Jou nal o Compu a ional and Applied Ma hema ics 453 (2025) 116158
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E icien exponen ial Rosenb ock me hods ill o de ou
B. Cano a,∗,1, M.J. Mo e ab
aIMUVA, Depa amen o de Ma emá ica Aplicada, Facul ad de Ciencias, Uni e sidad de Valladolid, Paseo de Belén 7, 47011 Valladolid, Spain
bIMUVA, Depa amen o de Análisis Económico y Economía Cuan i a i a, Facul ad de Ciencias Económicas y Emp esa iales, Uni e sidad
Complu ense de Mad id, Campus de Somosaguas, Pozuelo de Ala cón, 28223 Mad id, Spain
ARTICLE INFO
Keywo ds:
Exponen ial Rosenb ock me hods
Nonlinea eac ion–di usion p oblems
A oiding o de educ ion in ime
E iciency
ABSTRACT
In a p e ious pape , a echnique was desc ibed o a oid o de educ ion wi h exponen ial
Rosenb ock me hods when in eg a ing ini ial bounda y alue p oblems wi h ime-dependen
bounda y condi ions. Tha equi es o calcula e some in o ma ion on he bounda y om he
gi en da a. In he p esen pape we p o e ha , unde some assump ions on he coe icien s
o he me hod which a e mainly always sa is ied, no nume ical di e en ia ion is equi ed o
app oxima e ha in o ma ion in o de o achie e o de 4 o pa abolic p oblems wi h Di ichle
bounda y condi ions. Wi h Robin/Neumann ones, jus nume ical di e en ia ion in ime may be
necessa y o o de 4, bu none o o de ≤3.
Fu he mo e, as wi h his echnique i is no necessa y o impose any s i o de condi ions,
in sea ch o e iciency, we ecommend some me hods o classical o de s 2, 3 and 4 and we gi e
some compa isons wi h se e al me hods in he li e a u e, wi h he co esponding s i o de .
1. In oduc ion
I is well known ha s i o dina y di e en ial sys ems a e ypically hose o which an explici in eg a ion wi h s anda d me hods
is no possible due o a lack o 𝐴-s abili y [1]. When in eg a ing in space pa ial di e en ial equa ions, he space disc e ized sys em is
s i because he eigen alues associa ed o he disc e iza ion o he spa ial di e en ia ion ope a o can be in ini ely la ge in modulus
when he space g id is e ined.
Exponen ial me hods ha e been de eloped in he li e a u e in o de o ge a s able in eg a ion o s i sys ems in an ‘explici ’
way [2]. Al hough exponen ial- ype unc ions o ma ices applied o e ec o s ha e o be calcula ed wi h hese me hods, he ecen
imp o emen o K ylo echniques o app oxima e hem has made hese me hods a aluable ool o in eg a e some ini ial bounda y
alue p oblems [3–5].
Howe e , he phenomenon o o de educ ion which al eady u ns up when in eg a ing his ype o p oblems wi h s anda d
me hods also u ns up wi h exponen ial me hods. Mo e explici ly, when he bounda y condi ions a e no pe iodic o do no sa is y
enough condi ions o annihila ion on he bounda y, he o de o accu acy o he ime in eg a o s when in eg a ing he space
disc e ized sys em is smalle han he o de o accu acy which is obse ed when in eg a ing a nons i ODE. Some imes con e gence
is e en los [6]. Because o ha , es ic i e s i o de condi ions on he coe icien s o exponen ial Runge–Ku a me hods ha e been
i s ly sugges ed in he li e a u e so as o achie e he desi ed o de o accu acy when in eg a ing semilinea pa abolic p oblems wi h
anishing bounda y condi ions [7]. La e , ano he echnique has been gi en in o de o a oid ha o de educ ion wi hou ha ing
o impose es ic ions on he coe icien s [8–11]. Mo eo e , he la e echnique is alid o ime-dependen bounda y condi ions.
∗Co esponding au ho .
