVB parameter M . The L2

error norms of Cases D1, D2 and D3 are shown in Tables 2, 3 and 4, respec-tively. The rows of the tables labeled as “Fitted” are least square fit. The sameL2 error norms of the three cases are compared to each other in Figure 4. TheL2 error norms are computed with the solutions at t = 8s. The convergencerates of the non-slope-limiting case (D1) are approximately 1 for ζ and p, andapproximately 1.5 for q. The convergence rates close to 1 are satisfactory asone cannot expect a convergence rate greater than 1 if the solution containsa discontinuity. The dam break problem contains a discontinuity in the initialcondition. The convergence rate of L2(q) is considerably greater than 1. Thismay be because there is no discontinuity in the solution in y-direction.

The convergence rates degrade when the slope limiter is applied. The combi-nation of the PD operator and TVB limiter does seem to decrease the orderof convergence to less than 1, but it is within an acceptable range. Shown

17

Table 1Model parameters for the dam break problem and drying Riemann problem

Case ID Slope limiter M H0(m)

D1 Not Applied - 10−5

D2 Applied 1 10−5

D3 Applied 0.1 10−5

Table 2L2 error norms and convergence rates of the dam break problem: Case D1.

∆x L2(ζ) rate L2(p) rate L2(q) rate

20.00000 6.22e-04 - 6.91e-03 - 1.79e-04 -

10.00000 3.19e-04 9.65e-01 3.63e-03 9.30e-01 6.03e-05 1.57e+00

5.00000 1.59e-04 1.00e+00 1.79e-03 1.02e+00 2.04e-05 1.56e+00

2.50000 7.96e-05 1.00e+00 8.74e-04 1.04e+00 7.26e-06 1.49e+00

1.25000 4.01e-05 9.89e-01 4.33e-04 1.02e+00 2.64e-06 1.46e+00

Fitted - 9.91e-01 - 1.00e+00 - 1.52e+00



20.00000 6.34e-04 - 7.00e-03 - 3.29e-04 -

10.00000 3.25e-04 9.62e-01 3.68e-03 9.27e-01 6.78e-05 2.28e+00

5.00000 1.91e-04 7.69e-01 2.18e-03 7.54e-01 6.78e-05 7.78e-04

2.50000 1.08e-04 8.26e-01 1.26e-03 7.90e-01 1.71e-05 1.99e+00

1.25000 5.69e-05 9.20e-01 6.62e-04 9.31e-01 7.67e-06 1.16e+00

Fitted - 8.55e-01 - 8.35e-01 - 1.28e+00

in Figures 5, 6 and 7 are the numerical and exact solutions for each of thethree cases D1, D2 and D3. The horizontal axis corresponds to the horizontalvariable x. The values of the solutions at vertices (x, y) for each y are plottedon the vertical axis. Note that the numerical solutions vary slightly with y,unlike the analytical solution, which is independent of y in this case. In thesefigures, relatively large errors are found near the wetting front. Comparingthese figures, it is observed that the slope limiter effectively removes spuriousoscillations.

18



20.00000 6.81e-04 - 7.52e-03 - 3.31e-04 -

10.00000 3.82e-04 8.32e-01 4.28e-03 8.12e-01 1.45e-04 1.19e+00

5.00000 2.05e-04 8.96e-01 2.29e-03 9.00e-01 5.46e-05 1.41e+00

2.50000 1.09e-04 9.08e-01 1.22e-03 9.13e-01 2.18e-05 1.32e+00

1.25000 5.74e-05 9.33e-01 6.36e-04 9.38e-01 7.66e-06 1.51e+00

Fitted - 8.94e-01 - 8.94e-01 - 1.36e+00

1e-05

0.0001

0.001

1 10 100

L2 (ζ)

∆ x

Dam Break

D1D2D3

0.0001

0.001

0.01

1 10 100

L2 (p)

∆ x

Dam Break

D1D2D3

1e-06

1e-05

0.0001

0.001

1 10 100

L2 (q)

∆ x

Dam Break

D1D2D3

Fig. 4. L2 error norms of the dam break problem.

19

Fig. 5. The computed solutions of the dam break problem: Case D1.



20

4.3 Drying Riemann problem

In this section, we test the proposed method in a drying situation on a flatbottom. We set up a Riemann problem in which a dry region emerges, shownschematically in Figure 8. The analytical solution is described in [14]. Theconfiguration shown in Figure 8 satisfies the drying criterion

√gHl +

√gHr −

ul +ur < 0. Two expansion waves that propagate away from each other resultin a dry region, which appears at t > 0. The acceleration of gravity is set to10 m/s2 and the Courant number is set to 0.1.

u = 40 m/s

x

y

H = 5 m

100 m

x

ζ H = 10 m

-200 400 m0

u = 0l r

l

r

Bottom

Fig. 8. Schematic defining the drying Riemann problem. The ∆x of the computa-tional mesh shown above is 5 m.

