Energy conserving successive multi-stage method for the linear wave equation

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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

u_tt

= ∇ · (

^M

∇

^u

) ,

(1)

where M

=

^M

(

x

)

is asymmetric positivedefinite matrix and

κ = κ (

x

)

isa strictly positivefunction. Equation (1) might becompletedwithspecificboundaryconditionsinaboundeddomain

 ⊂ R

^d

(

d

=

¹

,

2

,

3

)

.Forsimplicity,weconsiderthe periodicboundaryconditionalongtheedgesofcomputationaldomainwhere

κ (

x

)

andM

(

x

)

areconstantsrepresentingthe homogeneous backgroundmedium.Homogeneous Neumannboundarycondition n

· (

^M

∇

^u

) =

^{0 or}homogeneous Dirichlet boundaryconditionu

=

^{0 canbeeasilyimposedusingevenoroddextensionofthesolution,}respectively.Thecorresponding energyof (1) canbewrittenasasumofthekineticandthepotentialenergiesofthesystem,

E

(

t

) =

¹ 2







κ

u²_t

+  

^M12

∇

^u

 

²



dx

,

(2)

*

Correspondingauthor.

E-mailaddress:[email protected](J.-Y. Lee).

https://doi.org/10.1016/j.jcp.2022.111098

0021-9991/©^{2022ElsevierInc.Allrightsreserved.}

where

 

^M12

∇

^u

 

²

= (∇

^u

,

M

∇

^u

)

.Itisworthnotingthatwhenconsideringtheperiodicorhomogeneousboundaryconditions, theenergy (2) isconservedwithrespecttotime,asdemonstratedbyintegrationbyparts:

d dtE

(

t

) =





( κ

u_tu_tt

−

^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-order}multi-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

=

¹

κ ^{∇ · (}

^M

^∇

^u

⁾

(4)

andtheenergy (2) inHamiltonianformas

H (

u

,

v

) =

¹

2







κ

v²

+  

^M12

∇

^u

 

²



dx

.

(5)

Forthesemi-discreteformulation, wedenoteuⁿ andvⁿ asapproximationsofu



·,

^tn



andv



·,

^tn



,wheretⁿ

=

ⁿ

Δ

t and

Δ

t isatimestepsize.Wefirstconsiderthewell-knownCrank–Nicolsonmethod

uⁿ⁺¹

−

^un

Δ

t

=

^vn+1

+

^vn

2

,

vⁿ⁺¹

−

^vn

Δ

t

=

¹

κ ^{∇ ·}



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⁺¹



2

− κ ^

vⁿ



2

+ 

M

∇

^un+1



²

− 

M

∇

^un



²dx

.

⁽⁸⁾

Forthefirsttwotermsof(8),wecanrearrangethemas 1

2





κ ^

vⁿ⁺¹



2

− κ ^

vⁿ



2

dx

=

¹ 2





κ ^

vⁿ⁺¹

−

^vn

 

vⁿ⁺¹

+

^vn



dx

= Δ

t 4







vⁿ⁺¹

+

^vn



∇ · 

M

∇ 

uⁿ⁺¹

+

^un



dx

= − Δ

t 4





∇ 

vⁿ⁺¹

+

^vn



· 

M

∇ 

uⁿ⁺¹

+

^un



dx

.

(9)

Forthelasttwotermsof(8),wecanexpandas 1

2





 

^M12

∇

^un+1

 

²

−  

^M12

∇

^un

 

^2dx

=

¹ 2





∇ 

uⁿ⁺¹

−

^un



· 

M

∇ 

uⁿ⁺¹

+

^un



dx

= Δ

t 4





∇ 

vⁿ⁺¹

+

^vn



· 

M

∇ 

uⁿ⁺¹

+

^un



dx

.

(10)

Adding(9) and(10),weobviouslyhave

H 

uⁿ⁺¹

,

vⁿ⁺¹



− H 

uⁿ

,

vⁿ



=

^0.



Remark1.Lemma1impliesthattheenergycorrespondingtothenumericalsolution



uⁿ

,

vⁿ



oftheCrank–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

R_s

=

^[r1

,

r₂

, · · · ,

^rs]

.

