The Courant–Friedrichs–Lewy number influences the accuracy of finite volume methods.

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PREFACE

All

praises are due to Allah, God Almighty, Who made this annual event

of

successful

of

"

The "4th Annual Basic Science International Conference 2014 in conjunction with The 5th International Conference on Global Resource Conservation2014, both annual scientific

events organized by the Faculty of Mathematics and Natural Sciences, Brawijaya Uniyersity.

ln

this year, the conference took a theme of "Applying science to conserve nature". These

conferences are concerned about our current challenge on how to explore, utilize and apply our knowledge and science to conserve water, soil, earth, air, plants, animals and

microorgan-isms that involved multi disciplines. As a conjunctive conferences, these covered a wide range

of topics on basic sciences: physics, biology, chemistry, mathematics and statistics as well as

conservation biology and applied science.

The conference in 2014 was the continuation of the preceding conferences initiated in 2011 as

the International Conference on Basic Science (ICBS) and International Conference on Global Resource Conservation initiade in 2010. Therefore the proceeding was also divided into two books, each with, each with a different ISSN. The proceedings were also published in

electronic forms that can be accessed from BaSIC website.

I

am glad that for the first time both types of publication can be realized. These international

conferences are held to to increase dissemination of applying science to conserve nafural

re-sources, to present new research findings, ideas and informations and to discuss topic related

to conference theme and to develop collaboration among multi discipline sciences and to find

potential young researchers. This is in line with university vision as a World Class Entrepre-neurial University.

I

am grateful to all the members of the program committee who contributed for the success in

farming the program. I also thank all the delegates who contributed to the success of this

con-ference by accepting our invitation and submitting paper for presentation in the scientific

pro-gam.

I

am also indebted to PT. Fajarmas, PT. Makmur Sejati, KPR[, CV. Gamma, PT. Tata

Bumi Raya and CV. Enseval Medica Prima for their support in sponsoring this event.

I

hope that you enjoy in this seminar. We wish for all of us a grand success in our scientific life and for the coming conferences and even better.

Thank you,

Malang, February 2014

The 4ft Annua! Basic Science lnternational Gonference February

t2-t3'd2o!4

Program Committee

Steering committee

Rector, University of Brawijaya

Dean, Faculty of Mathematics and Natural Sciences,

Associate Deans 1, 2 and3, Faculty of Mathematics and Natural Sciences Chairperson

Amin Setyo Leksono, S.Si.,M.Si.,Ph.D. Deputy-Chair

Lukman Hakim, S.Si.,MSc.Dr. Sc.

Secretary

Dr. Sri Rahayu, M.Kes.

Treasurers

Tri Ardyati, M.Agr.,Ph.D. Mrs. Sri Purworini

Mrs. Rustika Adiningrum

Publication & Proceedings

Muhaimin fufa'i, Ph.D. Med.Sc Luchman Hakim, M.Agr.,Ph.D.

Secretariats and registration

Mrs. Sri Aminin

Mrs. Trivira Meirany

Ms. ArikArubil

Mr. Ekwan

Scientific sections and program

Dr.k. Moch. Sasmito Djati, MS.

Dr. Suharjono

Ms. Dewi Satwika

Transportation, excursion & social Event Dr. Agung Pramana Warih Mahendra, M.Si.

Dr. Nunung Harijati Dr. Sri Widyarti

Brian Rahardi, S.Si., M.Sc. Mrs. Arnawati

Venue

Dr. Sofy Permana, M.Sc., D.Sc.

Drs. Aris Soewondo, M.Si. Mr. Karyadi Eka Putra

Mr. Agung Kumiawan Mr. Hasan Muhajir

Consumptions

Dra. Gustini Ekowati

Dr. Serafinah Indriyani

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Intemational Scientific Committee and Editors

Prof' Dr' Masashi _{Kato, Faculty of Agriculture, Department of Applied Biological Chemistry,}

Meijo University Nagoya

Prof' Dr. rer. Nat Helmut Erdmann, Flesnburg university of Applied Science

Prof. Dr. _{Maketab Mohamed, Malaysian Nature Society, tutataysia}

Prof' Dr' Chow Son _{Chen, School of Earth Science, National}

ientral

_{University, Taiwan prof.}

Fatchiyah, M.Kes., Ph.D, Department of Biolc,gy, _{Brawijaya Universifz}

Dr'Farah Aini Abdullah, School _{ofMathematicat sciencls, Universiti bains Malaysia}