The convergence study is conducted with the same sets of parameters usedfor the dam break problem in the previous section and shown in Table 1. TheL2 errors are shown in Tables 5, 6 and 7 and Figure 9. The convergence ratesof Case D1 are approximately 1 for ζ and p, and approximately 1.7 for q.These are reasonable convergence rates, considering the fact that this dryingRiemann problem also contains a discontinuity in the initial condition. It isalso observed that the slope limiter does not significantly affect the order ofaccuracy of ζ and p. Relatively large degradation occurs in L2(q) when theslope limiter is applied. It is presumed to be due to the fact that the slopelimiter induces a small gradient in the y-direction in the surface elevation andit in turn drives flow in the y-direction.

Shown in Figures 10, 11 and 12 are ζ and p computed with ∆x = 5. The numer-ical solutions agree well with the exact solution. The figures also demonstratethat the slope limiter effectively removes spurious oscillations.

21

Table 5L2 error norms and convergence rates of the drying Riemann problem: Case D1.


20.00000 1.53e-03 - 6.04e-02 - 3.96e-04 -

10.00000 6.75e-04 1.18e+00 2.62e-02 1.20e+00 8.67e-05 2.19e+00

5.00000 3.31e-04 1.03e+00 1.31e-02 1.01e+00 2.82e-05 1.62e+00

2.50000 1.72e-04 9.45e-01 6.93e-03 9.13e-01 9.95e-06 1.50e+00

1.25000 9.00e-05 9.36e-01 3.70e-03 9.05e-01 3.59e-06 1.47e+00

Fitted - 1.01e+00 - 9.98e-01 - 1.67e+00



20.00000 1.54e-03 - 6.06e-02 - 9.47e-04 -

10.00000 7.13e-04 1.11e+00 2.79e-02 1.12e+00 3.01e-04 1.66e+00

5.00000 3.88e-04 8.78e-01 1.51e-02 8.90e-01 1.97e-04 6.13e-01

2.50000 2.15e-04 8.53e-01 8.24e-03 8.71e-01 1.35e-04 5.45e-01

1.25000 1.18e-04 8.66e-01 4.47e-03 8.81e-01 6.99e-05 9.47e-01

Fitted - 9.14e-01 - 9.28e-01 - 8.68e-01



20.00000 1.58e-03 - 6.41e-02 - 1.03e-03 -

10.00000 7.28e-04 1.12e+00 2.85e-02 1.17e+00 8.66e-04 2.51e-01

5.00000 3.90e-04 8.99e-01 1.49e-02 9.34e-01 4.03e-04 1.10e+00

2.50000 2.09e-04 8.98e-01 7.88e-03 9.22e-01 1.55e-04 1.38e+00

1.25000 1.12e-04 9.05e-01 4.22e-03 9.00e-01 6.04e-05 1.36e+00

Fitted - 9.44e-01 - 9.71e-01 - 1.07e+00

22

1e-05

0.0001

0.001

0.01

1 10 100

L2 (ζ)

∆ x

Drying Riemann

D1D2D3

0.001

0.01

0.1

1 10 100

L2 (p)

∆ x

Drying Riemann

D1D2D3

1e-06

1e-05

0.0001

0.001

0.01

1 10 100

L2 (q)

∆ x

Drying Riemann

D1D2D3

Fig. 9. L2 error norms of the drying Riemann problem.

Fig. 10. The computed solutions of the drying Riemann: Case D1.

23



24

4.4 Carrier-Greenspan solution

The Carrier-Greenspan solution considers a periodic wave propagating up anddown a sloping beach [23]. This problem involves both wetting and drying ineach period. We use a configuration similar to that of [14]. The computationaldomain and a computational mesh with ∆x = 0.025 are shown in Figure 13.The acceleration of gravity is set to 1 to solve the dimensionless SWE. TheCourant number is 0.1. The slope is inclined at 45 . Parameters such as theamplitude and frequency of a periodic wave are selected such that the frontreaches infinite steepness against the slope at periodic instants [14]. Unliketwo test problems in the previous sections, the solution of this test problemdoes not contain a discontinuity. The boundary condition along the offshoreboundary is applied through the numerical flux.

