lecture on advanced mathematical technques
TRANSCRIPT
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Numerical Methods for Partial
Differential Equations
CAAM 452
Spring 2005
ecture !
"nstructor# $im %ar&urton
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Summar' of Small $heta Anal'sis
( $he dominant remainder term in this anal'sisrelates to a commonl' used) ph'sicall' moti*ated
description of the shortfall of the method#
( )
( ){ }
( ) ( ){ }
( ) ( ){ }
( )
33
3
3
2
5
2
5
1
2
2
2
2
0
1
42
5
4
3!
!
!
0
0
0
0
m
m
m
m
m
m
mtd
tm Oi ctdxL
m m
tm c Oi ctdxL
m m
itm c Oi ctdxL
m m
i im c ti c
i t
tdx
tdx
m
x
x
L
d
m
du
c u u u e edt
duc u u u e e
dt
duc u u u e e
dt
du
c u u u e edt
+
+
+
+
+
+
= =
= =
= =
= =
r
rr
r r
rr ( ){ }7m
tO
dx
Dissipati*e
+nsta&le
Dispersi*e
Dispersi*e
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Dissipation in Action
( Consider the right difference *ersion#
( %e are going to e,perimentall' determine ho-
much dissipation the solution e,periences.
( ) ( ){ }2
3312
0 mm
tm Oi ctdxL
m m
tdxdu c u u u e e
dt
+
+= =r
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$esting Methodolog'
/ 1educe the timestepping error to a secondar' effect &' choosing a 4
th
order 1unge3utta S$ timeintegrator and a small dt.
2 6i, the method choose one of the difference operators for the spatialderi*ati*e
7 Do a parameter stud' in M) i.e. -e as8 the questions# ho- doesincreasing the num&er of data points change the error.
4 %e need to understand -hat questions -e are as8ing#
/ "s the computer code sta&le as predicted &' the theor'92 Does the computer code con*erge as predicted &' the theor'9
7 %hat order of accurac' in M do -e achie*e9
4 %e h'pothesi:e that if the theor' holds then -e should achie*e#
5 %hat is the actual appro,imate p achie*ed9
( ) ( ), , pm mu T x u T x Cdx %
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"nitial Condition
( ;ecause -e do not -ish to introduce uncertaint'o*er the source of errors in the computation -e use
an initial condition -hich is infinitel' smooth.
( ) ( )2
4cos,0 with [0,2)xu x e x=
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Computing Appro,imate
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After 4 Periods
( $he numerical pulses are in the right place &utha*e se*erel' diminished amplitude.
du
c udt
+= r
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After 4 Periods
M Ma,imum error at t=> ?order
20 0.@/4/@45/!
40 0.@/7!5747@744! 0
>0 0.@>7240>/24>5@ 0.0/
/@0 0.@7/@7@042/>5 0.//
720 0.52!425@0@5040> 0.2@
After 4 periods the solution is totall' flattened in all &ut the last 2 results. "f -e &othered
to 8eep increasing M -e -ould e*entuall' see the error decline as /BM
duc u
dt+= r
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$he +nsta&le eft Stencil
( " repeated this -ith e*er'thing the same) e,cept the choice
of instead of
du
c udt
= r
+
( )( )
231 2
2!
2
0 m
mt
d
tm c O
i ct dxL
x M
m m
duc u u u e e
dt
+
+
= =r
r
%e clearl' see that there is initial gro-th
near the pulses) &ut e*entuall' the
dominant feature is the highl' oscillator'
and e,plosi*e gro-th
large m in the a&o*e red term.
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Cont (snapshot)
( )
( )2
31 2
2!
2
0
m
mt
d
tm c O
i ct dxL
x M
m m
du
c u u u e edt
+
+
= =
rr
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Dispersion in Action
( Consider the central difference *ersion#
( %ith the same time integrator &efore) M=/00)
( %e note that the remainder terms are dispersi*e corrections
i.e. the' indicate that modes of different frequenc' -ill
tra*el -ith different speed.
( 6urthermore) to leading order accurac' the higher order
modes -ill tra*el more and more slo-l' as m increases.