Dr'Ir. Estri Laras, MSt Sc., Department _{of Biology, Brawijaya university}

Dr. sasmito Djati, Deparhnent _{of Biology, univJisity of rirawijaya,Indonesia}

Dr. Setijono Samino, Department _{of Bior,ogy, univeisity of nrawi.iay4 Indonesia}

Dr. chomsin s. widodo, _{Department of} physic, _{University of Brawijaya, rndonesia} Dr. Endang _{Arisoesilaningsih, Department of Biology, University of Braw-rjaya, Indonesia}

Dr' Retno Mastuti, Department _of Biorogy, University of Brawijaya, rndonesia Dr. Sukir Maryanto, Department of physic, _{University of Brawijaya, lndonesia} Dr. Didik R santoso, Department _of physic, _{University of Brawijaya, rndonesia}

Dr. Ratno Bagus, Department of Mathematics, university of Brawijaya, rndonesia

Dr' Wuryansari Muharini K, Department of Mathematics, University of Brawijaya, lndonesia

Dr. Barlah _{Rumhayati, Department of chemistry, University of Brawijaya, lndonesia} Dr. Elvina, _{Department of Chemistry, University of Brawijaya, lndonesia}

Dr. _{Masruri, Department of Chemistry, University of Brawijaya, lndonesia}

Dr. Amin setyo Leksono, Department of Biorogy, University of Brawijaya, rndonesia

The 4s Annual Basic Science lnternational Conference

MESSAGE

FROM

RECTOR OF

BRAWIJAYA UNIVERSITY

Honorable Keynote Speaker

Prof. Dr. Masashi Kato, Dep. of Applied Biological Chemistry, Meijo University Nagoya

Prof. Dr. rer. Nat Helmut Erdmann, Flesnburg University of Applied Science

Prof. Dr. Maketab Mohamed, Malaysian Nature Society, Malaysia

Dr.Farah Aini Abdullah, School of Mathematical Sciences, Universiti Sains Malaysia prof. Dr. Chow Son Chen, School of Earth Science, National Central University, Taiwan

Dean Faculty of Mathematics and Natural Sciences

And

All

participants

Assalamualaikum Warahmatullahi Wabarakatuhu

First of all let us pray to Allah the Almighty for His blessings bestowed to all of us that today

we can all be here to attend the The 4th annual Basic Scienee International Conference and the

5th International Conference of Global Resources Conservation.

It

is indeed an honor and privilege for me to welcome you warmly at this Conference'

My

great appreciation

to

all

of you the distinguished participants

of

this important event' and

trust that this conference

will

be an valuable

input

to the empowerment of Applying science

to conserve our nature.

I

would like to take this opportunity to offer my appreciation to Dean

Faculty of Mathematics and Natural Sciences, and committee from Biology Department

Uni-versity of Brawijaya who have organized this Seminar.

Distinguished Guests, Ladies and Gentlemen,

University of Brawijaya (UB) as oneof the leading university in Indonesia with its mission

of

World Ctass University, should take actions to participate in conserving our nature where we

are living inside through innovation of science. We encourage all of academician here, as the backbone ofnation building for continuous learning to save our planet for best interest

ofhu-man being and

living

thing.

Therefore, We should work together, across institutions and

across discipline. We should start thinking how to be leader and control the world with our

resources, knowledge and technology to achieve an equitable welfare.

Distinguished Guests, Ladies

and

Gentlemen,

UB with

combination between lnternational standard and local culture has been educating

communities who have made positive impacts

in

their Communities-throughout Indonesia'

Hence we warmly welcomes collaborative works of mutual partnership with many other insti-tutions; especially universities, industr;r, government and other institution; both at national and

intemational level. Moreover UB has dedicated itself to be a world class university. Based on

spirit we belief that only with international partnership able pursue multinational connectivity

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The 4u Annual Basic Science lnternational Gonference

on business, established in the higher education institution, for future generations. In this case,

this occasion is really necessary to initiate cooperation beyond national border to supply

com-prehensive knowledge to be a winner in the global competition.

Distingukhed

Guests, Ladies and Gentlemen,

Science, technology and education

will

determine the well-being of people and nations in the

future. Therefore, academician and scientist as Scholar who

will

bear the future of this nation

have to work hard, and the ability to create _{brilliant innovation to supply environmental}

tech-nology

to

solve human problem.