Bottom

x

y

x

ζ1.6

1.0

1.6

Fig. 13. Schematic defining the Carrier-Greenspan problem. The ∆x of the compu-tational mesh shown above is 0.025 m.

We compute L2 error norms for five sets of parameters shown in Table 8. Wecompare Cases 1, 2 and 3 to find the influence of the slope limiter. We compareCases 1, 4 and 5 to check the influence of the threshold H0.

The L2 errors and convergence rates of five cases are shown in Tables 9 through13 and Figures 14 and 15. The computed order of accuracy is approximately1.1 for ζ and 1.3 for p if no slope limiter is applied and H0 = 10−5. Theconvergence rate of q is 1.6. The order of accuracy is greater than 1.0 andbetter than what we obtained in the previous problems with discontinuities.However, it is still less than the optimal order of accuracy, which is 2. Thismay be attributed to the use of a fixed mesh, or, as Bokhove discussed in[14], it may be due to the infinite steepness of the surface elevation solution.

25

Table 8Model parameters for the Carrier-Greenspan problem

Case ID Slope limiter M H0

CG1 Not Applied - 10−5

CG2 Applied 100 10−5

CG3 Applied 1 10−5



If we compare Cases CG1, CG2 and CG3, it is confirmed that accuracy is notaffected significantly by the slope limiter. Comparing Cases CG1, CG4 andCG5 in Figure 15, it is found that the convergence significantly slows down ifH0 is large. It is presumed that the threshold H0 must approach zero as ∆xapproaches zero in order to obtain convergence to an exact solution. On theother hand, it is also observed that, in practical applications, accuracy maynot be sensitive to a selection of H0 as long as element sizes are relativelylarge.

Table 9L2 error norms and convergence rates of the Carrier-Greenspan solution: Case CG1.


0.10000 9.54e-03 - 5.15e-03 - 3.71e-04 -

0.05000 5.59e-03 7.70e-01 1.41e-03 1.87e+00 7.83e-05 2.25e+00

0.02500 2.65e-03 1.08e+00 7.19e-04 9.68e-01 1.63e-05 2.26e+00

0.01250 9.41e-04 1.50e+00 2.74e-04 1.39e+00 2.35e-05 -5.25e-01

0.00625 4.67e-04 1.01e+00 1.19e-04 1.21e+00 2.90e-06 3.02e+00

Fitted - 1.13e+00 - 1.32e+00 - 1.57e+00



0.10000 1.05e-02 - 5.43e-03 - 2.55e-04 -

0.05000 5.73e-03 8.76e-01 2.39e-03 1.18e+00 8.50e-05 1.58e+00

0.02500 2.78e-03 1.04e+00 7.33e-04 1.71e+00 2.56e-05 1.73e+00

0.01250 1.02e-03 1.45e+00 2.61e-04 1.49e+00 8.53e-06 1.59e+00

0.00625 4.78e-04 1.10e+00 1.08e-04 1.27e+00 3.73e-06 1.19e+00

Fitted - 1.14e+00 - 1.45e+00 - 1.55e+00

Shown in Figures 16, 17 and 18 are the results of Cases CG1, CG3 and CG5,

26



0.10000 1.23e-02 - 8.13e-03 - 3.34e-04 -

0.05000 5.51e-03 1.16e+00 2.37e-03 1.78e+00 5.80e-05 2.53e+00

0.02500 2.73e-03 1.01e+00 7.49e-04 1.66e+00 3.85e-05 5.91e-01

0.01250 9.87e-04 1.47e+00 2.59e-04 1.53e+00 1.40e-05 1.46e+00

0.00625 4.82e-04 1.03e+00 1.05e-04 1.31e+00 2.43e-06 2.52e+00

Fitted - 1.18e+00 - 1.57e+00 - 1.63e+00



0.10000 9.68e-03 - 5.12e-03 - 2.73e-04 -

0.05000 4.76e-03 1.03e+00 1.90e-03 1.43e+00 8.07e-05 1.76e+00

0.02500 2.67e-03 8.34e-01 7.37e-04 1.36e+00 3.55e-05 1.19e+00

0.01250 1.04e-03 1.35e+00 2.69e-04 1.45e+00 5.44e-06 2.71e+00

0.00625 5.46e-04 9.35e-01 1.27e-04 1.08e+00 2.11e-06 1.37e+00

Fitted - 1.05e+00 - 1.35e+00 - 1.79e+00



0.10000 1.42e-02 - 4.66e-03 - 1.08e-04 -

0.05000 5.75e-03 1.31e+00 1.93e-03 1.27e+00 6.48e-05 7.39e-01

0.02500 3.36e-03 7.74e-01 8.04e-04 1.26e+00 1.20e-05 2.44e+00

0.01250 2.31e-03 5.39e-01 4.94e-04 7.03e-01 4.74e-06 1.34e+00

0.00625 1.75e-03 3.99e-01 4.43e-04 1.57e-01 2.13e-06 1.15e+00

Fitted - 7.35e-01 - 8.76e-01 - 1.51e+00

respectively. All the results agree well with the exact solution. A steep gradientin the solution at t = 1.5 is also captured well.