( ) ( ){ }3 52
3!
0 0
mm
itm i tc Oi ctdxL
m mdxdu c u u u e e
dt
+
= =r
0
duc u
dt= r
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( %e notice that the humps start to shed rear oscillations as
the higher frequenc' 6ourier components lag &ehind the
lo-er frequenc' 6ourier components.
2nd
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Con*ergence Stud' t=>
%hat should -e use as an error"ndicator 99
0
duc u
dt= r
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2nd ?order
20 0.4240>0/4!450@
40 0.7502>>50!77 /.42
>0 0.7>2>5/2!!@>5 0./7
/@0 0.///>/2!4@@>!> /.00
720 0.054!/!705244>2 /.>0
( %e do not see a con*incing 2ndorder accurac'
( " computed this &' log2errorMBerrorFM/G
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4th
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4th
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@th ?order
20 0./405!5@/5520
40 0.0457@5252@>7 /.!
>0 0.00/@2/55@07 4.>!
/@0 0.00002>/!24/05 5.>!
720 0.0000004@04/4/5 5.!
( %e see prett' con*incing @thorder accurac'
( " computed this &' log2errorMBerrorFM/G
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Summar' of $esting Procedure
/ +nderstand -hat 'ou -ant to test
2 3eep as man' parameters fi,ed as possi&le
7 "f possi&le) perform an anal'sis &efore hand
4 1un parameter s-eeps to determine if code agrees -ith anal'sis.
5 Estimate error scaling -ith a single parameter if possi&le.
@ "t should &e apparent here that the errors computed in the simulations
onl' appro,imate the tid' po-er la- d,Hp in the limit of small d,large M.
! "t is quite common to refer to the range of M &efore the error scalingapplied as the Ipreasymptotic convergence rangeJ.
> $he preas'mptotic range is due to the other factors in the errorestimate -hich are much larger than the d,Hp term i.e. the pKth
deri*ati*es ma' &e *er' large) or the constant ma' &e large "n this case -e heuristicall' set dt small) using a highorder 1unge
3utta time integrator) and then performed a parameter s-eep on M.%e could ha*e &een unluc8' and the timestepping errors ma' ha*e&een dominant -hich -ould mas8 the d, &eha*ior.
/0 $o &e careful -e -ould tr' this e,periment -ith e*en smaller dt and
chec8 if the scalings still hold.
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Summar' of our Approach to Designing
6inite Difference Methods
( %e ha*e s'stematicall' created finite difference methods &'separating the treatment of space and time deri*ati*es.
( $hen designing a sol*er consists of choosingBtesting# a time integrator so far -e co*ered Euler6or-ard) eap6rog)
Adams;ashford2)7)4) 1unge3utta a discreti:ation for spatial deri*ati*es a discrete differential operator -hich has all eigen*alues in the
left half of the comple, plane assuming the PDE onl' admitsnongro-ing solutions.
Possi&l' using LerschgorinKs theorem to locali:e theeigen*alues of the discrete differentiation operator
dt so that dtlargest eigen*alue &' magnitude of thederi*ati*e operator sits inside the sta&ilit' region sta&ilit'.
small d, spatial truncation anal'sis consistenc' andaccurac'.
6ourier anal'sis for classification of the differential operator.
%riting code and testing.
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Some Classical 6inite Difference Methods
( o-e*er) there are a num&er of classical methods-hich -e ha*e not discussed and do not quite fit
into this frame-or8.
( %e include these for completeness..
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eap 6rog space and time
( %e use centered differencing for &oth space and time.
( %e 8no- that the leap frog time stepping method is onl'sta&le for operators -ith eigen*alues in the range#
( o-e*er) -e also 8no- that the centered differencederi*ati*e matri, is a s8e- s'mmetric matri, -itheigen*alues#
( So -e are left -ith a condition# i.e.
