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Journal of Computational Physics
www.elsevier.com/locate/jcp
Energy conserving successive multi-stage method for the linear wave equation
Jaemin Shin
a, June-Yub Lee
b,∗
aDepartmentofMathematics,ChungbukNationalUniversity,Cheongju28644,RepublicofKorea bDepartmentofMathematics,EwhaWomansUniversity,Seoul03760,RepublicofKorea
a r t i c l e i n f o a b s t r a c t
Articlehistory:
Received18August2021
Receivedinrevisedform4February2022 Accepted21February2022
Availableonline25February2022
Keywords:
Linearwaveequation Energyconservation High-ordertimeaccuracy
SuccessiveMulti-Stage(SMS)method
We propose a new high-order multi-stage method to solve the linear wave equation in an unconditionally energy stable manner. ThisSuccessive Multi-Stage (SMS) method is extended from the Crank–Nicolsonmethod and unconditional energyconservation is guaranteed. We develop up to the sixth-orderSMS methodusing the order conditions for Runge–Kutta methods and provide mathematical arguments showing that the SMS method is adifferent branch from well-known high order energy preserving methods for Hamiltonian systems.We present aproof of theunique solvabilityand numerically demonstratetheaccuracyandstabilityoftheproposedmethodscomparedwithcompari- sons.
©2022ElsevierInc.Allrightsreserved.
1. Introduction
Linearwaveequationsappearinmanyfieldsofphysicsandhaveplayed asignificant roleinmathematicalmodelingfor transientphysicalphenomena.Forinstance,theyarisewhenconsideringtheMaxwellequationsforelectromagnetism [1,2], theacousticequationforsoundpropagation [3,4],andtheelastodynamicequationforwavepropagationinsolids [5,6].One ofthemostimportantpropertiesofwaveequationsisenergyconservation.
Inthispaper,weintroduceahigh-orderenergyconservingnumericalschemeforthelinearwaveequation,
κ
utt= ∇ · (
M∇
u) ,
(1)where M
=
M(
x)
is asymmetric positivedefinite matrix andκ = κ (
x)
isa strictly positivefunction. Equation (1) might becompletedwithspecificboundaryconditionsinaboundeddomain⊂ R
d(
d=
1,
2,
3)
.Forsimplicity,weconsiderthe periodicboundaryconditionalongtheedgesofcomputationaldomainwhereκ (
x)
andM(
x)
areconstantsrepresentingthe homogeneous backgroundmedium.Homogeneous Neumannboundarycondition n· (
M∇
u) =
0 orhomogeneous Dirichlet boundaryconditionu=
0 canbeeasilyimposedusingevenoroddextensionofthesolution,respectively.Thecorresponding energyof (1) canbewrittenasasumofthekineticandthepotentialenergiesofthesystem,E
(
t) =
1 2
κ
u2t+
M12∇
u2
dx
,
(2)*
Correspondingauthor.E-mailaddress:jyllee@ewha.ac.kr(J.-Y. Lee).
https://doi.org/10.1016/j.jcp.2022.111098
0021-9991/©2022ElsevierInc.Allrightsreserved.
where
M12
∇
u2
= (∇
u,
M∇
u)
.Itisworthnotingthatwhenconsideringtheperiodicorhomogeneousboundaryconditions, theenergy (2) isconservedwithrespecttotime,asdemonstratedbyintegrationbyparts:d dtE
(
t) =
( κ
ututt−
ut∇ · (
M∇
u))
dx=
0.
(3)Notethatnon-zeroDirichletorNeumannboundaryconditionscanbealsoconsideredusinganadditive patchtothesolu- tion,however,theenergydefinedin(2) isnolongerconservedwithoutfurthermodification.
Therehavebeenplentyofdiscussionsfornumericalmethodstosolvethewaveequations.Forthelong-timesimulation, consequently,energy-conservinghigh-ordermethodshavebeenattractingmuchattentiontodeterminethephaseandshape of thewavesasaccurately andstablyas possible.Also, therehas beenintensiveresearch on thenumerical treatmentof quasi-linearwave equationsorevenHamiltonianpartialdifferential equations(PDEs)overthepast decades.Forexample, theaveragevectorfieldmethod [7] hasbeenappliedtotheHamiltonianPDEswithconstantsymplecticstructure.And,the discretevariationalderivativemethod [8] hasbeenproposedtothefamilyofnonlinearwaveequationsbyconstructingthe discreteversion oftheenergyfunction.Therearealsostudiesfocused onfindingconditionsforenergyconservationinRK methods,forexample,asymplecticRKmethodwasdevelopedin [9–11].