E-mail add esses: [email p o ec ed] (B. Cano), [email p o ec ed] (M.J. Mo e a).
1This esea ch was suppo ed by Jun a de Cas illa y León and Fede h ough p ojec VA169P20
h ps://doi.o g/10.1016/j.cam.2024.116158
Recei ed 26 July 2023; Recei ed in e ised o m 12 Ma ch 2024
Jou nal o Compu a ional and Applied Ma hema ics 453 (2025) 116158
2
B. Cano and M.J. Mo e a
Fu he mo e, by assuming ha he Jacobian o he ec o ield can be easily calcula ed, ha ex a in o ma ion can be used
wi h Rosenb ock me hods so as o achie e a desi ed accu acy wi h less s ages han hei Runge–Ku a coun e pa s. Because o
ha , o he in eg a ion o eac ion–di usion ini ial bounda y alue p oblems, s i o de condi ions ha e also been s udied in [12]
o exponen ial Rosenb ock me hods. Mo eo e , many pa icula exponen ial Rosenb ock me hods ha e been cons uc ed sa is ying
hose condi ions in o de o achie e a desi ed accu acy while ying o be as e icien as possible [3,12–16].
In con as , in [17], a echnique is sugges ed o a oid o de educ ion wi h any exponen ial Rosenb ock me hod o any classical
o de wi hou imposing hose s i o de condi ions. Fo ha , some in e media e ini ial bounda y p oblems a e conside ed o which
he bounda y alues ha e o be calcula ed in e ms o da a. I analy ic exp essions o he la e a e known, in [17] i is s a ed ha
hose bounda y alues can be exac ly calcula ed in o de o ge local o de ≥2. Howe e , in o de o ge local o de ≥3, nume ical
di e en ia ion is in gene al equi ed ei he in space o in ime. When ha in space is necessa y, a weak CFL condi ion is needed o
p o e con e gence [17]. Al hough he la e condi ion is much less es ic i e han ha equi ed when in eg a ing wi h s anda d
explici Runge–Ku a me hods, i would be be e no o equi e i and no o eso o nume ical di e en ia ion o he ease o
implemen a ion.
Ou aim in his pape is o p o e ha , unde some simpli ying assump ions (which a e sa is ied by mainly all al eady buil
exponen ial Rosenb ock me hods), nume ical di e en ia ion in space is no equi ed o achie e local o de 3and 4, so ha
no CFL condi ion is necessa y hen. Mo eo e , nume ical di e en ia ion in ime is jus necessa y o achie e local o de 4wi h
Robin/Neumann bounda y condi ions, bu no wi h Di ichle ones and ha o de and nei he o achie e local o de 3. We ema k
ha , when in eg a ing wi h s anda d Rosenb ock me hods, a echnique was also gi en in [18] o a oid o de educ ion and no
nume ical di e en ia ion in space was ei he equi ed o achie e local o de ≤4. The e o e, he conclusions in his pape a e simila
o hose in [18] o s anda d Rosenb ock me hods. The ad an age o using exponen ial Rosenb ock me hods ins ead o s anda d ones
co espond o p oblems whe e a good p econdi ione is no a ailable o sol e linea sys ems in an e icien way and i is he e o e
mo e ecommendable o use K ylo echniques o app oxima e exponen ials o ma ices applied o e ec o s [16,19–21].
In any case, coming back o he abo e conside a ions on exponen ial Rosenb ock me hods, in his pape we also ecommend
some pa icula ones o ge global o de s 2, 3 and 4 ( he la e o pa abolic p oblems in which, by a summa ion-by-pa s a gumen ,
he local o de coincides wi h he global one). We emind ha he local e o co esponds o he e o a e jus one s ep while he
global e o is he e o a e he equi ed s eps o ge a inal ime. As no s i o de condi ions ha e o be imposed, he e a e mo e
pa ame e s o play wi h in o de o achie e a cheape compu a ion.