27

0.0001

0.001

0.01

0.1

0.001 0.01 0.1 1

L2 (ζ)

∆ x

Carrier-Greenspan

CG1CG2CG3

0.0001

0.001

0.01

0.001 0.01 0.1 1L2 (p

)

∆ x

Carrier-Greenspan

CG1CG2CG3

1e-06

1e-05

0.0001

0.001

0.001 0.01 0.1 1

L2 (q)

∆ x

Carrier-Greenspan

CG1CG2CG3

Fig. 14. L2 error norms of the Carrier-Greenspan problem.

28

0.0001

0.001

0.01

0.1

0.001 0.01 0.1 1

L2 (ζ)

∆ x

Carrier-Greenspan

CG1CG4CG5

0.0001

0.001

0.01

0.001 0.01 0.1 1

L2 (p)

∆ x

Carrier-Greenspan

CG1CG4CG5

1e-06

1e-05

0.0001

0.001

0.001 0.01 0.1 1

L2 (q)

∆ x

Carrier-Greenspan

CG1CG4CG5

Fig. 15. L2 error norms of the Carrier-Greenspan problem.

Fig. 16. The computed solutions of the Carrier-Greenspan problem: Case CG1.

29



30

4.5 Parabolic bowl

Here we test the proposed method for a two-dimensional axisymmetric phe-nomena. The bottom depth is a paraboloid and prescribed as h(x, y) = αr2

where α is a positive constant and r =√

x2 + y2. At an initial state, the watersurface is also in a parabolic shape and the velocity is zero. The exact solutionis periodic in time with a period τ . One can find the exact solution in [24].

The water depth H(r, t) is nonzero for r <√

(X + Y cos ωt)/α(X2 − Y 2) andthe exact solution is given within the range as follows:

H(r, t) =1

X + Y cos ωt+ α(Y 2 − X2)

r2

(X + Y cos ωt)2, (40)

(u(x, y, t), v(x, y, t)) = − Y ω sin ωt

X + Y cos ωt

(

x

2,y

2

)

, (41)

where ω2 = 8gα and X and Y are constants such that X > 0 and |Y | < X.The period is τ = 2π/ω.

We use a similar set of parameters to what Ern et al. used in [15]. The con-stants, α, X and Y are set to 1 × 10−7 m−1, 1 m−1, and −0.41884 m−1,respectively. The acceleration of gravity is set to 10 m2/s. The computationaldomain is a square with the side length of 8000 m. The Courant number is setto 0.12. The slope limiter is applied with M = 0.01.

We compute the L2 error norms for five different mesh resolutions. The errorsare computed at two times: t = τ/2 and t = τ . Since the exact solution isperiodic and wetting occurs in the first half of a period and drying occursin the second half of a period, the norms measured at t = τ/2 and t = τrepresent errors in wetting and drying, respectively. The obtained error normsand orders of convergence are shown in Tables 14 and 15 and Figure 19.

The computed convergence rate is approximately 1.1 for ζ and is 1.3 for p andq in both wetting and drying stages. This indicates that the proposed wettingand drying treatment yields a similar accuracy order in wetting and dryingstages at least in the tested situation.

Plotted in Figure 20 are the initial surface elevation and approximated surfaceelevations in the the half of the computational domain where x ≥ 0. Thetriangles drawn in blue are wet elements, in which the wet-or-dry status ω

(l)K is

1. The triangles drawn in light blue are dry elements. It is evident in the plotat t = 5τ/6 that some wet elements are left behind the correct drying front.However, note that water in these fictitiously wet elements is still flowing downthe slope and these wet elements will eventually become dry. Because of theseerroneously wet elements, it may not be appropriate to determine water frontsbased on the wet-or-dry status. One may want to introduce a threshold depth

31

different from H0 to determine water fronts according to practical necessity.We should emphasize that the water depths in these fictitiously wet elementsare very small and such a small water depth can be ignored in practical usage.In fact, the L2 errors in these artificially wet elements are at most less than1.5 % of the total.

Table 14L2 error norms and convergence rates of the parabolic bowl problem: t = τ/2.


