Lecture on Advanced Mathematical Technques

8/12/2019 Lecture on Advanced Mathematical Technques

1/39

Numerical Methods for Partial

Differential Equations

CAAM 452

Spring 2005

ecture !

"nstructor# $im %ar&urton


2/39

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


3/39

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


4/39

$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 %


5/39

"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=


6/39

Computing Appro,imate


7/39

After 4 Periods

( $he numerical pulses are in the right place &utha*e se*erel' diminished amplitude.

du

c udt

+= r


8/39

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


9/39

$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.


10/39

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


11/39

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


12/39

( %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


13/39

Con*ergence Stud' t=>

%hat should -e use as an error"ndicator 99

0

duc u

dt= r


14/39

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


15/39

4th


16/39

4th


17/39

@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


18/39

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.


19/39

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.


20/39

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..


21/39

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


22/39

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=


23/39

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

++ + =


24/39

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

++ +

+

= + +

= + +


25/39

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+

+ = + +


26/39

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

+


27/39

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

=


28/39

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

= +


29/39

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=


30/39

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 + = + +


31/39

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=


32/39

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

+

+

%

%

%

%

%


33/39

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

+ =

=


34/39

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

+ += + +

% % %


35/39

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

+= + +


36/39

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

+= + +

= + +


37/39

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


38/39

Ne,t $ime

( Definition of consistenc'

( a,1ichtm'er equi*alence theor'

( $reating higher order deri*ati*es


39/39

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

=

Lecture on Advanced Mathematical Technques

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