The pape is s uc u ed as ollows. Sec ion 2gi es some p elimina ies and pa icula izes he ob ained o mulas o he ull
disc e iza ion in [17] in o de o ge local o de 𝑝+ 1 o 𝑝= 1,2,3when he me hod has classical o de a leas 𝑝. Sec ion 3jus i ies
how hose o mulas g ea ly simpli y unde he p ecise condi ions on he coe icien s o he me hod which a e s a ed he e. (Tha
simpli ica ion mus no be seen in he leng h o o mulas, bu in he ac ha he equi ed bounda y alues a e easie o calcula e
and ha he linea combina ion o 𝜑𝑗- unc ions o ma ices applied o e ec o s is gi en, so ha K ylo sub ou ines can be di ec ly
applied.) Sec ion 4gi es some heo ems and ema ks which jus i y ha he simpli ying assump ions a e sa is ied by basically e e y
cons uc ed me hod. Sec ion 5gi es a ecommenda ion o me hods o o de 2,3and 4and, inally, Sec ion 6shows con e gence
ables and a nume ical compa ison in CPU ime wi h o he me hods in he li e a u e.
2. P elimina ies
We will assume ha he p oblem o in eg a e can be w i en as
𝑢(𝑡) = 𝐴𝑢(𝑡) + 𝛹(𝑢(𝑡)) + ℎ(𝑡),0≤𝑡≤𝑇 , (1)
𝑢(0) = 𝑢0,
𝜕𝑢(𝑡) = 𝑔(𝑡),
whe e ⋅deno es di e en ia ion wi h espec o ime and whe e 𝐴∶𝐷(𝐴)⊂ 𝑋 →𝑋is a linea di e en ial ope a o de ined in a
subse o he Banach space 𝑋=𝐿∞(
𝛺, C), whe e 𝛺is a bounded domain in R𝑑and he sup emum no m is conside ed. On he o he
hand, 𝜕∶𝑋→𝑌co esponds o a bounda y ope a o which a i es a ano he Banach space 𝑌=𝐿∞(𝜕
𝛺)wi h he same sup emum
no m. We will assume hypo heses (A1)–(A9) in [17] and, mo e pa icula ly, i (1) is eal and he solu ion s ays in he in e al 𝐼,
ha 𝛹∈𝐶2(𝐼, R)and, i (1) is complex, ha 𝛹is holomo phic in a egion whe e he solu ion s ays. Mo eo e , we will assume ha
ℎ∈𝐶2([0, 𝑇 ], 𝑋).
As s a ed in [17,18] h ough [22–24], (A1)–(A4) gua an ee ha p oblem (1) is well-posed. Howe e , mo e pa icula assump ions
on he egula i y o 𝑢and 𝛹a e s a ed in (A5)–(A9) and in Theo em 3.1 in [17] in o de o ge he desi ed local o de 𝑝+1 whene e
he me hod has non-s i global o de ≥𝑝. We do no s a e hem he e o he sake o b e i y, bu i is he e jus i ied ha all exp essions
which u n up in he es o he pape exis . Ano he issue would be o jus i y in e ms o he da a o he p oblem ha he exac
solu ion is egula enough. Al hough ha would be e y in e es ing, i is no an aim o his pape . In any case, we e e o [25–28]
o bounds o some de i a i es o he exac solu ion in he linea case. (The las pape s a e ocused on singula ly pe u bed p oblems,
which a e no necessa ily he case he e, bu he bounds which a e ob ained he e a e also alid o no so small pa ame e s.)