400.00000 3.56e-06 - 3.05e-06 - 2.54e-06 -

200.00000 1.85e-06 9.43e-01 1.39e-06 1.13e+00 1.13e-06 1.17e+00

100.00000 9.04e-07 1.04e+00 5.38e-07 1.37e+00 4.52e-07 1.33e+00

50.00000 4.09e-07 1.14e+00 2.08e-07 1.37e+00 1.72e-07 1.39e+00

25.00000 1.75e-07 1.22e+00 8.01e-08 1.38e+00 6.67e-08 1.37e+00

Fitted - 1.09e+00 - 1.32e+00 - 1.32e+00

Table 15L2 error norms and convergence rates of the parabolic bowl problem: t = τ .


400.0 3.39e-06 - 2.78e-06 - 2.43e-06 -

200.0 1.86e-06 8.68e-01 1.38e-06 1.02e+00 1.12e-06 1.12e+00

100.0 9.01e-07 1.04e+00 5.37e-07 1.36e+00 4.47e-07 1.32e+00

50.0 4.08e-07 1.14e+00 2.10e-07 1.35e+00 1.71e-07 1.39e+00

25.0 1.74e-07 1.23e+00 7.98e-08 1.40e+00 6.64e-08 1.37e+00

Fitted - 1.07e+00 - 1.30e+00 - 1.31e+00

32

1e-07

1e-06

1e-05

10 100 1000

L2 (ζ)

∆ x

Parabolic Bowl

t = 0.5 τt = 1.0 τ

1e-08

1e-07

1e-06

1e-05

10 100 1000L2 (p

)

∆ x

Parabolic Bowl

t = 0.5 τt = 1.0 τ

1e-08

1e-07

1e-06

1e-05

10 100 1000

L2 (q)

∆ x

Parabolic Bowl

t = 0.5 τt = 1.0 τ

Fig. 19. L2 error norms of the parabolic bowl problem.

33

t = 0

0

2000

4000

x (m)

-4000-2000

02000

4000

y (m)

0

1

2

3

4

5

Sur

face

Ele

vatio

n (m

)

t = τ/6

0

2000

4000

x (m)

-4000-2000

02000

4000

y (m)

0

1

2

3

4

5

Sur

face

Ele

vatio

n (m

)

t = 2τ/6

0

2000

4000

x (m)

-4000-2000

02000

4000

y (m)

0

1

2

3

4

5

Sur

face

Ele

vatio

n (m

)

t = 3τ/6

0

2000

4000

x (m)

-4000-2000

02000

4000

y (m)

0

1

2

3

4

5S

urfa

ce E

leva

tion

(m)

t = 4τ/6

0

2000

4000

x (m)

-4000-2000

02000

4000

y (m)

0

1

2

3

4

5

Sur

face

Ele

vatio

n (m

)

t = 5τ/6

0

2000

4000

x (m)

-4000-2000

02000

4000

y (m)

0

1

2

3

4

5

Sur

face

Ele

vatio

n (m

)

Fig. 20. The initial and computed surface elevations of the parabolic-bowl problem.

34

5 Conclusions

A wetting and drying treatment was proposed for the RKDG discretizationto the shallow water equations. The proposed method use fixed meshes. Apost-process operator, which is called the PD operator, was introduced as apart of the proposed wetting and drying treatment. A combination of theproposed wetting and drying treatment and a TVB slope limiter was tested.Conflicts between the wetting and drying treatment and TVB slope limiterwere observed, but the conflicts were able to be avoided by not applying theslope limiter on elements where the water depth is smaller than a certainvalue. The order of convergence was examined for four test problems. Thenumerically estimated order of accuracy is approximately 0.8 - 1 for solutionswith discontinuities or 1.1 - 1.6 for smooth solutions. The numerical flux iscontrolled in such a way that the positivity of the mean water level within eachfinite element is ensured with an arbitrary time step. This special treatment onthe numerical flux enabled us to compute all the test cases stably with Courantnumbers greater than or equal to 0.1. This indicates that the proposed wettingand drying treatment does not raise the time stepping restriction as would beexpected. This is important to solve long-term large-scale problems within arealistic computational time.

The proposed method was tested only with the linear triangular element.Extention to higher order interpolation is open to future work.

Acknowledgments

The first author was supported by the Japan Society for the Promotion of Sci-ence under grant KAKENHI(19860028). The second, third and fourth authorswere supported by the National Science Foundation grants DMS-0620697 and0620696 and by the Office of Naval Research, Award Number N00014-06-1-0285.

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Documents

A Wetting and Drying Treatment for the Runge-Kutta