1 1
1 1
2 2
n n n n
m m m mu u u ucdt dx
+ + =
[ ]1,1dt i
2sin
ic m
dx M
1cdt
dx
dxdt
c
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cont
( %e can perform a full truncation anal'sis in spaceand time#
( %e 8no- that dt O= d, so
( ) ( ) ( ) ( )
( ) ( )
1 1 1 1
3 3
, , , ,
2 2
n m n m n m n m nm
u x t u x t u x t u x t T cdt
dx
cdtO dt O dx
dx
+ + =
=
( ) ( )3 3nmT O dt cO dx=
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a,6riedrichs Method
( %e immediatel' conclude that the follo-ing method is notsta&le#
( ;ecause the Euler6or-ard time integrator onl' admits onesta&le point the origin on the imaginar' a,is) &ut thecentral differencing matri, has all eigen*alues on theimaginar' a,is.
( o-e*er) -e can sta&ili:e this formula &' replacing thesecond term in the timestepping formula#
1
0
n nnm mm
u uc u
dt
+ =
( )1 1 10
0.5n n nm m m nm
u u uc u
dt
++ + =
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cont
( $his formula does not quite fit into our constructi*eprocess method of lines approach.
( %e ha*e admitted spatial a*eraging into the
discreti:ation of the time deri*ati*e.
( %e can rearrange this#
( )1 1 10
0.5n n nm m m nm
u u uc u
dt
++ + =
( ) ( )1 1 1 1 1
1 1
1
2 2
1 11 1
2 2
n n n n n
m m m m m
n n
m m
dtu u u c u u
dx
dt dt c u c u
dx dx
++ +
+
= + +
= + +
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cont
( %e can immediatel' determine that this is a sta&le method as
long as cdtBd, O=/
( Li*en) this condition -e o&ser*e that the time updated
solution is al-a's &ounded &et-een the *alues of the left and
right neigh&or at the pre*ious time &ecause this is aninterpolation formula.
1
1 11 11 12 2
n n nm m mdt dt u c u c u
dx dx+
+ = + +
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cont
( A formal sta&ilit' anal'sis -ould in*ol*e#
( %hich gi*es sta&ilit' for each mode if#
( ) ( ){ }
1
1 1
1
1
1 11 1
2 2
1 11 1
2 2
1 11 1
2 2
cos sin
m m
n n n
m m m
n n n
i in n n
m m m
n
m m m
dt dt u c u c u
dx dx
dt dt u c u c u
dx dx
dt dt u c e u c e u
dx dx
dtc i udx
++
+
+
= + +
= + + = + +
= +
R Lr r r
% % %
%
( ) ( )cos sin 1m mdt
c idx
+
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cont
( $hus considering all the possi&le modes( %e note that the middle mode requires#
( $his condition gi*es a sufficient condition for all modes to &e
&ounded.
( ;' the in*erti&ilit' and &oundedness of the 6ourier transform
-e conclude that the original equation is sta&le.
cos sin 12 2
1
dtc i
dx
dtc
dx
+
2
m
m
M
=
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cont
( %e can recast the a,6riedrichs method again
( $his method consists of Euler 6or-ard) central
differencing for the space deri*ati*e and an e,tra
dissipati*e term i.e. a grid dependent ad*ection
diffusion equation#
( )
1
1 1
1 2 2
0
1 11 1
2 2
11
2
n n n
m m m
n n n
m m m
dt dt u c u c u
dx dx
u cdt u dx u
++
+
= + +
= + +
2 2
2
2
u u dx uc
t x x
= +
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C6 Num&er
( $he ratio appears repeatedl') in particular inthe estimates for the ma,imum possi&le time step.
( %e refer to this quantit' as the C6 Courant
6riedrichse-' num&er.
( ;ounds of the form# -hich result from a
sta&ilit' anal'sis are frequentl' referred to as C6conditions.
dt
dx
=
dtC
dx=
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a,%endroff Method
( %e are fairl' free to choose the parameter in the sta&ili:ingterm#
( $he artificial viscosity term acts to shift the eigen*alues ofthe spatial operator into the left halfplane.
