Numerical entropy production of shallow water flows along channels with varying width.

NUMERICAL ENTROPY PRODUCTION OF SHALLOW WATER FLOWS ALONG
CHANNELS WITH VARYING WIDTH
Sudi Mungkasi
Department ol Mathematics, Faculty ol Science and Technology, Sanata Dharma University,
Mrican, Tromol Pos 29, Yogyakarta 55002, lndonesia, Ernail: sudi@usd"ac.id

- Our aim is to indicate the smoothness of solulions to the shallow water equations involving uarying width.
achieve the aim, we extend the applicat;cn of numerical ent[opy production. as the smoothness indicatror. from
consider a radial dam break with the whole spatial domain is B/et. The resulting dam break flow is a solution

o{ balance laws. To
a numerical test. we
are interested in finding the

Abstract

discontinuities can be an aid tor the improvement of the numerical solution in ierms of its accuracy. The numerical entropy production is found to be accurate to detect
discontinuities. The discontinuity of the water surface is clearly indicated by iarge values of the numerical entropy produclioo as the smoothness indicator- ln short, the
numerical entropy production is simple to irnplement, bul gives an accurate detection of the smoothness of solutions.

Keywords: shallow water equations, finite volume method, numerical entropy production, smoothness indicator

THE CONSIDERED PROBLEM

INTRODUCTION

We cor,sider waier flows along channels with varying vridth.

Waler {lows are wellmodelled by shaliow water equations. These llows
o{ten involve discontinuities or rough regions on iis free surtace lll- A
smoothness indicaior is needed, when we want 10 know lhe positions ol

---a

smooih and rough regions.

I

,{r. f)

Puppo and Semplice l2l proposed the use ol numerical eniropy prodt,ction

to measure the smoothness o{ numerical solutions to conservation laws.
The order of accuracy ol the numerica, entropy produclion ai

i

I

------r--.-u- --'--

rjl.,,

./L.--

discontinuilies is lower than that at smooth regions.

ln this work, we extend lhe application of nlrmerical entropy production
lrom conservation laws to balance laHs [3]" A conservation law is a

balance law witl'lout source teIms" The balance law thal we consider is lhe
one-dimensional shallow waier equalions involving varying width. The

varying width csntribules in the source term ol lhe balance law [4]. To our
knowledge, ideniifying discontinuifles along channels with varyi&g width

Frcm ihe top le{t wilh the clockwise

direciion, the figures show

a

schematic
setling ot water {lows viewed lrom (a). aside,
(bi. {ro!11, and (c). above. respectively. Here
the figilres are from Balbas and Karni [4].

has never been invesiigated by other authors.

METHOD

MATHEMATICAL MODEL
We assume a horizoillal topagraphy io simplity ihe problem. The bottom o{ the

chanr,ei is the harizo*lal re{erence- The mathematical model

ior water llows

To solve the mathenralical model, we use a ilnite volume method. The smoothness
indicaior ol the solution is the numerical entropy production.

is

nA d{)
At dx
-+3=(J.
to t{o: l
\ ! .,do
3+_l
Ldx .
il f,rld-+_si-rr:l=
2" ) =9h-

The entropy and the entropy inequalily are respectively given by

t4.\.tt _ritt

\-1r,

ln this model:

.r:
/

is the spaoe variable
is the lime variable,

o - oi,r'lt

is ihe channei width,
is the discharge.
is lhe wel cross-section,
represeDts the depih averaged water valociiy. and
is ihe acceleralion due to gravity.

t tl
.-:
,'l

fi = Jr(,v.ll represe!:ts the heighl ol waler above the bottom ol the channel.

- 1tt
-1 - olt

Q

r/ = rii.\'. / )

.q

i

---r u
.'r i

JJ

*

I

l'

_f*r,
!lt--

-\ t)

The nume.ica, entropy production is the local truncation error ol the enlropy. Note
thal the entropy inequality above is conect in ihe weak sense.

NUMERICAL RESULTS

DISCUSSION

Consider the circular dam break problem, wher€ waler depih o, 10 ,n is inside a drum with rad;us ol 50 m.
the waier depth is 1 m. A1 tima zero, the druin is remr:ved. For positive time, we solve
the problem using lhe tinile volume melhod- We measure the smoothness of the solutjon using Numerical
Entropy Production {NEP}. Three figures below show the initral water surface (in 3D viewJ, the initial water
surface and its NEP (in ?D view), and the results o{ waier sudace and its NEP at time 4 s (in 2D view).

lhe numerical Puppo and Semplice [2] proved
entropy production behaves that {or conservation laws, the
excellently as the smooihness numerical entropy production is
indicator {or the radial (circuiail large at rough regions and

Yr'e oblain that

Outsi.-Je the drum.

dam break problem. We

E_

El

"1
;.1
"l

U

,/^l \
,/\l
r----,, ..--1

observe that large vaiues

I

]

numerical enlropy produotion

are generated at the shock

ol rhe slage
(free water surlace). The

discontinuilies

values at smooth regions

very small crmpared to

values at the

Conclusion
Our cof,cluding remarks are as followsi
Numerical ertropy production has been found to be acamte as a ffioothne$ iRdi€tor lor solutions to the
shallowwaler equations involvirg varying widttr.
ii is merely the lo€al
The amputer programming work for numrical mtropy production is also simple,
truncation error or lhe triropy.

For luture work, re rmmrEnd the use of nxmiel entropy produclion as the rsrinerenl anayor
coarsening indiEtorwhen an adaptive nuruicalmlhod is desired to solle the ghallffiwatel equatjofis.

.

.

.

s

References
[] I Fl. J. Leveque,

ol

are
the

discontinuities,

small ai smooth regions. In fact
this is also lrue for balance
laws as we see
these
numerical results. This gives
the flexlbility of the use ol the
numerioal snlropy produclion
as smoothness indicator. Nole
that the resillts that we obtain
here is very accurale.

in

Finite-volume nethods tot hypeft]alic problems, Camblidge University Press.

Cambridge,2004.

[2] G. Puppo, M, Semplice, Numerical enlropy and adaptivity tor linite voiure schemes,
Com*unialions ia Computationat Physics 10 (2011) 1132-1160,
I3l S. Mungkasi, Ao accurate smoothness indicator for shallow Mter flos along channels with
varying sidth, submified lo Applied Mechanics and Materials"
A central scheme lor shallop water tlows along channels with iregular
qeomelry, ESAIM: M2AN43 (200S) 333-351.

[4] J, Balbas, S. Karni,