UB

rely on and encourage increased intemational

partner-ship, as well as greater staff mobility. The partnership,

it will

be able to supports economic

and social development.

In

Another word, whatever challenges that

will

arise,

it

should be

challenges for all of us, and it should be solved with partnership.

As a conclusion,

I

propose to strengthen our collaboration to create synergism at any levels, especially all stakeholders, locally, regionally, as well as internationally. By working together

all over the world through sharing information and resources, we can make the bright future.

Finally, let us hope that the

fruitful

points aimed from this conference

will

develop the new

concept and networking based on science and technology to save our nature.

During the organization and execution of this conference, and the one

I

am cunently opening

"The 4th annual Basic Science International Conference and the 5th lnternational Conference

of Global Resources Conservation".

Ladies and gentlemen, _I hereby wish you a fruitful Conference.

lvabillahitauliqwalhidayah, wossalamualuikumwarahmatullahiwabarakatuh

Thank you for your attention.

The 4e Annual Basic Science lnternational Conference

February

t2-13'd2OL4

TABLE OF CONTENT

Field of Study: Biology

Ileld

code

Author(s) Paqe

Bil

l

Sumartini, Heru

Kuswantoro

Resistance evaluation oflarge seeded soybean

lines to rust disease

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19

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39

43

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Ruuy A.

Nugroho.

il'si%,?mffiffiff*rs

ln Agricurturar Distribution and Accumulation ofMercury in

Haikal Prima

Fadholi

Kerang Bulu (Anadara sp.) at Sekotong Region, West Lombok, Nusa Tenggara Barat.

Bi7

Bil0

Bil2

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Fauziah Hafidha,

Mohamad

Nur-zaman, andNurullia Fihiani Teguh Husodo

Siti Sumarmi, Retno Peni,

Sancayaningsih,

Sebastian Margino,

RC. Hidayat

Soesilohadi

Nurul Isnaini

Supartini Syarif,

Heru Kuswantoro RCH. Soesilohadil,

S. Sumarmi, S.

Marginio danR.

Susandarini

Hary Widjajanti

Sar-no, and Novida

Rosalia Sinaga

Srihardyastutie, A,

Soeatmadji,

D.W.,

Fatchiyah, and

Aulanni'am

Leaf characteristics of some trees species in

ac-cumulating SOz and NO2 pollutant in park area

of Padjadjaran University Jatinangor

Biological Control of Cabbage Head Caterpillar Crocidolomia binotalis, Zeller by Using Fusants

Bacillus thuringiensis vw. kurstaki and Bacillus thuringiensis var. Israelensis Cultured in Whole

Coconut Fruits

The Effect of Trehalose Levels on Post-Thaw

Sperm Membrane Integrrty of Boer Goat Semen

antimutagenic effect from brassicaceae extracts on swiss-webster mice induced by lead acetate

Yield And Yield Components Of Acid-Tolerant

Soybean Promising Lines On Ultisols

Biological control and managemen of insect pest

on'strobe.ry community: 1,' lnsect diversity in

strawbery community

Screening and identification of cellulose-degrading bacteria from the mangrove areas

of

Sembilang National Park South Sumatera

Lipase/amylase ratio as the indication of

pancre-atic exocrine inflammation and the correlation with insulin resistance in type 2 diabetes mellitus

29

a

F.

-Bi3

Bi24 Bi23

February

Lz-t3td2014

The 4s Annual Basic Science lnternational Conference

Bi32

Bi33

Bi34

Bi35

Bi45

Bi50

Eko Agus Suyono,

Dr. Ida Kinasih Ateng Supriatn4

Ramadhani

EkaPu-trq

Robert

Ma-nurung, and Rizkita

R.E

Ramadhani Eka

Pu-tra

Khairunnisa

Arumsa-ri, RindraAryandari, Haikal Prima Fadholi

Atit Kanti, Nampiah Sukarno,Latifah

K

Darusman, Endang

Sukar4I Made Sudianq and Kyria Boundy-Mills

The effect of salinity on growth, dry weight and

lipid

content

of

the

mixed microalgae culture

isolated from glagah as biodiesel substrate

Soil mesofauna diversity in two different ages

of

cocoa (Theobroma cacao L.) Plantation

Study on Bioconvertion

of

Cassava Tuber Skin

and

Dry

Rice

Stalk

By

Black

Soldier Flies (Hermetia illucens)