Ourultimategoalis topresenta newRK-basedhigh-order energypreservingmethodforquasi-linearwave equations orwiderangeofHamiltoniansystems.Todemonstratethatourmethodisinadifferentbranchfromexistingmethods,we focus onthe linearwave equations forthe simplicityofargument inthispaper. Consideringenergy-conservingmethods forlinearwaveequations,itisnoteworthyreferringtotheapproach,basedontheleap-frogmethod,tobecomparedwith ourmethod.Theleap-frog methodisawell-knownmulti-stepmethodtosolve thewaveequation andhasbeenused for manyapplicationsandsimulations [12].Itisaccuratetothesecond-order,butstability isnotguaranteedwithlargertime stepsizes. Manystudieshaveintroduced energy-conservingmethods,such asthetamethods [13–15],by generalizing the leap-frogmethod.Ontheotherhand,itiswellknownthatthestandardCrank–Nicolsonmethodpreservestheconservation lawsofthe linearwave equation,butitjust hasthe second-orderaccuracy. Sowe extendtheCrank–Nicolsonmethodto constructthehigh-ordermethodwithguaranteeingenergyconservationanddemonstratethattheCrank–Nicolsonmethod basedRKmethodisdifferentfromtheexistingRKones.
Inthispaper,we proposeasuccessivemulti-stage(SMS)methodasahigh-order energyconservingnumericalscheme.
We start by constructing a framework that guarantees energy conservation.The proposed methodcan be considered as an RK method, making it is easy to find a proper coefficient set for high-order accuracy. We note that SMS methods are different from the existing symplectic RK methods. Specifically, our proposed methods do not satisfy the condition bibj
−
biai j−
bjaji=
0 in [11].Thatis,SMSmethodsconstituteanewclassofhigh-ordermulti-stagemethodsthatguarantee energyconservation.We first provide a proof that the Crank–Nicolsonmethod is unconditionally energy conserving inSection 2, andwe extend thisidea to an s-stageSMS methodin Section 3. In Sections4 and5, we numerically demonstratethe order of theaccuracyaswell asenergyconservationinoneandhigherdimensions.Finally,conclusionsare drawninSection 6.In addition,we brieflyintroduce thederivationoforderconditionsforthe linearRunge–KuttamethodintheAppendix.It is worth notingthat weonlyconsidertemporalaccuracyinthispaper,thussemi-discretenumericalschemesare presented.
Fourierspectralmethodsareusedforspatialderivatives,andallsimulationsareexecutedusingtheMATLABprogramviaa fastFouriertransform.
2. ClassicalCrank–Nicolsonmethod
Bydefininganauxiliaryvariablev
=
ut,wecanrepresentthelinearwaveequation (1) incanonicalformas ut=
v,
vt
=
1κ ∇ · (
M∇
u)
(4)
andtheenergy (2) inHamiltonianformas
H (
u,
v) =
12
κ
v2+
M12∇
u2
dx
.
(5)Forthesemi-discreteformulation, wedenoteun andvn asapproximationsofu
·,
tn andv·,
tn,wheretn
=
nΔ
t andΔ
t isatimestepsize.Wefirstconsiderthewell-knownCrank–Nicolsonmethodun+1
−
unΔ
t=
vn+1+
vn2
,
vn+1
−
vnΔ
t=
1κ ∇ ·
M
∇
un+1+
un 2.
(6)
Lemma1.TheCrank–Nicolsonmethod (6) isunconditionallyenergyconserving,meaningthatforanytimestepsize
Δ
t,H
un+1
,
vn+1= H
un,
vn.
(7)Proof. Wefirstcalculatethedifferenceinenergyfunctionals,
H
un+1
,
vn+1− H
un,
vn=
1 2
κ
vn+12− κ
vn2+
M∇
un+12−
M∇
un2dx.
(8)Forthefirsttwotermsof(8),wecanrearrangethemas 1
2
κ
vn+12− κ
vn2dx
=
1 2
κ
vn+1−
vnvn+1
+
vn dx= Δ
t 4vn+1
+
vn∇ ·
M∇
un+1
+
undx
= − Δ
t 4
∇
vn+1
+
vn·
M∇
un+1
+
undx
.