(12)

TheSMS(R_s)isaone-steps-stagemethodthatcomputesthenextapproximation



uⁿ⁺¹

,

vⁿ⁺¹



from



uⁿ

,

vⁿ



.Wesetu₀

=

^un andv0

=

^vnfortheinitial-stage,andthencalculatetheintermediatevalue

(

u_i

,

v_i

)

bysolving

u_i

−

^ui−¹

Δ

t

=

^ri

(

v_i

+

^vi−1

) ,

v_i

−

^vi−¹

Δ

t

=

^ri

κ ^{∇ · (}

^M

^{∇ (}

^ui

⁺

^ui−1

^{)) ,}

(13)

fori

=

¹

,

2

, · · · ,

^{s.Finally,wehaveun+1}

=

^usandvⁿ⁺¹

=

^vs.WenotethatSMS(R1)withacoefficientvector R₁

=

₁

2



(14)

is identicalto theCrank–Nicolson method(6), whichprovides second-order accuracy. Forhigherorder accuracy, we will describetheorderconditionsforthecoefficientvector R_s inSection3.3.Beforeintroducingaspecificexample,weprovide proofsoftheuniquesolvabilityandenergyconservationinSections3.1and3.2,respectively.

3.1. Uniquesolvability

Theorem2.TheSMS(R_s)method (13) isuniquelysolvableforanytimestepsize

Δ

t.

Proof. Foreachstagei

=

¹

,

2

, · · · ,

^{s,weneedtosolve} ui

−

^ri

Δ

t vi

= ϕ

i

,

vi

−

^ri

κ ^Δ

^t

^{∇ · (}

^M

^∇

^ui

^{) = ψ}

ⁱ

^,

(15)

where

ϕ

i

=

^ui−¹

+

^ri

Δ

t vi−¹ and

ψ

i

=

^vi−¹

+

^ri

Δ

t

∇ · (

^M

∇

^ui−¹

)

.Thesystem(15) canberepresentedas u_i

−

^r2i

Δ

t²

κ ^{∇ · (}

^M

^∇

^ui

^{) =} ϕ

i

+

^ri

Δ

t

ψ

i

,

(16)

whichhasauniquesolutionu_i foranytimestepsize

Δ

t,since I

−

^ri2

Δ

t²

κ ^{∇ · (}

^M

^∇)

⁽¹⁷⁾

is an invertibleoperator withall eigenvalues bigger than 1 for givenpositive function

κ

and negative definite operator

∇ · (

^M

∇)

^{.Usingthesolutionu}i from(16),wecaneasilyobtainauniquesolutionofviby(15).



3.2. Energyconservation

Theorem3.ForeachintermediatestageoftheSMS(R_s)method (13),theenergyisconservedforanytimestepsizer_i

Δ

t,

H (

u_i

,

v_i

) = H (

u_i−1

,

v_i−1

) .

(18)

Moreover,theproposedmethod (13) isunconditionallyenergyconserving,meaningthat

H 

uⁿ⁺¹

,

vⁿ⁺¹



= H 

uⁿ

,

vⁿ



foranytime stepsize

Δ

t.

Proof. SimilarlytotheproofofLemma1,wefirstconsiderthedifferenceofadjacentdiscreteenergyfunctionals,

H (

u_i

,

v_i

) − H (

u_i−1

,

v_i−1

)

=

¹ 2





κ

v²_i

− κ

v²_i−1

+  

^M12

∇

^ui

 

²

−  

^M12

∇

^ui−¹

 

^2dx

.

⁽¹⁹⁾

Thetwopartsof(19) canbeexpandedas 1

2





κ

v²_i

− κ

v²_i−1dx

= −

^ri

Δ

t 2





∇ (

^vi

+

^vi−1

) · (

^M

∇ (

^ui

+

^ui−1

))

dx

,

(20)

1 2





 

^M12

∇

^ui

 

²

−  

^M12

∇

^ui−1

 

^2dx

=

^ri

Δ

t 2





∇ (

^vi

+

^vi−1

) · (

^M

∇ (

^ui

+

^ui−1

))

dx

.

(21)

Thenwehave

H (

^ui

,

vi

) = H (

^ui−¹

,

vi−¹

)

foranyri

Δ

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. Temporalaccuracy

WecanconsideraframeworkoftheRunge–Kutta(RK)methodtoshowthetimeaccuracyfortheSMSmethod.Summing (13) uptothei-thstage,thedifferencebetweeni-thstage

(

ui

,

vi

)

andinitial-stage

(

u0

,

v0

)

valuescanbewrittenasalinear combinationofthepreviousstagevalues:

u_i

−

^u0

Δ

t

=



i j=¹

r_j



v_j

+

^vj−1

 ,

v_i

−

^v0

Δ

t

=



i j=¹

r_j

κ ^{∇ ·}



M

∇ 

u_j

+

^uj−1

 .