Pollination Agents

of

Coffee

at

Small Urban

Plantations in Sumedang, West Jav4 Indonesia

Diversity and

Abundance

of

Zooplankton in

Rawa Jombor, Klaten, Central Java

Novel Cellulolytic Yeast Isolated from South

East

67 Sulawesi, Indonesia

47

51

55

59

63

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The 4e Annual Basic Science lnternational Conference February

!2-t3'd2o44

Field of Study: Chemistry

Field

code

Authods)

Title

Page

Effect Of Agricultural Waste On The Growth And

Manganese Peroxidase Production

Of

Phanero-chaete Chrysosporium

the study of adsorytion on Cr(VI) in natural clay

surface modified with surfactant CTAB

(Cetyltri-m ethy I ammonium Br om i de)

Selective Hydrogenation Of Furfrral Using

Ni/!-Alzo: Catalyst

A New Oxalate Ion Sensors Based On Chitosan Membrane

Evi Susanti

Maksum, SusiNu-rul Khalifah,

An-ton Prasetyo,

Siti Mariyah Ulfa

and Elvina Dhiaul Iftitah

Atikah

75

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Field of Study: PhYsics

Field code

Ph4

Ph9

Phl0

Ph12

Ph13

Ph15

Munawwarah, Ahmad Marzuki S,

and Iven Ganesja

Firdy Yuana

Abdurrouf

Supriyanto, A. Ha-ris, and Y. Sofyan

T.B.

Susilo,

U.

B.

L.

Utami,

R.

Nur-masari,

R.

Mu-jianti, _{M. Habibi,} S.

Mawadah,

A.

Pradanq S. Hayati,

Y.

Seftiawan, O.

Soesanto,

B.

I.

Prayogo and Satrio Wasis, Sukir Maryanto, and

Dahlia Kurniawati

Relocation of Hypocentrum Earthquake in Mentawai Island Region as Data Support

Determination of Earthquake Early Warning

Measurement of bioefficacy and its effects on one

push aerosol insecticide by using glass chamber The effect of probe pulse on the dynamic signal in a

double pulses experiment

Gravity data analysis

of

Teluk Mandar, Makassar Strait

Modeling

of

Relative Dating Using Spectroscopy

Analysis of Tamban's Subfossil

Particle

Motion

Of

Seismic Waves Recorded

From Hydrothermal Area

At

Cangar, East Java,

Indonesia

93

97

101

105

113

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L-The 4tr Annual Basic Science lnternational Conference

Field of Study: Mathematic and Statistics

Field code Title

tvtonitoring of High-Yielding Varieties of Rice

MS2

CandraDewi _{Plants Using Multi Temporal Lands at 8 Data}

in

statistics

in

Tripoli,

125

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131

135 Serag M.

Ali

Elwan, Ni Wayan

Surya Wardhani,

and Maria

Bernna-detha Theresia

Mitakda

Sudi Mungkasi

Preatin, Iriawan, N.

,

Z-ain,I., and

Hartanto, W

Diana

Kurnia

Sari

Sudirman,

Ris-mawati

Ramdani,

and Siti Julaeha

lnu Laksito Wibowo,

Suhariningsih, and

Basuki Widodo

Rismawati

Ramda-ni

Ratna Dwi

Christyanti Ratno Bagus Edy Wibowo

2), _{Abdul Rouf}

Alghofari 2) Caroline

Lilik

Hidayati

Factors affecting

the

Performance

subject among trigtr sctroot students

Libya

(part 2: all factors)

The Courant-Friedrichs-Lewy number influences

the convergence rate of finite volume methods Bayesian migration schedule model: an aplication

to migration in east java

A Totally Irregular Total Labelling Disjoint Union of Wheel GraPh

The Application Of Meshless Local Petrov

Gaierkin On CalculatingVolume Of River

Sedimentation ln Confluence Two Rivers

On the Total Irregularity Strength of Friendship

Existence and Uniqueness Solution of

Euler-Lagrange Equation in O

c

IRn

Aplication Barro Model as Eflorts to lncrease

Eco-nomic Growth Central Java Through Health

Spend-ing

The Development of

GtlI

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MS10

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MS14

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MSIT

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The dh Annual Basic Science lnternational Conference

The

Courant*Friedrichs-Lewy

Numb

er

Influences

the

Accuracy of Finite

Volume

Methods

Sudi Mungkasi, S.Si., M.Math.Sc., Ph.D.''-) and Drs. Noor Hidayat, M.Si.''')