(9)
Forthelasttwotermsof(8),wecanexpandas 1
2
M12
∇
un+12
−
M12∇
un2dx
=
1 2
∇
un+1
−
un·
M∇
un+1
+
undx
= Δ
t 4
∇
vn+1
+
vn·
M∇
un+1
+
undx
.
(10)
Adding(9) and(10),weobviouslyhave
H
un+1
,
vn+1− H
un,
vn=
0. Remark1.Lemma1impliesthattheenergycorrespondingtothenumericalsolutionun
,
vnoftheCrank–Nicolsonmethod isaphysicalconstantE
(
t)
,H
un,
vn= H
u0,
v0=
E t0.
(11)3. Successivemulti-stagemethods
We nowpropose a successivemulti-stage (SMS)methodasan extension ofthe Crank–Nicolsonmethodto an s-stage method,referredtoasSMS(Rs),withagivencoefficientvector
Rs
=
[r1,
r2, · · · ,
rs].
(12)TheSMS(Rs)isaone-steps-stagemethodthatcomputesthenextapproximation
un+1
,
vn+1 fromun
,
vn.Wesetu0
=
un andv0=
vnfortheinitial-stage,andthencalculatetheintermediatevalue(
ui,
vi)
bysolvingui
−
ui−1Δ
t=
ri(
vi+
vi−1) ,
vi−
vi−1Δ
t=
riκ ∇ · (
M∇ (
ui+
ui−1)) ,
(13)
fori
=
1,
2, · · · ,
s.Finally,wehaveun+1=
usandvn+1=
vs.WenotethatSMS(R1)withacoefficientvector R1=
12
(14)is identicalto theCrank–Nicolson method(6), whichprovides second-order accuracy. Forhigherorder accuracy, we will describetheorderconditionsforthecoefficientvector Rs inSection3.3.Beforeintroducingaspecificexample,weprovide proofsoftheuniquesolvabilityandenergyconservationinSections3.1and3.2,respectively.
3.1. Uniquesolvability
Theorem2.TheSMS(Rs)method (13) isuniquelysolvableforanytimestepsize
Δ
t.Proof. Foreachstagei
=
1,
2, · · · ,
s,weneedtosolve ui−
riΔ
t vi= ϕ
i,
vi
−
riκ Δ
t∇ · (
M∇
ui) = ψ
i,
(15)
where
ϕ
i=
ui−1+
riΔ
t vi−1 andψ
i=
vi−1+
riΔ
t∇ · (
M∇
ui−1)
.Thesystem(15) canberepresentedas ui−
r2iΔ
t2κ ∇ · (
M∇
ui) = ϕ
i+
riΔ
tψ
i,
(16)whichhasauniquesolutionui foranytimestepsize
Δ
t,since I−
ri2Δ
t2κ ∇ · (
M∇)
(17)is an invertibleoperator withall eigenvalues bigger than 1 for givenpositive function
κ
and negative definite operator∇ · (
M∇)
.Usingthesolutionui from(16),wecaneasilyobtainauniquesolutionofviby(15). 3.2. EnergyconservationTheorem3.ForeachintermediatestageoftheSMS(Rs)method (13),theenergyisconservedforanytimestepsizeri
Δ
t,H (
ui,
vi) = H (
ui−1,
vi−1) .
(18)Moreover,theproposedmethod (13) isunconditionallyenergyconserving,meaningthat
H
un+1
,
vn+1= H
un,
vnforanytime stepsize
Δ
t.Proof. SimilarlytotheproofofLemma1,wefirstconsiderthedifferenceofadjacentdiscreteenergyfunctionals,
H (
ui,
vi) − H (
ui−1,
vi−1)
=
1 2
κ
v2i− κ
v2i−1+
M12∇
ui2
−
M12∇
ui−12dx
.
(19)Thetwopartsof(19) canbeexpandedas 1
2
κ
v2i− κ
v2i−1dx= −
riΔ
t 2
∇ (
vi+
vi−1) · (
M∇ (
ui+
ui−1))
dx,
(20)1 2
M12
∇
ui2
−
M12∇
ui−12dx
=
riΔ
t 2
∇ (
vi+
vi−1) · (
M∇ (
ui+
ui−1))
dx.