Gene al Rosenb ock exponen ial me hods a e de e mined by some coe icien s 𝑐1,…, 𝑐𝑠and some coe icien unc ions in a
Bu che ableau
𝑎𝑖,𝑗 (𝑧) =
𝑟
∑
𝑙=1
𝜆𝑖,𝑗,𝑙𝜑𝑙(𝑐𝑖𝑧), 𝑖 = 1,…, 𝑠, 𝑗 = 1,…, 𝑖 − 1,
Jou nal o Compu a ional and Applied Ma hema ics 453 (2025) 116158
3
B. Cano and M.J. Mo e a
𝑏𝑖(𝑧) =
𝑟
∑
𝑙=1
𝜇𝑖,𝑙𝜑𝑙(𝑧), 𝑖 = 1,…, 𝑠, (2)
whe e
𝜑𝑙(𝑧) = ∫1
0
𝑒(1−𝜃)𝑧𝜃𝑙−1
(𝑙− 1)! 𝑑𝜃, 𝑙 ≥1,
and whe e i is assumed ha
𝑖−1
∑
𝑗=1
𝑎𝑖𝑗 (0) = 𝑐𝑖.(3)
Fo an au onomous ODE di e en ial sys em o he ype
𝑈(𝑡) = 𝐹(𝑈(𝑡)),(4)
he nume ical solu ion 𝑈𝑛+1 a ime 𝑡𝑛+1 =𝑡𝑛+𝑘is gi en om he nume ical solu ion 𝑈𝑛a ime 𝑡𝑛 h ough he ollowing o mulas
𝐾𝑛,𝑖 =𝑒𝑐𝑖𝑘𝐽𝑛𝑈𝑛+𝑘
𝑖−1
∑
𝑗=1
𝑎𝑖𝑗 (𝑘𝐽𝑛)𝐺𝑛(𝐾𝑛,𝑗 ), 𝑖 = 1,…, 𝑠, (5)
𝑈𝑛+1 =𝑒𝑘𝐽𝑛𝑈𝑛+𝑘
𝑠
∑
𝑖=1
𝑏𝑖(𝑘𝐽𝑛)𝐺𝑛(𝐾𝑛,𝑖),(6)
whe e
𝐽𝑛=𝐹′(𝑈𝑛), 𝐺𝑛(𝑈) = 𝐹(𝑈) − 𝐽𝑛𝑈. (7)
When he p oblem is non-au onomous, by ew i ing
𝑈(𝑡) = 𝐹(𝑡, 𝑈(𝑡))
as an au onomous one, he me hod eads
𝐾𝑛,𝑖 =𝑒𝑐𝑖𝑘𝐽𝑛𝑈𝑛+𝑐𝑖𝑘𝑡𝑛𝜑1(𝑐𝑖𝑘𝐽𝑛)𝑉𝑛+𝑘
𝑖−1
∑
𝑗=1
𝑟
∑
𝑙=1
𝜆𝑖,𝑗,𝑙[𝜑𝑙(𝑐𝑖𝑘𝐽𝑛)[𝐹(𝑡𝑛,𝑗 , 𝐾𝑛,𝑗 ) − 𝑡𝑛,𝑗 𝑉𝑛−𝐽𝑛𝐾𝑛,𝑗 ] + 𝑐𝑖𝑘𝜑𝑙+1(𝑐𝑖𝑘𝐽𝑛)𝑉𝑛],
𝑈𝑛+1 =𝑒𝑘𝐽𝑛𝑈𝑛+𝑘𝑡𝑛𝜑1(𝑘𝐽𝑛)𝑉𝑛+𝑘
𝑠
∑
𝑖=1
𝑟
∑
𝑙=1
𝜇𝑖,𝑙[𝜑𝑙(𝑘𝐽𝑛)[𝐹(𝑡𝑛,𝑖, 𝐾𝑛,𝑖) − 𝑡𝑛,𝑖𝑉𝑛−𝐽𝑛𝐾𝑛,𝑖] + 𝑘𝜑𝑙+1(𝑘𝐽𝑛)𝑉𝑛],(8)
whe e 𝑡𝑛,𝑗 =𝑡𝑛+𝑐𝑗𝑘, and
𝑉𝑛=𝜕𝐹
𝜕𝑡 (𝑡𝑛, 𝑈𝑛), 𝐽𝑛=𝜕𝐹
𝜕𝑈 (𝑡𝑛, 𝑈𝑛).
