AM213B Numerical Methods for the Solution of …hongwang/AMS213B/Notes/...AM213B Numerical Methods...
Transcript of AM213B Numerical Methods for the Solution of …hongwang/AMS213B/Notes/...AM213B Numerical Methods...
- 1 -
AM213BNumericalMethodsfortheSolutionofDifferentialEquations
Lecture05CopyrightbyHongyunWang,UCSC
Listoftopicsinthislecture
• ConsistencyofLMM,characteristicpolynomials,consistencycondition
• Stabilityandzero-stabilityofLMM,rootcondition
• Stabilitytheorem:relationbetweenstabilityandzero-stability
• Dahlquistequivalencetheorem:consistency+stability=convergence
• Stiffproblems,verydifferenttimescalesinaproblem
ConsistencyofLMM(linearmulti-stepmethods)LocaltruncationerrorofanLMMisdefinedthesamewayasbefore:
Localtruncationerror
=residualtermwhensubstitutinganexactsolutionintonumericalmethod.
en(h)= α j u(tn + jh)
j=0
r
∑ −h β j f u(tn + jh),tn + jh( )j=0
r
∑
ConsistencyofLMMisdefinedthesamewayasbefore:
AnLMMisconsistentifen(h)=O(hp+1)withp>0.
Consistencycondition
Weinvestigatetheconditiononcoefficients{αj,βj}forconsistency.
WeintroducecharacteristicpolynomialsofanLMM(therearetwoofthem).RecallthegeneralformofLMM
α j un+ j
j=0
r
∑ = h β j f (un+ j ,tn+ j )j=0
r
∑ , αr =1
Definition:ThetwocharacteristicpolynomialsofanLMMaredefinedas
AM213BNumericalMethodsfortheSolutionofDifferentialEquations
- 2 -
ρ(ξ)= α j ξj
j=0
r
∑ ← coefficientsontheleftsideofLMM
σ(ξ)= β j ξj
j=0
r
∑ ← coefficientsontherightsideofLMM
Theorem(consistencycondition):AnLMMisconsistentifandonlyif
ρ(1)=0ρ′(1)= σ(1)
⎧⎨⎪
⎩⎪
Proof:
RecallthatconsistencymeansO(1)andO(h)termsdisappearinthelocaltruncationerror.Intheexpressionoflocaltruncationerror,wedoTaylorexpansionaroundtnandcalculatecoefficientsoftermsuptoO(h).
en(h)= α j u(tn + jh)
j=0
r
∑ −h β j f u(tn + jh),tn + jh( )j=0
r
∑
= α j u(tn)+u′(tn) jh( )
j=0
r
∑ −h β j f u(tn),tn( )j=1
r
∑ +O h2( )
=u(tn) α j
j=0
r
∑ +h u′(tn) α j jj=0
r
∑ − f u(tn),tn( ) β jj=1
r
∑⎛
⎝⎜⎞
⎠⎟+O h2( )
Usef u(tn),tn( ) =u′(tn) , α j
j=0
r
∑ = ρ(1) , α j jj=0
r
∑ = ′ρ (1) , β jj=1
r
∑ = σ(1)
=u(tn)ρ(1)+h ′u (tn) ρ′(1)−σ(1)( )+O(h2) Thus,en(h)=O(h
2) ifandonlyifthecharacteristicpolynomialssatisfy
ρ(1)=0ρ′(1)= σ(1)
⎧⎨⎪
⎩⎪
StabilityofLMM
Recallthestabilityweintroducedearlierforsingle-stepmethodsun+1 = Lnum(un) :
Lnum(un)−Lnum(vn) ≤(1+C ⋅h)un − vn forsmallh (E01)
AM213BNumericalMethodsfortheSolutionofDifferentialEquations
- 3 -
whereconstantCisindependentofh,unandvn.
Applyingthenumericalmethodovernstepsleadsto
Lnum( )n(u0)− Lnum( )n(v0) ≤CT u0 − v0 forsmallhandnh≤T (E01B)
whereCTisindependentofh,n,u0andv0(aslongasnh≤T).