(23)

Next,(23) canbefurthersimplifiedastheRKmethod,

u_i

−

^u0

Δ

t

=



i j=0

ai jvj

,

v_i

−

^v0

Δ

t

=



i j=0

ai j

κ ^{∇ · (}

^M

^∇

^ui

^{) ,}

(24)

wherea_i0

=

^r1,a_ii

=

^ri,anda_{i j}

=

^rj

+

^rj+¹ for j

=

¹

,

2

, · · · ,

ⁱ

−

^1.Furthermore,theSMS(R_s)method (13) canbedescribed bytheButchertablefortheRKmethod,

c A b^T

=

0 0 0 0

· · ·

⁰

c₁ r₁ r₁ 0

· · ·

⁰

c₂ r₁ r₁

+

^r2 r₂

· · ·

⁰

.. . .. . .. . .. . . . . .. .

c_s r₁ r₁

+

^r2 r₂

+

^r3

· · ·

^rs

r1 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-thorderaccuracyisb^TA^q−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-orderconditionb^T1

=

^1,weneedr1

=

¹

/

2,whichalsosatisfiesthesecond-orderconditionb^TA1

=

¹

/

2.

Thus,theSMSmethodbeginswithsecond-orderaccuracyandR1

=

₁

2



isonlyonecaseofthesecond-orderSMSmethod.

Wenotethatthereisnosolutionforthetwo-stagecoefficientvectorR₂

=

^[r1

,

r₂] thatsatisfiestheorderconditionsupto third-orderaccuracy.WenowchooseapossiblechoiceofthecoefficientvectorR3

=

^[r1

,

r2

,

r3],whichimpliesathree-stage SMSmethod,andconsidercorrespondingmatrixA andvectorb as

A

=

⎡

⎢ ⎢

⎢ ⎢

⎢ ⎣

0 0 0 0

r1 r1 0 0

r₁ r₁

+

^r2 r₂ 0 r₁ r₁

+

^r2 r₂

+

^r3 r₃

⎤

⎥ ⎥

⎥ ⎥

⎥ ⎦ ,

b

=

⎡

⎢ ⎢

⎢ ⎢

⎢ ⎣

r₁ r1

+

r2

r₂

+

^r3

r₃

⎤

⎥ ⎥

⎥ ⎥

⎥ ⎦

.

(27)

Tosatisfythefirst-orderconditionb^T1

=

^{1,thecoefficientsshouldsatisfy} r₁

+

^r2

+

^r3

=

¹

2

,

(28)

whichalsosatisfiesthesecond-orderconditionb^TA1

=

¹

/

2,justasinthecaseoftheone-stageSMSmethod (26).Withthe identity (28) andthethird-orderconditionb^TA²1

=

1

/

6,wehavethefollowingidentity:

1

2b^TA²1

=

^r3₁

+

^r3₂

+

^r₃²

+

²



r²₁r₂

+

^r₁^2r3

+

^r2₂^r1

+

^r2₂^r3

+

^r2₃^r1

+

^r₃^2r2

+

^4r1r₂r₃

= (

^r1

+

^r2

+

^r3

)

³

− (

^r1

+

^r2

)(

r₂

+

^r3

)(

r₃

+

^r1

)

=

^r1

+

^r2

+

^r3

4

−



1 2

−

r3

 

1 2

−

r1

 

1 2

−

r2



=

¹

12

.

(29)

Thus,

r₁r₂

+

^r1r₃

+

^r2r₃

−

^2r1r₂r₃

=

¹

12

.

(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) andtheconditionb^TA³1

=

¹

/

24 inducethesameresultasin (30) since

1

2b^TA³1

= (

^r1

+

^r2

+

^r3

)

⁴

−

²

(

r₁

+

^r2

+

^r3

)(

r₁

+

^r2

)(

r₂

+

^r3

)(

r₃

+

^r1

)

=

^r1

+

^r2

+

^r3

4

−

¹

16

−



1 2

−

r3

 

1 2

−

r1

 

1 2

−

r2



=

¹

48

.

(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 andr₃satisfy

r₂

+

^r3

=

¹

−

²

γ

2

,

r₂r₃

=

¹

−

⁶

γ +

¹²

γ

²

12

(

1

−

²

γ ) .