1)

Department of Mathematics, Faculty of Science and Technologs, Sanata Dharma (Jniversity,

Mrican, Ttornol Pos 29, Yogvakarta 55002, Indonesia

2)

Docloral Program, Facttlty of science and rechnologl', Airlangga utiversity, surabaya' Indonesia

3)

Department of Mathematics, Brawijaya t)niversi4t, Malang, lndonesia

-)

Couespondiug author: sudi@usd.ac.id

Abstract- The shallow water (wave) equatlons govern

shallow water flows. We solve the shallow water equations

using a finite volume method. A necessary condition for a

consistent linite volume method

to

be stable (hence'

convergent)

is

thet the method satislies the Courant-Friedrichs--Lewy (CfL) condition. Numbers representing

this condition are called CFL numbers. In this paper, the

effects of CFL numbers to the convergence rate of the finite volurnr method are investigated. Setting a CFL number to

the method gives varying time steps

in

the numerical cvolufion. We compare results betwcen those produced by imposing a CFL numtrer and imposing a fixed time step to

the numerical method. We shall show rvhich stretegy is more

cflicicnt and produces morc accurate solutions in solving the shallorv water equations.

Kelnrords----t,onvergence ratg Courant-Friedrichrlewy',

flnite volume method, shallow water equations.

I.

INTRoDUCTTON

rFHE

_system of shallow water equations is a

well-l.

koo*n mathematical model that describes shallow water waves and flows. We are interested in solving

these equations as the solutions are usef,rl

in

the simulations of rsal world problems such as dam break floods and tsunamis. Ia this paper we implement a flnite volume method to solve the shallow water equations. The method is chosen due to its robustness in dealing with

srnooth and non-smooth solutions [9, l0].

In finite volume methods, a necessary condition for

convergelrce is that the Courant-Friedrichs-Lewy (CFL) condition be satisfied [3, 9, l0]. This condition is related

to the time stepping in the integration of the shallow water equations with respect to time after the equatioas

are discretized with respect to spacs. This means thal we

can use either a fixed time step as long as the CFL condition is satis{ied at every time step or a varying time step based on a lixed CFL number. Here a CFL nutnber

represents a positive nuurber such that the CFL condition is satisfied.

This paper investigates the influence of CFL number to the accuracy ofnumerical solutions produced by the finite

volume method. The accuracv

of

the frnite volume

method,

of

cowse, depends on the accuracy

of

the

integration teclnique implemented to the space and time. To focus on our investigation, we use a single integration technique for the space variable, that is, we use a second

order method for the space integralion. Then we compare the performance of a second order method for the tilue

integration

by

presenting

the

errors

between implementing a fixed time step and a fixed CFL number.

This paper is organized as follows, Tn Section

II

we recall the shallow water equations in one dimension. The finite volume method

is

presented

in

Sectiou III. Nurnerical results are given in Section [V. Finally some

concluding retnarks are drawn in Section V'

II.

GownNnvoEquenoNs

The shallorv water equations are

h, + (hu), =0

,

(1)

thu\,+$gh2

+huz),--ghB,.

(2)

where

/

denotes the time varjable,

x

denotes the space

variable,

*(x,l)

is water height or depth,

n(x,l)

is

veiocity,

8(x)

represents

flre

bottom elevation or topography, and

g

is the acceleration due to gravity. The

absolute water level (stage) is defined as

A number of authors lrave proposed numerical techniques to solve these shallow water equations

(l)

and (2). Some

ofthem are _[l, 2, 5-8, I

l,

12, 15, l6].

III.

NUMERICALMETHOD

As we mentioned, we use a finite t'olume nurnerical.

method to solve the shallow waler equations. [n a semi-discrete form, the finite volume method is

fio,rl=-*(l.rr,)-{,-.rr))+si

$)

where

Q

is an approximation of the conserved qualitify,

F

is an approximation of the analytical flux and S is a

discretiz-ation

of

the analytical source term. See the References _{[1, 6,} 15] for more details of this lype of

scheme. This scheme is called semi-discrete because we have discretize the shallow water equations u'ith respect

to space, but the tirne variable is still continuous [3, 9,

The 4th Annual Basic Science lnternational Conference February 12-13'd2014

1 01.

To get a second order method in spacc, rve use a linear reconstruction tbr quanilties stage, height, bed, velocity

and momentum. Then

in

order

to

suppress arlilicial

trscillation due to the space reconstruction, we implement the minmod limiter. This limiter gives a limitation to the values of the gradients in the linear reconstruction of the aforernentioned quantities. After that, uumerical fluxes

are computed based on these reconstructions. We use the Lax-Friedrichs numerical flux function. We refer to _[9,

I 0] for the formulation of this flux flurction.