(21)Thenwehave
H (
ui,
vi) = H (
ui−1,
vi−1)
foranyriΔ
t,i=
1, · · · ,
s.Therefore,H
un+1
,
vn+1= H (
us,
vs) = H (
u0,
v0) = H
un,
vn(22) whichprovesconservationofthediscreteenergyfunctional
H
un
,
vn . 3.3. TemporalaccuracyWecanconsideraframeworkoftheRunge–Kutta(RK)methodtoshowthetimeaccuracyfortheSMSmethod.Summing (13) uptothei-thstage,thedifferencebetweeni-thstage
(
ui,
vi)
andinitial-stage(
u0,
v0)
valuescanbewrittenasalinear combinationofthepreviousstagevalues:ui
−
u0Δ
t=
i j=1rj
vj
+
vj−1,
vi
−
v0Δ
t=
i j=1rj
κ ∇ ·
M∇
uj
+
uj−1.
(23)
Next,(23) canbefurthersimplifiedastheRKmethod,
ui
−
u0Δ
t=
i j=0ai jvj
,
vi
−
v0Δ
t=
i j=0ai j
κ ∇ · (
M∇
ui) ,
(24)
whereai0
=
r1,aii=
ri,andai j=
rj+
rj+1 for j=
1,
2, · · · ,
i−
1.Furthermore,theSMS(Rs)method (13) canbedescribed bytheButchertablefortheRKmethod,c A bT
=
0 0 0 0
· · ·
0c1 r1 r1 0
· · ·
0c2 r1 r1
+
r2 r2· · ·
0.. . .. . .. . .. . . . . .. .
cs r1 r1+
r2 r2+
r3· · ·
rsr1 r1
+
r2 r2+
r3· · ·
rs,
(25)whereA
∈ R
(s+1)×(s+1),b∈ R
s+1,andc=
A1 with1= (
1,
1, . . . ,
1)
T∈ R
s+1.Becauseofthelinearityandautonomyofthegivenwaveequation (1),weconsiderthelinearRKmethod,whichisbriefly explainedinAppendix.Theorderconditionforthe p-thorderaccuracyisbTAq−11
=
1/
q!
for1≤
q≤
p.Fornumerical simulations,we need a specifictable forthe desiredaccuracy. We nowconstruct thecoefficient vector Rs fromtherelationship betweentheSMSandlinearRKmethods. Wefirst considerasingle coefficient vector R1
=
[r1].ConsideringthestructureoftheSMSmethod,wehavethecorrespondingmatrixA andvectorb as A
=
0 0
r1 r1
,
b=
r1
r1
.
(26)Tosatisfythefirst-orderconditionbT1
=
1,weneedr1=
1/
2,whichalsosatisfiesthesecond-orderconditionbTA1=
1/
2.Thus,theSMSmethodbeginswithsecond-orderaccuracyandR1
=
12
isonlyonecaseofthesecond-orderSMSmethod.Wenotethatthereisnosolutionforthetwo-stagecoefficientvectorR2
=
[r1,
r2] thatsatisfiestheorderconditionsupto third-orderaccuracy.WenowchooseapossiblechoiceofthecoefficientvectorR3=
[r1,
r2,
r3],whichimpliesathree-stage SMSmethod,andconsidercorrespondingmatrixA andvectorb asA
=
⎡
⎢ ⎢
⎢ ⎢
⎢ ⎣
0 0 0 0
r1 r1 0 0
r1 r1
+
r2 r2 0 r1 r1+
r2 r2+
r3 r3⎤
⎥ ⎥
⎥ ⎥
⎥ ⎦ ,
b=
⎡
⎢ ⎢
⎢ ⎢
⎢ ⎣
r1 r1+
r2r2
+
r3r3
⎤
⎥ ⎥
⎥ ⎥
⎥ ⎦
.
(27)Tosatisfythefirst-orderconditionbT1
=
1,thecoefficientsshouldsatisfy r1+
r2+
r3=
12
,
(28)whichalsosatisfiesthesecond-orderconditionbTA1
=
1/
2,justasinthecaseoftheone-stageSMSmethod (26).Withthe identity (28) andthethird-orderconditionbTA21=
1/
6,wehavethefollowingidentity:1
2bTA21
=
r31+
r32+
r32+
2r21r2
+
r12r3+
r22r1+
r22r3+
r23r1+
r32r2+
4r1r2r3= (
r1+
r2+
r3)
3− (
r1+
r2)(
r2+
r3)(
r3+
r1)
=
r1+
r2+
r34
−
1 2−
r31 2
−
r11 2
−
r2=
112
.