On he o he hand, we will conside a gene al space disc e iza ion o he di e en ial ope a o 𝐴, such ha , when applied o he
ellip ic p oblem
𝐴𝑢 =𝐹 , 𝜕𝑢 =𝑔,
he nodal alues on he g id a e gi en by he solu ion 𝑈ℎ∈C𝑁o his sys em
𝐴ℎ,0𝑈ℎ+𝐶ℎ𝑔=𝑃ℎ𝐹+𝐷ℎ𝜕𝐹 ,
whe e 𝐴ℎ,0disc e izes 𝐴 es ic ed o Ke (𝜕),𝑃ℎis he nodal p ojec ion and 𝐶ℎ, 𝐷ℎ∶𝑌→C𝑁a e o he linea ope a o s o e
unc ions on he bounda y.
We will assume hypo heses (H1)–(H3) in [17] and will conside he ma ix
𝐽𝑛,ℎ,0=𝐴ℎ,0+𝛹′(𝑈𝑛
ℎ),
which disc e izes he Jacobian wi h espec o 𝑈o he ec o ield which de ines (1) a 𝑡=𝑡𝑛=𝑛𝑘, whe e 𝑘is he imes epsize.
In [17], a echnique is sugges ed o achie e local o de 𝑝+ 1 wi h a me hod which has classical global o de a leas 𝑝. Fo he
p ecise alues 𝑝= 1,2,3, he modi ied exponen ial Rosenb ock me hod o achie e ha goal when in eg a ing he non-au onomous
p oblem (1) eads as ollows.
Fo 𝑝= 1,
𝐾𝑛,𝑖,ℎ =𝑒𝑐𝑖𝑘𝐽𝑛,ℎ,0𝑈𝑛