The next step is to integrate the serni-discrete fonn (4)

with respect to time. We acrually can use any standard

method

of

Oldinary Differential Equations (ODEs) solver. However, because we have used a second order method in space, it is better to use either a tirst or second

order rnethod in time. This is because we will never get a

firite volume method of order higher than two, eveu if we

use higher order method

in

time.

In

this paper we

implement the second order Runge-Kutta method to integrate the semi-discrete fonn (4) with respect to time.

IV.

NT,MF,RICALRESULTS

This section provides numerical results regarding two different strategies for the numerical evolution. The fust

sh?tegy is imposing a tlxed time step in the second order Runge-Kutta integration. The second sh'at€gy is imposing

a fixed CFL number where in our simulations we use

CFL number to be 1.0 in one case and 0.01 in another

case. Details about CFL conditions and CFL nunrbers can

be found in the References _{[9, 10,} 17].

Numerical settings are as follow. We use SI units for

measured quantities, so wr; omit the writing of rurits.

Errors are quantified using absolute

Ll

formula as used in [13, l4]. In this paper we consider one test casc. Stamdard test cases are avaiiable in the References _[4, l8].

As a test case we consider the darn break problern. W"e

assume that the topography is given by a flat horizontal

bottom B(-t)=6, _{where -1}

<x<1.

There{bre we have

that stage equals to v'ater height. The water height is

initially given by

fto,

x < o, i

4,

x>0.

The anaiyical solution of this problem has been lbund

by Stoker

[8]

and an extension to the debris avalanche problem has been solved by Mungkasi and Roberts

[3,

l4l,

TABLE I

CoM?,{RIsoN oF ST,\GE ERRoRS BETV/EEIi IMPoSINC A F[{ED T1ME STEP

AND {MP0SING FTXED CFL NUMBERS. TTE FTXED TiME STEP Is O.O5

TIilIES THE CELL.WIDTH, WIIERE.A.S FD(ED CFL NTIMBER ,{RE I.O AND

0_01.

Ccll

number

Fixed time step

Eror RC

CFL=i.0

Error RC

CFL:0,01

Eror RC

100 20t) 400 80t)

1 600

u.0589

0.030s 0.9343

0.0144 1.0940

0.tx|72 l _0014

0.0036 0.9925 0.0582

0.0304 0.9359

0.0143 r.0917

0.0071 L0014

0.0036 0.9900

0.0296 0.9405

0.0140 1.0823

0.0070 0.9994

0.0035 0.9860

verage rate 1.0055 ,OM .0020

Average rate of convergorce

TABLE II

Coiv{pARrs(lN oF DTScHARGE ERRoRS BETWEEN INtPoslNG A FIxED TIN{E

STEP .{ND It{PoSIi{G FIXED CFLNUTIBERS. THE FLXED TIME STEP IS O-05 TIMES TI{E CELL-WIDTH, WHtsREAS FDGD CFL NLA,{BER A&E I.() AND

TABLE III

CoMPARISoN oF VELOCITY ERIOR.S BETWEEN IMPOSINC A FIXED TIME

STEP AND II\fPoShIG FTXED CFLMJMBERS. THE FTX.ED TIME STEP IS O.O5

TIMES THE CELI-.WIDTH, WHEREAS FTXED CFL NLMBER ARE 1 .O ANI)

0.0r.

Cell number

Fixed time step

Eror RC

CFL=1.0

Enor RC

CFL:O,OI

Eror RC

r00

2M

400 800

I 600

0.0732

0.0388 0.9157

0.0178 t.1276

0.0088 r.0090

0.0044 r.0032 0.0721

0.0384 0.9156

0.0i76 1.1268

0.0087 r.0088

0.004.1 1.0035 0.0705

0.0373 0.9r66

0.0172 1.t209

0.0085 1.0067

0.0043 0.9966

rate L0139 1.0137 I .0102 Average fate

of conveigence

Our simulation results are summarized in Tables 1-3.