(E01B)ismoregeneralinthesensethat(E01)implies(E01B).
Foranr-stepLMM,wewriteitinthevector-operatorform:
!un+1 = LLMM(
!un)
where
!un ≡
unun+1"
un+r−1
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
, !u0 ≡
u0u1"ur−1
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
Applyingthenumericalmethodovernstepsleadsto
!un = LLMM( )n(!u0)
While(E01)isconvenienttouseinanalysis,itistoonarrowandisinadequateforLMM.Wedemonstratethisinthesimpleexamplebelow.
Example:WecasttheEulermethodintotheformofanr–stepLMM
un+r =un+r−1 + β j f (un+ j ,tn+ j )
j=0
r
∑ , β j =1 , j = r −10 , otherwise
⎧⎨⎪
⎩⎪
WeapplyittoODEu′=0.Itbecomesun+r=un+r–1 forn≥0.
Invector-operatorform,thissimplenumericalmethod(forODEu′=0)is
!un+1 = LLMM(!un)=
!un(2)!un(3)"!un(r)!un(r)
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟
whereu ⃗n(i)denotesthei-thcomponentofvectoru ⃗n.
Toexaminethevalidityof(E01)and(E01B),weapplyLLMMtotwospecialvectors:
AM213BNumericalMethodsfortheSolutionofDifferentialEquations
- 4 -
!u0 = ( 0 " 0 1 )T , !v0 = ( 0 " 0 )T .
Wehave
LLMM(!u0)= ( 0 " 0 1 1 )T
LLMM( )n(!u0)= ( 1 " 1 1 )T forn≥(r −1)
LLMM( )n(!v0)=0
Itfollowsthat
•
LLMM( )n(!u0)− LLMM( )n(!v0)( 1 " 1 1 )T−0
# $%%%%% &%%%%%≤ CT
!u0 −!v0
( 0 " 0 1 )T−0#$% &%
istrue.
•
LLMM(!u0)−LLMM(
!v0)( 0 " 0 1 1 )T−0
# $%%% &%%%≤ 1+C ⋅h( ) !u0 −
!v0( 0 " 0 1 )T−0#$% &%
isNOTtrue.
Itisclearthatthissimplemethod(forODEu′=0)shouldbeclassifiedas“stable”.
Therefore,weselect(E01B)todefinethestabilityforLMM.
DefinitionofstabilityforLMM
AnLMM !un+1 = LLMM(
!un) isstableif
LLMM( )n(!u0)− LLMM( )n(!v0) ≤ CT
!u0 −!v0 forsmallhandnh≤T
whereCTisindependentofh,n,u0andv0(aslongasnh≤T).
Remark:
Thisstabilityisdifficulttocheckdirectly.Weneedtoworkwithalternativeconditionsthataremoreconvenienttocheck.
Zero-stabilityofLMMConsideranLMMappliedtothemodelODE: u’=0
Exactsolution:u(t)=const
Numericalsolution:
α j un+ jj=0
r
∑ = 0
AM213BNumericalMethodsfortheSolutionofDifferentialEquations
- 5 -
Motivation:
Wewantnumericalsolutionuntoremainboundedasn→∞.
Definitionofzero-stability:
Ifallsolutionsof α j un+ jj=0
r
∑ = 0 remainboundedasn→∞,
thenwesaythecorrespondingLMMiszero-stable.
Remark:
Thezero-stabilityisnotaffectedbycoefficients{βj}.
Nextweconnectthezero-stabilityofanLMMtoitscharacteristicpolynomialρ(ξ).
Inequation α j un+ jj=0
r
∑ = 0 ,weconsidersolutionsoftheform{uk=ξk,k=0,1,2,…}.
Substitutinguk=ξkintotheequation,weget
α j ξ
n+ j
j=0
r
∑ =0 ⎯→⎯ α j ξj
j=0
r
∑ =0
==> ρ ξ( ) = 0
Forpolynomialρ(ξ),let
{ξj}denotesimplerootsand
{qi}denoterootswithmultiplicity>1.