( 1ecall the Euler6or-ard sta&ilit' region is the unit circle
centered at / in the comple, plane. So pushing theeigen*alues off the imaginar' a,is allo-s us to choose a dt
small enough to push the eigen*alues of the discrete spatial
operator into the sta&ilit' region
( )1 2 20
11
2
n n n
m m mu cdt u cdt u + = + +
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Class E,ercise
( Please pro*ide a C6 condition for#
( "n terms of#
( %hat is the truncation error9
( ) ( )1 1 1 1 2 24 1
3 6
n n n n n n
m m m m m mu u c u u c u u +
+ + = +
dt
dx=
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on Neumann Anal'sis
( ;' stealth " ha*e introduced the classical on
Neumann anal'sis.
( $he first step is to 6ourier transform the finite
difference equation in space.
( A short cut is to ma8e the follo-ing su&stitutions#
1
1
2
2
2
2
......
m
m
m
m
n n
m m
in n
m m
in n
m m
in n
m m
in n
m m
u u
u e u
u e u
u e u
u e u
+
+
%
%
%
%
%
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Anal'sis cont
( %e can also ma8e the su&stitutions for thedifference operators#
( )
( )
{ } ( )
2
2
1
1
2 0
2
2
22
1 11 2 sin
2
1 11 2 sin
2
1 1sin
2
......
m
m
m
mm
m
m m
m
m
ii m
n n
m mi
i min n
m m
in n
m mi i
in n m
m m
in n
m m
e iedx dx
u ue ieu e u
dx dx
u e ue e i
u e u dx dxu e u
+
+
+
+ =
= =
%
%
%
%
%
( ){ }
2
2
2
14sin
2
2
1 cos
m
m
dx
dx
+ =
=
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Example
( $he a,%endroff scheme
( &ecomes
( )1 2 20
11
2
n n n
m m mu cdt u cdt u + = + +
2 2
1 2
2
1 21
2 2
m m m mi i i i
n n n
m m m
e e e eu cdt u cdt u
dx dx
+ += + +
% % %
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Example cont
( So the a,%endroff scheme &ecomes a neat onestep method for each 6ourier coefficient#
( %e ha*e neatl' decoupled the 6ourier coefficients
so -e are &ac8 to sol*ing a recurrence relation in n
for each m#
( ence -e need to e,amine the roots of the sta&ilit'pol'nomial for the a&o*e
2 2
1 2
2
1 21
2 2
m m m mi i i in n n
m m m
e e e eu cdt u cdt u
dx dx
+ += + + % % %
n
mu%
2 2
2
2
1 21
2 2
m m m mi i i ie e e ez cdt cdt
dx dx
+= + +
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E,ample cont
( $his case is tri*ial as -e Qust need to &ound thesingle root &' /
( Plot it this as a function of theta
( ) ( )( ){ }
2 2
2
2
2
1 21
2 2
1 sin cos 2 1
m m m mi i i i
m
e e e ez cdt cdt
dx dx
c i c
+= + +
= + +
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6inal on Neuman Shortcut
( %e can s8ip lots of steps &' ma8ing the directsu&stitution#
( ere g is referred to as the amplification factor and
imtheta is our pre*ious thetam
n n im
mu g e
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Ne,t $ime
( Definition of consistenc'
( a,1ichtm'er equi*alence theor'
( $reating higher order deri*ati*es
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ome-or8 7
R/ ;uild a finitedifference sol*er for
R/a use 'our Cash3arp 1unge3utta time integrator from %2 for time stepping
R/& use the 4thorder central difference in space periodic domain
R/c perform a sta&ilit' anal'sis for the timestepping &ased on *isual inspection of
the C3 13 sta&ilit' region containing the imaginar' a,is
R/d &ound the spectral radius of the spatial operator
R/e choose a dt -ell in the sta&ilit' region
R/f perform four runs -ith initial condition use M=20)40)>0)/@0 and compute ma,imum error at t=>
R/g estimate the accurac' order of the solution.
R/h e,tra credit# perform adapti*e timestepping to 8eep the local truncation errorf ti t i & d d & t l
u uc
t x
=
( ) ( )
[ ]
24cos
,0 , 0,2
x
u x e x
=