(29)
Thus,
r1r2
+
r1r3+
r2r3−
2r1r2r3=
112
.
(30)Fig. 1. Coefficients of R3for fourth-order accuracy.
We note that the coefficientssatisfying up to third-order conditions also satisfy the fourth-order condition. In fact, the identity (28) andtheconditionbTA31
=
1/
24 inducethesameresultasin (30) since1
2bTA31
= (
r1+
r2+
r3)
4−
2(
r1+
r2+
r3)(
r1+
r2)(
r2+
r3)(
r3+
r1)
=
r1+
r2+
r34
−
116
−
1 2−
r31 2
−
r11 2
−
r2=
148
.
(31)
Inconclusion,weneedonlytworelations(28) and(30) toachievethefourth-orderaccuracy.
Now, for finding R3 to construct three-stage fourth-order SMS methods, we set a free parameter r1
= γ
. Then the parametersr2 andr3satisfyr2
+
r3=
1−
2γ
2
,
r2r3=
1−
6γ +
12γ
212
(
1−
2γ ) .
(32)Fig.1showsthecoefficientsof R3 withrespecttotheparameter
γ >
1/
2,whichisthesolvableregion.Thisfigureimplies that thereare infinitelymanythree-stage fourth-order SMSmethods.Choosingγ =
0.
8 forthe numericalsimulation,we haveR3
( γ ) ≈
[0.
8,
0.
599258893, −
0.
899258893].
(33)Asweobserveintheidentity (28) and(30),anypermutationofthesolutionr1,r2,andr3 inR3 satisfiesthesameorder conditionsuptofour.Infact,thefollowingtheoremprovesthatthispermutationinvariantpropertyisgenerallytrueforall s-stageSMSmethods.
Theorem4.Forans-stageSMSmethodwithacoefficientvectorRs,thevalueofbTAp1 forp
≥
0 isinvariantunderthepermutation ofRs=
[r1,
r2, · · · ,
rs].Proof. Without loss of generality, we consider a swapped coefficient vector R
ˆ
s:=
r1
, · · · ,
rk−1,
rk+1
,
rk,
rk+2, · · · ,
rs for 0<
k<
s andprovethatbTAp1
= ˆ
bTAˆ
p1 (34)whereA,
ˆ
b,ˆ
andc areˆ
thematrixandthevectorsintheButchertablecorrespondingtotheswappedcoefficientvector Rˆ
s. Weobservethat entriesai j ofA andaˆ
i j ofA areˆ
differedbyd=
rk+1−
rk onlyata fewpoints,sowe denoteAˆ =
A+
dD whereD∈ R
(s+1)×(s+1) havezeroentriesexcept Di,k−1=
1,
i≥
k, Dk,k=
1,andDi,k+1= −
1,
i>
k. Alsoentriesofb,ˆ
c areˆ
samewithb,c exceptb
ˆ
k−1=
bk−1+
d, ˆ
bk+1=
bk+1−
d,
and cˆ
k=
ck+
2d.
(35) Inthecaseof p=
0,(34) is obviousbT1= ˆ
bT1.Letc(p):=
Ap1 and cˆ
(p):= ˆ
Ap1,thenweclaimthat thefollowingidentity holdsforp≥
1,ˆ
c(p)
=
c(p)+ β
pe, β
p=
d rk+
rk+1c(kp+)1
−
c(kp−)1,
(36)wheree
∈ R
s+1 isthestandardunitvector withek=
1.For p=
1,itisatrivialstatementsincecˆ
k=
ck+
2d,ˆ
ci=
ci,
i=
k, andβ
1=
rk+drk+12rk
+
rk+1=
2d.Weprovethisassertion(36) for p>
1 bymathematicalinduction.ˆ
c(p+1)
= (
A+
dD)
c(p)
+ β
pe=
c(p+1)+
dDc(p)+ β
pAe+
dβ
pDe.