Thegeneralnumericalsolutionhastheform:
un = c1ξ1n + c2ξ2
n +!( )+ b1(0) +b1
(1)n+!( )q1n + b2(0) +b2
(1)n+!( )q2n +!
unremainsboundedasn→∞ifandonlyif
|ξj|≤1and|qi|<1
Thisiscalledtherootcondition.
Definition(rootcondition)
Forapolynomial,if
• allrootssatisfy|ξj|≤1and
• allrootswithmultiplicityabove1 satisfy|qi|<1,
thenwesaythepolynomialsatisfiestherootcondition.
AM213BNumericalMethodsfortheSolutionofDifferentialEquations
- 6 -
Wesummarizetheresultweobtainaboveintoatheorem.
Theorem(conditionforzero-stability):
AnLMMiszero-stableifandonlyifitscharacteristicpolynomialρ(ξ)satisfiestherootcondition.
Nextwelookattheconnectionbetweenthezero-stabilityandthestability.Theorem(thestabilitytheorem):
Supposeƒ(u,t)inu′=ƒ(u,t)isLipschitzcontinuous.IfanLMMiszero-stable,thenitisstable.
Thatis,foranyT>0,thereexistsCTsuchthat
LLMM( )n(!u0)− LLMM( )n(!v0) ≤ CT
!u0 −!v0 forsmallhandnh≤T
whereCTisindependentofh,n,u0andv0(aslongasnh≤T).
Proof:Wewillnotgothroughtheproofindetails.AnoutlineofkeystepsintheproofispresentedinAppendixB.
Thistheoremconnectsthezero-stabilitytothestability.Withthistheoremasthestepping-stone,wenowintroducetheDahlquistequivalencetheorem(whichissimilartotheequivalencetheoremweprovedforsingle-stepmethods:
consistency+stability=convergence
Theorem(Dahlquistequivalencetheorem)
AnLMMmethodisconvergentifandonlyifitiszero-stableandisconsistent.Proof:Wewillnotgothroughtheproofindetails.ThekeystepsforprovingtheDahlquist
equivalencetheoremaresimilartothestepsusedforprovingthestabilitytheorem(AppendixB).TheadaptationofthesestepsforprovingtheDahlquistequivalencetheoremisdiscussedinAppendixC.
Example:Themidpointmethod:
un+2 −un =2h f (un+1 ,tn+1) Itisa2-stepLMM.Thetwocharacteristicpolynomialsare
ρ(ξ)= ξ2 −1 , σ(ξ)=2ξ
Checkingtheconsistencycondition:
AM213BNumericalMethodsfortheSolutionofDifferentialEquations
- 7 -
ρ(1)=0, ρ′(1)=2=σ(1)
==> Itisconsistent.Checkingtherootcondition:
ρ(ξ)hastwosimpleroots:
ξ1=1, ξ2=–1
==> Itsatisfiestherootcondition.==> Itiszero-stable.
BytheDahlquistequivalencetheorem,themidpointmethodisconvergent.
Remark: Innumericalsimulations,wefindthatthemidpointmethodhasanexponentiallygrowingerrormodeevenforthesimpleODE:u’=–u.Theexponentiallygrowingerrormodequicklyruinsthenumericalsolutionastimeincreases.ThisbehaviordoesnotcontradicttheDahlquistequivalencetheorem.Atafinitetime,ifweuseaverysmalltimestep,themidpointmethodwillbewellbehaved.
h→0whileT isfixed
⎛⎝⎜
⎞⎠⎟
vs T→∞whilehisfixedalthoughsmall
⎛⎝⎜
⎞⎠⎟
Example:
AllAdamsmethods(bothAdams-BashforthandAdams-Moulton)arezero-stable.Thegeneralformofr-stepAdamsmethods:
un+r =un+r−1 +h β j f (un+ j ,tn+ j )
j=0
r
∑
Characteristicpolynomialρ(ξ):
ρ(ξ)= ξr −ξ(r−1) = ξ(r−1)(ξ−1)
ρ(ξ)hasonesamplerootandonerootofmultiplicity(r–1).