ℎ+𝑐𝑖𝑘𝜑1(𝑐𝑖𝑘𝐽𝑛,ℎ,0)[𝑡𝑛𝑃ℎ
ℎ(𝑡𝑛) + 𝐶ℎ𝜕𝑢(𝑡𝑛)]
+𝑘
𝑖−1
∑
𝑗=1
𝑟
∑
𝑙=1
𝜆𝑖,𝑗,𝑙[𝜑𝑙(𝑐𝑖𝑘𝐽𝑛,ℎ,0)𝐺𝑛,𝑗,ℎ +𝑐𝑖𝑘𝜑𝑙+1(𝑐𝑖𝑘𝐽𝑛,ℎ,0)𝑃ℎ
ℎ(𝑡𝑛)],
𝑈𝑛+1
ℎ=𝑒𝑘𝐽𝑛,ℎ,0𝑈𝑛
ℎ+𝑘𝜑1(𝑘𝐽𝑛,ℎ,0)[𝑡𝑛𝑃ℎ
ℎ(𝑡𝑛) + 𝐶ℎ𝜕𝑢(𝑡𝑛) − 𝐷ℎ𝜕
𝐽(𝑡𝑛)𝑢(𝑡𝑛)]
+𝑘2𝜑2(𝑘𝐽𝑛,ℎ,0)𝐶ℎ𝜕[
𝐽(𝑡𝑛)𝑢(𝑡𝑛) + 𝑡𝑛
ℎ(𝑡𝑛)]
Jou nal o Compu a ional and Applied Ma hema ics 453 (2025) 116158
4
B. Cano and M.J. Mo e a
+𝑘
𝑠
∑
𝑖=1
𝑟
∑
𝑙=1
𝜇𝑖,𝑙[𝜑𝑙(𝑘𝐽𝑛,ℎ,0)𝐺𝑛,𝑖,ℎ
+𝑘𝜑𝑙+1(𝑘𝐽𝑛,ℎ,0)[𝑃ℎ
ℎ(𝑡𝑛) + 𝐶ℎ𝜕
𝐺𝑛]](9)
whe e
𝐺𝑛,𝑗,ℎ =𝛹(𝐾𝑛,𝑗,ℎ) + 𝑃ℎℎ(𝑡𝑛,𝑗 ) − 𝑡𝑛,𝑗 𝑃ℎ
ℎ(𝑡𝑛) − diag(𝛹′(𝑈𝑛
ℎ))𝐾𝑛,𝑗,ℎ,
wi h 𝑡𝑛,𝑗 =𝑡𝑛+𝑐𝑗𝑘, and
𝐺𝑛=𝛹(𝑢(𝑡𝑛)) + ℎ(𝑡𝑛) − 𝑡𝑛
ℎ(𝑡𝑛) − 𝛹′(𝑢(𝑡𝑛))𝑢(𝑡𝑛).(10)
As o 𝑝= 2, we ge
𝐾𝑛,𝑖,ℎ =𝑒𝑐𝑖𝑘𝐽𝑛,ℎ,0𝑈𝑛
ℎ
+𝑐𝑖𝑘𝜑1(𝑐𝑖𝑘𝐽𝑛,ℎ,0)[𝑡𝑛𝑃ℎ
ℎ(𝑡𝑛) + 𝐶ℎ𝜕𝑢(𝑡𝑛) − 𝐷ℎ𝜕
𝐽(𝑡𝑛)𝑢(𝑡𝑛)]
+ (𝑐𝑖𝑘)2𝜑2(𝑐𝑖𝑘𝐽𝑛,ℎ,0)𝐶ℎ𝜕[
𝐽(𝑡𝑛)𝑢(𝑡𝑛) + 𝑡𝑛
ℎ(𝑡𝑛)]
+𝑘
𝑖−1
∑
𝑗=1
𝑟
∑
𝑙=1
𝜆𝑖,𝑗,𝑙[𝜑𝑙(𝑐𝑖𝑘𝐽𝑛,ℎ,0)𝐺𝑛,𝑗,ℎ +𝑐𝑖𝑘𝜑𝑙+1(𝑐𝑖𝑘𝐽𝑛,ℎ,0)[𝑃ℎ
ℎ(𝑡𝑛) + 𝐶ℎ𝜕
𝐺𝑛]],
𝑈𝑛+1
ℎ=𝑒𝑘𝐽𝑛,ℎ,0𝑈𝑛
ℎ+𝑘𝜑1(𝑘𝐽𝑛,ℎ,0)[𝑡𝑛𝑃ℎ
ℎ(𝑡𝑛) + 𝐶ℎ𝜕𝑢(𝑡𝑛) − 𝐷ℎ𝜕
𝐽(𝑡𝑛)𝑢(𝑡𝑛)]