(37)Fori
<
k,ˆ
c(ip+1)−
c(ip+1)=
0 since Di,j=
ai,k=
0.Incaseofi>
k,cˆ
(ip+1)−
c(ip+1)=
dc(kp−)1
−
c(kp+)1+ β
pai,k=
0 duetothe mathematicalinductionassumptionforβ
p.Fori=
k,usingak,k+
dDk,k=
rk+1,wegetˆ
c(kp+1)
−
c(kp+1)=
dck(p−)1
+
c(kp)+ β
prk+1=
drk
+
rk+1rkc(kp−)1
+ (
rk+
rk+1)
c(kp)+
rk+1ck(p+)1.
(38)
Ontheotherhand,bythedefinitionofc(p+1)
=
Ac(p),wehave c(kp++11)−
c(kp−+11)=
s j=0 ak+1,j−
ak−1,j c(jp)=
rkc(kp−)1+ (
rk+
rk+1)
ck(p)+
rk+1c(kp+)1.
(39)Bycombining(38) and(39),weconcludethemathematicalinduction(36) forp
+
1β
p+1:= ˆ
c(kp+1)−
ck(p+1)=
drk
+
rk+1c(kp++11)
−
c(kp−+11).
(40)Finally,(35) and(36) impliestheinvariant, b
ˆ
TAˆ
p1−
bTAp1=
s i=0 bˆ
icˆ
(ip)−
bic(ip)= (ˆ
bk−1−
bk−1)
ck(p−)1+
bk(ˆ
c(kp)−
ck(p)) + (ˆ
bk+1−
bk+1)
c(kp+)1=
dc(kp−)1
−
c(kp+)1+ (
rk+
rk+1)β
p=
0.
(41)
The p-thorderSMS(Rs)involvesthesystemofpolynomialequationsbTAq−11
=
q1! ofdegreeq=
1, · · · ,
p withrespect tor1,
r2, · · · ,
rs,thus finding Rs bydirectlysolving thesystemofp-thorderpolynomial equationsisnota trivialtaskfor p>
2.Wenowbrieflyexplainhowtofindas-stageSMScoefficientsvector Rs=
[r1,
r2, · · · ,
rs] forsixth-orderaccuracy.The first-ordercondition israthertrivial.For s
≥
1, givenr1, · · · ,
rs−1,we choosers=
rs(
r1, · · · ,
rs−1) =
1/
2−
s−1 i=1ri, satisfyingthefirst-orderconditionbT1=
1.Thisisjustasimpleextensionof(28).ThenitisalsoeasytoshowthatSMS(Rs) withRs=
[r1, · · · ,
rs−1,
rs(
r1, · · · ,
rs−1)
] satisfiesthesecond-orderconditions,bTA1=
1/
2.Nextstep istofind lasttwo coefficientsrs−1 andrs forgivenr1
, · · · ,
rs−2 satisfyingboth ofthefirst- and third-order conditionscorrespondingtobT1=
1 andbTA21=
1/
6,respectively.Ifwechooserstosatisfythefirst-orderconditionthen thethird-orderconditionisjust apolynomialequationforrs−1,whichisagainan extensionof(30).Herewedonotshow thedetails butwehavean explicitformulaforrs−1 andrsasa functionofr1, · · · ,
rs−2 fors≥
3 andthe solutionpairis uniqueuptopermutationifitexists.Thisissimilarto(32) whichisthecaseofs=
3.Aratherlengthyalgebraiccalculation proves that a third-orderSMS(Rs) wherers−1=
rs−1(
r1, · · · ,
rs−2)
andrs=
rs(
r1, · · · ,
rs−2)
alsosatisfies thefourth-order condition,bTA31=
1/
24.Finally we tryto seekan s-stageSMS(Rs) satisfyingthefirst-,third-,andfifth-order conditions.Forgivenr1
, · · · ,
rs−3 withs≥
3,weformafifthpolynomialequationasafunctionofγ =
rs−2 tosolvebTA41=
1/
120 withRs
=
[r1, · · · ,
rs−3, γ ,
rs−1,
rs] (42)where rs−1
=
rs−1(
r1, · · · ,
rs−3, γ )
andrs=
rs(
r1, · · · ,
rs−3, γ )
are given by the explicit formula forthe first- and third- conditions.Thoughthereisnosimplealgebraic formulaonγ
,we numericallyimplementthissolverfordoubleprecision accuracyandfigureoutthatthereisnosolutionwiths≤
4.Fig.2showsthedottedplotof(
r1,
r2)
whereR5
(
r1,
r2) =
[r1,
r2,
r3(
r1,
r2),
r4(
r1,
r2,
r3),
r5(
r1,
r2,
r3)
] (43) is a solution for the fifth-order condition. For the numerical observation, we consider a domain for the coefficients as(
r1,
r2) ∈
[−
1.