ξ1=1, asimpleroot,
q1=0, arootofmultiplicity(r–1).
==> Itsatisfiestherootcondition.
==> Itiszero-stable.
Question: Iszero-stabilityenoughformakingamethodwellbehaved?
ConsidernumericalsolutionsofODE
u′ = −λsinh u− cos(t)( ) , λ = large>0 (E02)
AM213BNumericalMethodsfortheSolutionofDifferentialEquations
- 8 -
InAssignment#1,youobservedtheresultsbelow.
ThenumericalsolutionofEulermethodisnotevenboundedunlessthetimestepistiny.TheimplicittrapezoidalmethodperformsmuchbetterthanEulermethod.TheimplicitbackwardEulerisevenbetter.
Thebottomline:
Zero-stabilityimpliesthatwhenthetimestepissmallenough,thenumericalsolutioniswellbehaved.Butforacertaincategoryofproblems,wecannotaffordverytinytimesteps.Weneedothertypesofstabilitytoensurethatthenumericalsolutioniswellbehavedevenwhenthetimestepisnotverysmall.
Stiffproblems
Weconsideralinearversionof(E02)asamodelproblem.
u′ = −λ u− cos(t)( ) , λ = large>0u(0)=0
⎧⎨⎪
⎩⎪ (E03A)
Exactsolution:
u(t)= λ2
1+λ2 cos(t)+λ
1+λ2 sin(t)−λ2
1+λ2 e−λt
(SeeAppendixAforderivation)
Therearetwoverydifferenttimescalesinthisproblem:
• Slowevolutionofcos(t),and
• Veryfastdecayofexp(−λt)
Definition(stiffproblem):Aproblemiscalledstiffifithas(atleast)twoverydifferenttimescales.
InODE(E03A),thetwotimescalesaretangledtogether.Tosimplifythediscussion,weconsideranevensimplerprobleminwhichthetwotimescalesareseparated.
u′ = −λu , λ = large>0′v = −v
⎧⎨⎪
⎩⎪ (E03B)
Wefocusontheu-componentof(E03B)anduseitasamodelproblemforexaminingtheperformanceofvariousnumericalmethods.
Asimplifiedmodel:
AM213BNumericalMethodsfortheSolutionofDifferentialEquations
- 9 -
u′ = −λu , λ = large>0u(0)=1
⎧⎨⎪
⎩⎪
Exactsolution:
u(t)= exp(−λt) Twopropertiesoftheexactsolution:
1. u(t)decreasestozeroast→∞.
Forlargeλ,u(t)decreasesveryfast.
2. Forafixedvalueoft>0(nomatterhowsmallitis),u(t)→0asλ→∞.Thesearethepropertieswewanttopreserveinnumericalsolutions.
Behaviorsofnumericalsolutionsofu′=–λu
Eulermethod:
un+1 =un +h(−λun)
==> un+1 =un(1−λh)
==> un =u0(1−λh)n
Asn→∞,undecreasesinabsolutevalueifandonlyif 1− λh <1 ,
ifandonlyif0 < λh < 2 ,
ifandonlyifh < 2λ.
Forlargeλ,thisconditionisveryrestrictive.
λ=108 ==> h<2×10-8
Foruntodecreasewithoutoscillatinginsign,weneed h ≤1λ.
ConclusionforEulermethod:
• Forlargeλ,Eulermethodhastouseatinytimestepjusttoensurethatthenumericalsolutiondecreasesinabsolutevalueasn→∞.Inotherwords,thenumericalsolutioniswellbehavedONLYwhenthefastevolutioncomponentisresolved,whichrequiresatinytimestep.
• Inastiffproblem,therequirementofatinytimestepiscoupledwiththeneedofsimulatingtheslowevolutioncomponentoveralongtimeperiod.
==> Extremelylargenumberoftimestepsisneeded.