Table

I

shows error comparison for stage (water surface)

befween tfuee scenarios

of

simulations. Errors for

dischalge (momenhrm) and velocity are summarized in Table I and Table 2 respectively. From these tlu'ee tables, the highest convergence rate is achieved by setti.ng a fixed dme step, rather than imposing a fixed CFL number. We

should note that the average rate of convergence for

imposing CFL number 1.0 produces a very close average

rate

of

convergence

to

the fixed time step setting. Furthermore inrposing CFL number to be 1.0 gives the

most efficient computation as it takes the shortest nu:ning time. Setting CFL to be too small such as 0.01 gives a low

rate of convergence. Of course setting CFL number too small makes the computation be expensive, so the rurming

coilputation is long. Here the fixed time step scertario

used is & :0.05 Ax, with the time step dJ atrd the cell

width Ar.

Stags Dfdafttreakraler al tire FA 0m

I

>1

:-l

E"-'

:J

}..-J

-l

F,-

-l

}--r

_t

}---r

-l

l--r

:-I

lL

-_I

l--r

-l

Fr

-l

li

-l

f----_r

-T

t-=J

-l

l-_=-r

_l

l-_ .

-l

L=J

_l

r--__J

-J

l---J

'-l

\t

_J

L.--J

"J

l-r

-I

a--J

_t

-__l

=-

t

--_

=-q s

{8 {.6 {.4 {2 B 0.2 04 0.6 ti.B

Mol@ium oi dambDak Mtef al lime r-0.m0

{8 {]6 .84 {? 0 0.2 04 86 0.8 V€locdy ofdsmbEak kler.l tims t=0.0m

20 10

B

-l

1 -0.8 -0.6 -0.4 {.2 0 0.2 0.1 0.6 0.8 I

Fig. l. The initirl cooditiotr of the dam brcalr problem (at tirnc

/ = 0 using 100 cells. Solid line represcnls the exaet solution. Dotted line represenls the numerical solutjnn.

Fixed time step

Error RC

CFL:I.O

Error RC

0.4652

0.2416 0.9453

0.1 164 t.0533

0.0581 0.9964

0.0296 0s778

0.4714

0.2448 0.9455

o.tt7'1 l.0562

0.0589 0.9984

0.0299 0.9791

0.4520 0.234r

0.1 138 0.057r 0.0291

L32 | Botu, East lava, lndonesio

[image:14.612.303.518.53.357.2]

February 12-13"12014 The 4th Annual Basic Science lnternational Gonference

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$

20

1B

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Slage oldambreal vdler rl lims l-0.050

40 {5 {r {j2 0 n.2 B.r 05 0.8 Llomentum of drmb,eek Hlpr d llme t4 053

R-EFERENCES

J. Balbas md S. Karai. A centtal schemc lor sirallow water florvs

ulong churncls with irregular geometry, ESAIM: Mathenuoicul llodallhry nnd Numeical Ana$'sis,43 (2009), 333-351.

F. Biiilrco, G. Puppo and G. Russo, I{igh order centlal schemes for

hyperbolic systt'ms of consenation laws, SZ{M,Iournal on

Scientilic Computirg 2'l f 1999), 294-322.

F. Bouchut Nonlinear stability oJ _fnite volume melhods fitr

h.tperbolir: consen'ation laws and w,ell-balanced schemes Jbr salrres, Birkhauser, Bassel, 20M.

N. Goutal ard F. Maulel, Proceedings of the 2nd Workthop on

DanrB reak Wuw Si nniation, No. I{E-43l97l01 6.ts, Departement

Iaboratoire National d' Fr ytrau li que, Groupe Hydrauli que Fh,l vi ale. EDF, Chatou, 1997.

[5] A. I{arten, Hig}r resolution schemes for hypertrolic conservation

laws, J ounta I ol C omp u tu t i or u I P hysic:s, I 35 ( I 997), 26G-2?i,.

16] A. Kurgmov and D. Lery, Cmtral-upwird schemes for the

Saint-VEtrmt system, ESAIM: Mathemoticol Moclelling und Ntmerit:al

,4nalysis, 36 (2002)- j97425.

[7] A. Kurganov, S. Noelle and G. Pelror.a, Semidiscrete

central-upu,iad schenres for hlperbolic consellation laws and

]Iarnilion-Jacubi cquations, SLITLI Jowtal on ScientiJic Compuling, 23 (?001''t.'707-740.

t8] A. Kurganov and E. Tadmor, New highresolution cenral schemes

for nonlinear conservation la*s ald convcction-dit'fusion

equations, Jounnl of Cotrputational Physrcs, 160 (2000),

241-282.

t9] R. J. kVeque, Numeriral metho(ts _Jor anservation lou's, 2"t

Edition, Birkhauser, Basel. I 992.