(32)

Fig.1showsthecoefficientsof R₃ withrespecttotheparameter

γ >

1

/

2,whichisthesolvableregion.Thisfigureimplies that thereare infinitelymanythree-stage fourth-order SMSmethods.Choosing

γ =

⁰

.

8 forthe numericalsimulation,we have

R3

( 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,thevalueofb^TA^p1 forp

≥

^{0 isinvariantunderthe}permutation ofRs

=

^[r1

,

r2

, · · · ,

^rs].

Proof. Without loss of generality, we consider a swapped coefficient vector R

ˆ

_s

:=

r₁

, · · · ,

^rk−¹

,   

r_k+1

,

r_k

,

r_k+2

, · · · ,

^rs



for 0

<

k

<

s andprovethat

b^TA^p1

= ˆ

^bTA

ˆ

^p1 (34)

whereA,

ˆ

_b,

ˆ

_andc are

ˆ

thematrixandthevectorsintheButchertablecorrespondingtotheswappedcoefficientvector R

ˆ

s. Weobservethat entriesa_{i j} ofA and^a

ˆ

i j ofA are

ˆ

differedbyd

=

^rk+¹

−

^rk onlyata fewpoints,sowe denoteA

ˆ =

^A

+

^dD whereD

∈ R

^{(s+1)×(s+1) havezeroentriesexcept D}i,k−¹

=

¹

,

i

≥

^{k, D}k,k

=

^1,andDi,k+¹

= −

¹

,

i

>

k. Alsoentriesofb,

ˆ

^{c are}

ˆ

samewithb,c except

b

ˆ

_k−1

=

^bk−¹

+

^d

, ˆ

b_k+1

=

^bk+¹

−

^d

,

and c

ˆ

_k

=

^ck

+

^2d

.

(35) Inthecaseof p

=

^{0,(34) is obviousbT1}

= ˆ

^bT1.Letc(p)

:=

^{Ap1 and} c

ˆ

^(p)

:= ˆ

^{Ap1,thenweclaimthat thefollowingidentity} holdsforp

≥

^1,

ˆ

c^(p)

=

^c(p)

+ β

pe

, β

_p

=

^d r_k

+

^rk+1



c⁽_k^p₊⁾₁

−

^c(_k^p₋⁾₁

,

(36)

wheree

∈ R

^{s+1 isthestandardunitvector withe}k

=

^{1.For p}

=

^{1,itisatrivialstatementsince}c

ˆ

k

=

^ck

+

^2d,

ˆ

ci

=

^ci

,

i

=

^k, and

β

1

=

_rk+^d_rk+1²



r_k

+

^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

+

d

β

pDe

.

(37)

Fori

<

k,

ˆ

^c(_i^p+1)

−

^c(_i^p+1)

=

^{0 since D}i,j

=

^ai,k

=

^0.Incaseofi

>

k,^c

ˆ

⁽_i^p+1)

−

^c(_i^p+1)

=

^d



c⁽_k^p₋⁾₁

−

^c(_k^p₊⁾₁

+ β

pa_i,k

=

^{0 duetothe} mathematicalinductionassumptionfor

β

p.Fori

=

^k,usingak,k

+

^dDk,k

=

^rk+1,weget

ˆ

c⁽_k^p+1)

−

^c(_k^p+1)

=

^d



c_k^(p₋⁾₁

+

^c(_k^p)

+ β

pr_k+1

=

^d

r_k

+

^rk+1



r_kc⁽_k^p₋⁾₁

+ (

^rk

+

^rk+1

)

c⁽_k^p)

+

^rk+1c_k^(p₊⁾₁

.

(38)

Ontheotherhand,bythedefinitionofc^(p+1)

=

^Ac(p),wehave c⁽_k^p₊⁺₁¹⁾

−

c⁽_k^p₋⁺₁¹⁾

=



s j=0



ak+¹,j

−

ak−¹,j



c⁽_j^p)

=

rkc⁽_k^p₋⁾₁

+ (

rk

+

rk+¹

)

c_k^(p)

+

rk+¹c⁽_k^p₊⁾₁

.

(39)

Bycombining(38) and(39),weconcludethemathematicalinduction(36) forp

+

¹

β

p+1

:= ˆ

^c(_k^p+1)

−

^c_k^(p+1)

=

^d

r_k

+

^rk+¹



c⁽_k^p₊⁺₁¹⁾

−

^c(_k^p₋⁺₁¹⁾

.