5,
2.
5]2 anduniformspacingwith0.
05.Forexamples,numericalvaluesuptosingleprecisionaccuracyR5
(
1,
1.
5) ≈
[1,
1.
5,−
1.
474930077, −
1.
109591016,
0.
584521093],
(44) R5(
2,
1) ≈
[2,
1, −
1.
074758082,−
1.
995305362,
0.
570063445],
(45)Fig. 2. Pairs of r1and r2where R5has a solution of the fifth-order accuracy.
andtheir permutationsare thesolutionssatisfyingall orderconditionsup to5.Wealsonumericallyobservethat afifth- ordermethodofSMS(R5)alwayssatisfiesthesixth-ordercondition,bTA51
=
1/
720.Remark2.We have presented the numerical search algorithm for R3
( γ )
and R5(
r1,
r2)
satisfying bTAq−11=
1/
q!,
q=
1, · · · ,
s fors=
3,
5.Similarly,onecantrytosearch R7= (
r1,
r2,
r3,
r4,
r5(
r1, · · · ,
r4),
r6(
r1, · · · ,
r5),
r7(
r1, · · · ,
r5))
forgiven(
r1,
r2,
r3,
r4)
satisfyingbTAq−11=
1/
q!,
q=
5,
6,
7 whiler5,
r6,
r7arechosentofixbTAq−11=
1/
q!,
q=
1,
2,
3,
4.Insteadof tryingabruteforcesearchforthenumericalsolutionsofthesystemof3polynomialequationsup toseventhorderwhich maynot exist,we will presentmoreidentities usefultosimplifythe systemofpolynomial equationsrelatedto the SMS methodinfutureworks.Beforewefinishthissection,wewouldliketointroducesomeproperties,whicharecomparablewithexistingnumerical methods. First, implicit RK methods usually consider the singly diagonally implicit type for the efficient computations, however,thisstrategyfortheSMSmethodsisnotavailable.
Lemma5.ThereisnoasinglydiagonalcoefficientvectorRs
=
[γ , γ , · · · , γ
] forhigherthansecond-orderSMSmethods.Proof. Supposethatwe haveasinglydiagonalcoefficientvector Rs
=
[γ , γ , · · · , γ
] andwe trytofindaspecificvalue ofγ
tosatisfy theorderconditions.Wedenoteasr1=
r2= · · · =
rs= γ
andwehavecorrespondingcoefficientsA and b as in (25).Forthefirst-orderaccuracy,wehavebT1
=
2 s i=1ri
=
2sγ =
1,
(46)andthus
γ =
1/(
2s)
.Forthethird-orderaccuracy,wehave bTA(
A1) =
2γ
s−1
i=1
⎛
⎝
i−1j=1
4
γ
2j+
2γ
2i⎞
⎠ + γ
⎛
⎝
s−1j=1
4
γ
2j+
2γ
2s⎞
⎠
=
4γ
3s−1
i=1
i2
+
2γ
3s2=
2 3γ
3s2s2
+
1=
2s2+
1 12s2=
16
.
(47)
Therefore,eventhesinglydiagonalstrategyfortheSMSmethodcannotsurpassthethird-orderconditions.
Remark3.Infact,aSMSmethodwithasinglydiagonalcoefficient vector Rs
=
[γ , γ , · · · , γ
] whereγ =
1/(
2s)
alsoisthe second-orderaccuratemethod,becauseitalwayssatisfiesthesecond-orderconditionbTA1=
1/
2.Next, we need to remark that the proposed SMS methods differ from symplectic RK methods in [16,17]. It is well knownthatthealgebraic stabilityconditionbibj
−
biai j−
bjai j=
0 isasufficientconditionforthesymplecticRKmethods.However, the SMS methods do not satisfy this condition. For the simplicity, we define the algebraic stability matrix as G
=
bbT−
BA−
ATB,referredtoasAG-matrix.Theorem6.ASMSmethoddoesnotsatisfythealgebraicstabilitycondition.