I0l R. J. [-eVeque, Fittitc-volune nethods.fbr hypcrbolic pt'oblcms.

Cambridge University Press, Cambridge, 20M.

[l1] D. Levy, G. Puppo and G. Russo, Central \\.ENO schemes for

hyperbolic systems of conservation laws, ESzll,\r': Mathenatical lt{odelling ond Numpriul Anal.ysis,33 (1999), 541-571.

[2] X. D. Liu and E. Tadmor; Third order nonoscillatory central

scherne lbr hyperbolic conssrvation laws, i'\'unterische Muhemati k,

79 (t998),39742s.

[3] S. Mungkasi, A snrdy of well-balanced finite volume methods and

relinernent indicators for rhe shallow water equations, Thesis ol

Dodor of Philosophy, The Australiu Natiolal University,

Cmberra" 2012;

Bulletin of the Australian Mathematical Society. 88 (20i3),

351-352, Cambridge Univeniry Press.

[14] S. Mulghasi and S. G. Robera, Analytical solutiors involving

shock waves for testing debris avalanche numerical models, Praz

and Applied Geophysics, 169 (2012), 187-1858.

[5] R. Naidoo and S. Baboolal, Application of the Kurganov-Lery

senri-discrete numerical scheme to hyperbolic problems with

nonlinear source terms, Future Geteration Contputer Systems,2O

(zoo4),465_4'73.

U6] H. Nessyahu and E. Tadmor, Non-oscillatory central diti'erorcing

for hypertrolic conserrration laws, Journal of (hmpatalional

Phys ics, 87 ( I 990), 408-463.

[7] S. Osher atd E. Tadmor, On the convmgence-.of dillbrence approximation to scalar consen'ation laws, Mathzmatics oJ Computatiun, 50 (1988), 19-5.1.

U8] J. J. SLokct, Wder Wm,es: The Mathematical Theory with Application, Intascience Publishers, New York, 1957,

I {r8 {5 ,6.r {2 0 0.2 0r 06 08

Velocity oldahbreak wa1e, d tifte td.(Eo

1 0.8 0.5 .0.1 €.2 0 0.7 0.{ 0.€ 0.8 1

Pig. 2. Solution to tfie dam brcak problem at time ,:0.05 using

[image:15.612.139.317.55.189.2]

100 cslls with the fixcd time step. Solid line represents the exact solution. Dotted line represents the nurnq'ical solution.

Figure

I

shows the initial condition for the test case.

The first subfigure is the stage or water level (free

surface). The second and third subfigures are the

momentum and velocity respectively.

It

is

clear that

initially we have only discontinuity in the stage, while the

momeffum and velocity are continuous.

Figure 2 shows the stage, discharge and velocity of

water after' 0.05 seconds of dam break using the fixed

tirne step. The numerical solutious approximate the

analytical solution well based on this Figure 2. Here we

see discontinuities in the slage, momentum and velocify.

The convergence rate in our simulation is about 1.0

everl though we have implemented a second order {inite volume method, that is, second order in spaoe and secotrd

order in time. This is because the discontinuities of the

solution occur. The discontinuity appears in the rneasured

quantities as well as the derivative of the quantities. Again, see Figure 2 for these discontinuities.

It is worth to note that the formal convergence rate of

our numerical method is two, because we use a second

order method in space as well as in time. However this formal order is true only when the solution of the shallow water equatiors is smooth _[9, 10]. As our solution in this

paper

is

nonsmooth due

to

discontinuiries,

it

is

not

suprising that we obtain that the rate of convergence is less than two, that is, about one.

Even though we have

a

fixed

folnal

order, the

numerical order or rate

of

convergence

is

obviously

dependent on the numerical strategy that we use. This has

been shown in this paper. Taking a fixed tirne step in the

finite volume method gives different convergence rate

from taking a varying time step with imposing a CFL

number. In addition, imposing a specific CFL number gives dilferent convergence rate liom imposing another CFL number.

V.

CoNcrustoN

We have investigatetl

the

CFL effects

on

the convergence rate of finite volume methods used to solve the shallow watfl'equations. Our simulations indicate that

the use of CFL number L0 for solving the dam break

problem gives the best combination betwesn effrciency aud accuracy. Note that setting the CFL number greater than 1.0 may make the numerical method unstable when we solve the shallow v/ater equations in general.