AM213BNumericalMethodsfortheSolutionofDifferentialEquations
- 10 -
Forexample,toaccommodateλ=108andtosimulatetoT=10,weneed
h<2/λ=2×10–8
N=T/h>5×108=500milliontimesteps
BackwardEulermethod:
un+1 =un +h(−λun+1) , λ = large>0
==> un+1(1+λh)=un
==> un+1 = un1
1+ λh( )
==> un = u01
1+ λh( )n
Property1:
Asn→∞,undecreasesto0withoutoscillatinginsign,foranyvalueofh>0.
Property2:
Whenhisfixed,wehaveu1 = u01
1+ λh( )→ 0 asλ→∞.
ConclusionforBackwardEulermethod:
• Evenforverylargeλ,thenumericalsolutionofbackwardEulermethoddecreasesto0withoutoscillatinginsign,asn→∞,foranytimesteph>0.ThebackwardEulermethodpreservesbothproperties1and2.
• Inotherwords,thenumericalsolutionisalwayswellbehavedevenwhenthefastevolutioncomponentisnotresolved.Wecanselectthetimestepbasedontheneedofresolvingtheslowevolutioncomponent.
Trapezoidalmethod:
un+1 = un +h2
−λun − λun+1( )
==> un+1 1+λh2
⎛⎝⎜
⎞⎠⎟ = un 1−
λh2
⎛⎝⎜
⎞⎠⎟
==> un+1 = un1− λh
21+ λh
2
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
AM213BNumericalMethodsfortheSolutionofDifferentialEquations
- 11 -
==> un = u01− λh
21+ λh
2
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
n
Themultiplicationfactorsatisfies
1− λh2
1+ λh2
<1 for all values of h > 0
Property1:
Asn→∞,undecreasesinabsolutevalueto0,foranyvalueofh>0.
However,forλh>2,themultiplicationfactorisnegative.
==> Forλh>2,unoscillatesinsignwhiledecreasingto0asn→∞.
Property2:
Whenhisfixed,wehave
u1 =1− λh
21+ λh
2
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟u0→ (−u0) asλ→∞.
Conclusionfortrapezoidalmethod:
• Thenumericalsolutionoftrapezoidalmethoddecreasesto0asn→∞,foranytimesteph>0.Sothenumericalsolutionwillalwaysremainboundedforanytimestep.Wedon’tneedtorestricttheselectionoftimesteptomakethenumericalsolutionbounded.
• However,forlargeλ,asn→∞,thenumericalsolutionoftrapezoidalmethodoscillatesinsignwithaveryslowdecayinamplitude.Thetrapezoidalmethodpreservesproperty1,butnotproperty2.
BasedontheperformancesofEuler,backwardEulerandtrapezoidalmethodsanalyzedabove,weintroducetheA-stabilityandtheL-stabilitytomeasuretheperformanceofanumericalmethodforsolvingstiffproblems.
Intuitively,wedefineA-stabilityandL-stabilityasfollows:A-stabilitydescribeswhetherornotproperty1ispreserved.
L-stabilitydescribedwhetherornotbothproperties1&2arepreserved.
AM213BNumericalMethodsfortheSolutionofDifferentialEquations
- 12 -
AppendixA: ExactsolutionoftheIVP
u′ = −λ u− cos(t)( )u(0)=0
⎧⎨⎪
⎩⎪
WewritetheODEas
u′+λu= λcos(t) Weusetheintegratingfactormethod.Multiplyingbyeλt,leadsto
eλtu′+λeλtu= λeλt cos(t)
==> eλtu( )′ = λeλt cos(t)
Integratingfrom0tot,weget
eλtu(t)= λ eλ s cos(s)ds
0
t
∫
Weusetheintegrationformula
eλ s cos(s)ds
0
t
∫ =Real e(λ+i )s ds0
t
∫⎡
⎣⎢⎢
⎤
⎦⎥⎥=Real 1
λ+ ie(λ+i )t −1( )⎡
⎣⎢
⎤
⎦⎥
=Real λ− i
λ2 +1 eλt cos(t)−1+ ieλt sin(t)( )⎡
⎣⎢
⎤
⎦⎥ =
λλ2 +1 eλt cos(t)−1( )+ 1
λ2 +1eλt sin(t)
ThesolutionoftheIVPis
u(t)= e−λtλ eλ s cos(s)ds
0
t
∫
= λ2
1+λ2 cos(t)+λ
1+λ2 sin(t)−λ2
1+λ2 e−λt
AppendixB: Proofofthestabilitytheorem
Theorem(thestabilitytheorem):
Supposef(u,t)inu’=ƒ(u,t)isLipschitzcontinuous.
