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Finite Diference

MethodNHET RA 23214701

GAMA TITIS ANURAGA 23214061JATI PHARMADITA 23214053

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Introduction

T! "nit! di#!r!nc! $!tod%&DM' i( )n)**ro+i$)t! $!tod ,or (o-.in/ *)rti)-di#!r!nti)- !u)tion( Inc-udin/ -in!)r)nd non-in!)r ti$! ind!*!nd!nt )ndd!*!nd!nt *ro-!$(

R!, %Nu$!ric)- An)-(i( o,E-!ctro$)/n!tic &i!-d('

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Di(cr!ti)tion $od! ,or t8odi$!n(ion)- *ro-!$(

T! 9)(ic Id!) o, &DM i( to r!*-)c!t! d!ri.)ti.!( o, )n un:no8n,unction t! di#!r!nc! uoti!nt(o, un:no8n ,unction(

%1'

%2'

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Diference equation in 2-DCartesiancoordinates

Su)r!/rid

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Di#!r!nc! uoti!n( ,ro$ t)-or

=!r! i( ) ($)-- incr!$!nt o, t!.)ri)-! +

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>!t

=!r! i( ) ,or8)rd (t!* )nd)c:8)rd

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Su$ @ ti$! E%5' )nd @ ti$!(E%6'

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Diference equation in 3-DCartesiancoordinates

%15'

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Discretization equation in PolarCoordinates

r,

x,

y

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Discretization equation inCylindrical coordinates

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Discretization Formula o he!on-"inear Ma#netic Field $%&2'

I, t! r!-ucti.it .B1F o, /i.!n $)t!ri)- i( ) ,unction o, t!$)/n!tic u+ d!n(it Poi((on( E ,or t! $)/n!tic .!ctor*ot!nti)- in ) 2D c)(! i(

A((u$in/ J )nd . con(t)nt in t! )r!) o, !)c $!( t!di(cr!ti)tion ! c)n ! d!ri.!d t! ()$! $!todr!(u-tin/

%1'

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In t! (o-ution o, ) non -ini!r *ro-!$ )n und!rr!-)+)tion,)ctor o, . $) ! introduc!d in (o$! c)(!( to )cc!-!r)t!t! it!r)tion )( in

Discretization Formula o he!on-"inear Ma#netic Field $2&2'

%20'

t

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erence quat ons orime-De)endent

Pro*lems $%&+'

Di(cr!ti)tion o, 1D Ti$!D!*!nd!ntPro-!$

T! !+*r!((ion( o, t! T)-or( (!ri!( ,or t )nd + )r!

%21

'

%22'%23

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(x)licit teration Formulation

ime-De)endent

Pro*lems $2&+'%24'

%25'

( )211

1 21

x

uuu

at

uu njn

j

n

j

n

j

n

j

+=

+

+

n

j

n

j

n

j

n

j CuuCCuu 111 2)21(

+

+++= 2)( xa

tC

=

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m)licit teratie Formulation

ime-De)endent

Pro*lems $3&+'

%26

'%27'

( )2

1

1

11

1

1 21

x

uuu

at

uu njn

j

n

j

n

j

n

j

+=

+

++

+

+

n

j

n

j

n

j

n

j uCuuCCu =++ +

+

++

1

1

11

1 )21( 2)( xat

C

=

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Cran.-!icholson (q/

ime-De)endent

Pro*lems $0&+'Pro.id!( )n )-t!rn)ti.! i$*-icit (c!$! 8ic )( t! )ccur)co, t! (!cond ord!r in ot (*)c! )nd ti$!

%2?'%2'

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Cran.-!icholson (q/

ime-De)endent Pro*lems$+&+'

=!n B1

=!r!

%30'

%31'

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1! 456