(40)

Finally,(35) and(36) impliestheinvariant, b

ˆ

^TA

ˆ

^p1

−

^bTAp1

=



s i=0



b

ˆ

_i^c

ˆ

⁽_i^p)

−

^bic⁽_i^p)

= (ˆ

^bk−¹

−

^bk−¹

)

c_k^(p₋⁾₁

+

^bk

(ˆ

^c(_k^p)

−

^c_k^(p)

) + (ˆ

^bk+¹

−

^bk+¹

)

c⁽_k^p₊⁾₁

=

^d



c⁽_k^p₋⁾₁

−

^c(_k^p₊⁾₁

+ (

^rk

+

^rk+¹

)β

p

=

⁰

. 

(41)

The p-thorderSMS(R_s)involvesthesystemofpolynomialequationsb^TA^q−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, givenr}1

, · · · ,

^rs−¹,we choosers

=

^rs

(

r1

, · · · ,

^rs−¹

) =

¹

/

2

− 

s−¹ i=1r_i, satisfyingthefirst-orderconditionb^T1

=

1.Thisisjustasimpleextensionof(28).ThenitisalsoeasytoshowthatSMS(Rs) withRs

=

^[r1

, · · · ,

^rs−¹

,

rs

(

r1

, · · · ,

^rs−¹

)

] satisfiesthesecond-orderconditions,b^TA1

=

¹

/

2.

Nextstep istofind lasttwo coefficientsrs−¹ andrs forgivenr1

, · · · ,

^rs−² satisfyingboth ofthefirst- and third-order conditionscorrespondingtob^T1

=

1 andb^TA²1

=

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,b^TA³1

=

¹

/

24.

Finally we tryto seekan s-stageSMS(R_s) satisfyingthefirst-,third-,andfifth-order conditions.Forgivenr₁

, · · · ,

^rs−3 withs

≥

^{3,weformafifthpolynomialequationasafunctionof}

γ =

^rs−² tosolveb^TA⁴1

=

¹

/

120 with

Rs

=

^[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 formulaon

γ

,we numericallyimplementthissolverfordoubleprecision accuracyandfigureoutthatthereisnosolutionwiths

≤

^{4.Fig.2showsthedottedplotof}

(

r1

,

r2

)

where

R5

(

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,numericalvaluesuptosingleprecisionaccuracy

R5

(

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,b^TA⁵1

=

¹

/

720.

Remark2.We have presented the numerical search algorithm for R3

( γ )

and R5

(

r1

,

r2

)

satisfying b^TA^q−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

)

satisfyingb^TA^q−11

=

¹

/

q

!,

^q

=

⁵

,

6

,

7 whiler5

,

r6

,

r7arechosentofixb^TA^q−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 of

γ

tosatisfy theorderconditions.Wedenoteasr₁

=

^r2

= · · · =

^rs

= γ

andwehavecorrespondingcoefficientsA and b as in (25).Forthefirst-orderaccuracy,wehave

b^T1

=

²



s i=¹

r_i

=

^2s

γ ₌

1

,

(46)

andthus

γ =

¹

/(

2s

)

.Forthethird-orderaccuracy,wehave b^TA

(

A1

) =

²

γ

s−1



i=¹

⎛

⎝ 

ⁱ⁻¹

j=¹

4

γ

²j

+

²

γ

²i

⎞

⎠ + γ

⎛

⎝ 

^s−1

j=¹

4

γ

²j

+

²

γ

²s

⎞

⎠

=

⁴

γ

³

s−1



i=¹

i²

+

²

γ

³s²

=

² 3

γ

³s



2s²

+

¹

=

^2s2

+

¹ 12s²

=

¹

6

.

(47)

Therefore,eventhesinglydiagonalstrategyfortheSMSmethodcannotsurpassthethird-orderconditions.



Remark3.Infact,aSMSmethodwithasinglydiagonalcoefficient vector Rs

=

^[

γ , γ , · · · , γ

] where

γ =

¹

/(

2s

)

alsoisthe second-orderaccuratemethod,becauseitalwayssatisfiesthesecond-orderconditionb^TA1

=

¹

/

2.

Next, we need to remark that the proposed SMS methods differ from symplectic RK methods in [16,17]. It is well knownthatthealgebraic stabilityconditionb_ib_j

−

^bia_{i j}

−

^bja_{i 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.