IfanLMMiszero-stable,thenforanyT>0,thereexistsCTsuchthat
AM213BNumericalMethodsfortheSolutionofDifferentialEquations
- 13 -
LLMM( )n(!u0)− LLMM( )n(!v0) ≤ CT
!u0 −!v0 forsmallhandnh≤T
whereCTisindependentofh,n,u0andv0(aslongasnh≤T).
Outlineofkeystepsintheproof:
1. WefirstlookattheLMMappliedtosolvingu’=0.Ithastheform
α jun+ jj=0
r
∑ = 0 , αr = 1
Inthematrix-vectorform,wecanwriteitas
!un+1 = A
!un
wherevectoruandmatrixAare
!un ≡
unun+1"
un+r−1
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟, A ≡
0 1 0" # #0 $ 0 1
−α0 −α1 $ −αr−1
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
MatrixAisaconstantmatrix,independentofh.ThecharacteristicpolynomialofmatrixAisρ(ξ).Polynomialρ(ξ)determinesthezero-stabilityoftheLMM.WhentheLMMiszero-stable,theeigenvaluesofmatrixAsatisfytherootcondition.BywritingmatrixAintheJordancanonicalform,wecanshowthatthereexistsCA≥1suchthat
An ≤CA for all n ≥ 0
2. Next,welookattheLMMappliedtosolvingu’=ƒ(u,t).Ithastheform
α jun+ jj=0
r
∑ = h β j f un+ j , tn+ j( )j=0
r
∑ , αr = 1
Forsimplicity,wefocusonexplicitmethods.Wewriteitinthematrix-vectorform
!un+1 = A
!un + h!φ !un , tn( ) (E05)
wherefunctionφ(u,t)satisfiesLipschitzcontinuitywithrespecttou:
!φ !u, t( )− !φ !v, t( ) ≤CL
!u − !v
NotethatwhilemappingAuislinear,functionφ(u,t)isnon-linear.Let
Δ!un ≡!un −!vn and Δ
!φn ≡!φ !un , tn( )− !φ !vn , tn( )
TheLipschitzcontinuitygivesus
AM213BNumericalMethodsfortheSolutionofDifferentialEquations
- 14 -
Δ!φn ≤CL Δ!un
Takingthedifferenceof(E05)betweenuandvyields
Δ!un+1 = AΔ
!un + hΔ!φn (E06)
Substituting(E06)atn=0into(E06)atn=1,andtheninto(E06)atn=2,…weobtain
Δ!u2 = A
2Δ!u0 +h AΔ!φ0 +Δ
!φ1( )
Δ!un+1 = A
n+1Δ!u0 +h An− jΔ!φ j
j=0
n
∑ (E07)
Takingnormofbothsides,usingAn ≤CA andtheLipschitzcontinuity
Δ!φn ≤CL Δ!un ,
wearriveatarecursiveinequalityfor Δ!un
Δ!un+1 ≤CA Δ!u0 + hCACL Δ!uj
j=0
n
∑ (E08)
3. Nowwesolvetherecursiveinequality(E08).
Weintroduce{gn}recursivelyas
g0 = CA Δ!u0
gn+1 = g0 + hCACL gjj=0
n
∑ (E09)
Itisstraightforwardtoshowthat
Δ!un ≤ gn for all n ≥ 0
Tocalculategn,were-writetherecursiveequation(E09)as
gn+1 = gn +hCACL gn = (1+hCACL)gn Applyingthisrelationsuccessivelyfromindex0toindex(n–1),yields
gn = (1+hCACL)n g0 ≤ exp CACLT( ) Δ!u0 fornh≤T
Using Δ!un ≤ gn andg0 =CA Δ!u0 ,wehave
Δ!un ≤CAexp CACLT( ) Δ!u0 fornh≤T
Therefore,wefinallyarriveat
LLMM( )n !u0 − LLMM( )n !v0 ≤CAexp CACLT( ) !u0 − !v0 fornh≤T
AM213BNumericalMethodsfortheSolutionofDifferentialEquations
- 15 -
whichimpliestheLMMisstable,bydefinition.
