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

Abstract

-

Our aim is

to

indicate the smoothness of solulions

to the

shallow water equations involving uarying width. o{ balance laws. To

a numerical test. we

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

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

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 smooih and rough regions.

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

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 has never been invesiigated by other authors.

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.

.

The amputer programming work for numrical mtropy production is also simple,

s

ii is merely the lo€al 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.

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

---a

I

,{r. f)

I

i

---r--.-u-rjl.,,

--'--

./L.--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].

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

The entropy and the entropy inequalily are respectively given by

t4.\.tt

_ritt

1r,

_f*r,

\-t \-tl

i

I

.-:

---r

u

JJ

* !lt--

_{-\ t)}

,'l

.'r

i

l'

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.

Yr'e oblain

that

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

dam

break

problem.

We

small ai smooth regions. In fact

observe

that

large vaiues

ol

this

is

also

lrue

for

balance

numerical enlropy

produotion laws

as we

see

in

these

are

generated

at

the

shock

numerical

results.

This

gives

discontinuilies

ol

rhe slage

the flexlbility

of the

use

ol

the

(free water

surlace).

The

numerioal

snlropy

produclion

values

at

smooth regions

are

as

smoothness indicator. Nole

very

small crmpared

to

the

that

the

resillts that

we

obtain values at the

discontinuities,

here is very accurale.

References

[]

I

Fl. J. Leveque, 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"

[4] J, Balbas, S. Karni, A central scheme lor shallop water tlows along channels with iregular

qeomelry, ESAIM: M2AN43 (200S) 333-351.

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 is

nA

d{)

-+3=(J.

At

dx

to t{o:

l

\

3+_l

-+_si-rr:l=

il f,rld 2" )

! .,do

=9h-

.

Ldx

ln this model:

.r:

is the spaoe variable

/

is the lime variable,

fi =

Jr(,v.ll

represe!:ts the heighl ol waler above the bottom ol the channel.

o

-

oi,r'lt

is ihe channei width, Q

- 1tt

is the discharge.

-1

- olt

is lhe wel cross-section,

r/ = rii.\'. /

)

represeDts the depih averaged water valociiy. and

.q

is ihe acceleralion due to gravity.

Consider the circular dam break problem, wher€ waler depih o, 10 ,n is inside a drum with rad;us ol 50 m.

Outsi.-Je the drum. 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).

E_

El

^l

"1

,/ \

I

;.1

"l

,/\l

r----,,

..--1

]

U

are interested in finding the

INTRODUCTION

THE CONSIDERED

PROBLEM

MATHEMATICAL MODEL

METHOD