AppendixC: KeystepsforprovingDahlquistequivalencetheorem
BelowwediscussbrieflythekeystepsforprovingtheDahlquistequivalencetheorem.ThesestepsareadaptedfromthestepsinAppendixBforprovingthestabilitytheorem.
Again,wewritetheLMMinthematrix-vectorform
!un+1 = A
!un + h!φ !un , tn( )
Weintroducethevectorversionsofexactsolutionandlocaltruncationerror.
!vn ≡!u tn( ) =
u tn( )u tn+1( )"
u tn+r−1( )
⎛
⎝
⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟
, !en h( ) ≡00"
en h( )
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
=O hp+1( )
Numericalsolutionuandexactsolutionvsatisfy,respectively
!un+1 = A!un + h
!φ !un , tn( )
!vn+1 = A!vn + h
!φ !vn , tn( ) + !en h( )
(E25)
Let Δ!un ≡!un −!vn andΔ
!φn ≡!φ !un ,tn( )− !φ !vn ,tn( ) .Takingthedifferencein(E25)givesus
Δ!un+1 = AΔ
!un +hΔ!φn −!en(h) (E26)
Substituting(E26)atn=0into(E26)atn=1,andtheninto(E26)atn=2,…weobtain
Δ!u2 = A
2Δ!u0 +h AΔ!φ0 +Δ
!φ1( )− A!e0(h)+
!e1(h)( )
Δ!un+1 = A
n+1Δ!u0 + h An− jΔ!φ j
j=0
n
∑ − An− j !en h( )j=0
n
∑ (E27)
Takingnormofbothsides,usingAn ≤CA andtheLipschitzcontinuity
Δ!φn ≤CL Δ!un ,
wearriveatarecursiveinequalityfor Δ!un
Δ!un+1 ≤CA Δ!u0 +hCACL Δ!uj
j=0
n
∑ + n+1( )CACehp+1 (E28)
Tosolverecursiveinequality(E28),weintroduce{gn}recursivelyas
g0 = CA Δ!u0
AM213BNumericalMethodsfortheSolutionofDifferentialEquations
- 16 -
gn+1 = g0 +hCACL gj
j=0
n
∑ + n+1( )CACehp+1 (E29)
whichensuresthat
Δ!un ≤ gn for all n ≥ 0
Tocalculategn,were-writerecursiveequation(E29)as
gn+1 = 1+hCACL( )gn +CACehp+1 ==> 1+hCACL( )−(n+1) gn+1 ≤ 1+hCACL( )−n gn + 1+hCACL( )−(n+1)CACehp+1
==>1+hCACL( )−n gn ≤ g0 +CACehp+1 1+hCACL( )−(n+1)
j=0
n−1
∑
≤ g0 +CACeh
p+1 1+hCACL( )−1 1− 1+hCACL( )−n1− 1+hCACL( )−1
==>gn ≤ 1+hCACL( )n g0 +CACehp+1
1+hCACL( )n −1CACLh
≤ exp(nhCACL)g0 +
exp(nhCACL)−1CL
Cehp
Using Δ!un ≤ gn andg0 =CA Δ!u0 ,weobtain
Δ!un ≤CAexp CACLT( ) Δ!u0 +
exp CACLT( )−1CL
Cehp fornh≤T
Whentheinitialvalueu0isatleastp-thorderaccurate: Δ!u0 = !u0 −
!v0 ≤ cinthp ,the
differencebetweenthenumericalsolutionuandexactsolutionvisboundedby
!un −!vn ≤ CAexp CACLT( )cint +
exp CACLT( )−1CL
⎛
⎝⎜⎜
⎞
⎠⎟⎟hp fornh≤T
whichimpliestheLMMisconvergent,bydefinition.