+𝑘2𝜑2(𝑘𝐽𝑛,ℎ,0)[𝐶ℎ𝜕[
𝐽(𝑡𝑛)𝑢(𝑡𝑛) + 𝑡𝑛
ℎ(𝑡𝑛)] − 𝐷ℎ𝜕[
𝐽(𝑡𝑛)2𝑢(𝑡𝑛) + 𝑡𝑛
𝐽(𝑡𝑛)
ℎ(𝑡𝑛)]]
+𝑘3𝜑3(𝑘𝐽𝑛,ℎ,0)𝐶ℎ𝜕[
𝐽(𝑡𝑛)2𝑢(𝑡𝑛) + 𝑡𝑛
𝐽(𝑡𝑛)
ℎ(𝑡𝑛)]
+𝑘
𝑠
∑
𝑖=1
𝑟
∑
𝑙=1
𝜇𝑖,𝑙[𝜑𝑙(𝑘𝐽𝑛,ℎ,0)𝐺𝑛,𝑖,ℎ +𝑘𝜑𝑙+1(𝑘𝐽𝑛,ℎ,0)[𝑃ℎ
ℎ(𝑡𝑛) + 𝐶ℎ𝜕
𝐺𝑛−𝐷ℎ𝜕
𝐽(𝑡𝑛)
𝐺𝑛]
+𝑘2𝜑𝑙+2(𝑘𝐽𝑛,ℎ,0)𝐶ℎ𝜕[
𝐽(𝑡𝑛)
𝐺𝑛+
ℎ(𝑡𝑛)]].(11)
Finally, o 𝑝= 3, we ha e
𝐾𝑛,𝑖,ℎ =𝑒𝑐𝑖𝑘𝐽𝑛,ℎ,0𝑈𝑛
ℎ
+𝑐𝑖𝑘𝜑1(𝑐𝑖𝑘𝐽𝑛,ℎ,0)[𝑡𝑛𝑃ℎ
ℎ(𝑡𝑛) + 𝐶ℎ𝜕𝑢(𝑡𝑛) − 𝐷ℎ𝜕
𝐽(𝑡𝑛)𝑢(𝑡𝑛)]
+ (𝑐𝑖𝑘)2𝜑2(𝑐𝑖𝑘𝐽𝑛,ℎ,0)[𝐶ℎ𝜕[
𝐽(𝑡𝑛)𝑢(𝑡𝑛) + 𝑡𝑛
ℎ(𝑡𝑛)] − 𝐷ℎ𝜕[
𝐽(𝑡𝑛)2𝑢(𝑡𝑛) + 𝑡𝑛
𝐽(𝑡𝑛)
ℎ(𝑡𝑛)]]
+ (𝑐𝑖𝑘)3𝜑3(𝑐𝑖𝑘𝐽𝑛,ℎ,0)𝐶ℎ𝜕[
𝐽(𝑡𝑛)2𝑢(𝑡𝑛) + 𝑡𝑛
𝐽(𝑡𝑛)
ℎ(𝑡𝑛)]
+𝑘
𝑖−1
∑
𝑗=1
𝑟
∑
𝑙=1
𝜆𝑖,𝑗,𝑙[𝜑𝑙(𝑐𝑖𝑘𝐽𝑛,ℎ,0)𝐺𝑛,𝑗,ℎ +𝑐𝑖𝑘𝜑𝑙+1(𝑐𝑖𝑘𝐽𝑛,ℎ,0)[𝑃ℎ
ℎ(𝑡𝑛) + 𝐶ℎ𝜕
𝐺𝑛−𝐷ℎ𝜕
𝐽(𝑡𝑛)
𝐺𝑛]
+(𝑐𝑖𝑘)2𝜑𝑙+2(𝑐𝑖𝑘𝐽𝑛,ℎ,0)𝐶ℎ𝜕[
𝐽(𝑡𝑛)
𝐺𝑛+
ℎ(𝑡𝑛)]].
𝑈𝑛+1
ℎ=𝑒𝑘𝐽𝑛,ℎ,0𝑈𝑛
ℎ+𝑘𝜑1(𝑘𝐽𝑛,ℎ,0)[𝑡𝑛𝑃ℎ
ℎ(𝑡𝑛) + 𝐶ℎ𝜕𝑢(𝑡𝑛) − 𝐷ℎ𝜕
𝐽(𝑡𝑛)𝑢(𝑡𝑛)]
+𝑘2𝜑2(𝑘𝐽𝑛,ℎ,0)[𝐶ℎ𝜕[
𝐽(𝑡𝑛)𝑢(𝑡𝑛) + 𝑡𝑛
ℎ(𝑡𝑛)] − 𝐷ℎ𝜕[
𝐽(𝑡𝑛)2𝑢(𝑡𝑛) + 𝑡𝑛
𝐽(𝑡𝑛)
ℎ(𝑡𝑛)]]
+𝑘3𝜑3(𝑘𝐽𝑛,ℎ,0)[𝐶ℎ𝜕[
𝐽(𝑡𝑛)2𝑢(𝑡𝑛) + 𝑡𝑛
𝐽(𝑡𝑛)
ℎ(𝑡𝑛)] − 𝐷ℎ𝜕[
𝐽(𝑡𝑛)3𝑢(𝑡𝑛) + 𝑡𝑛
𝐽(𝑡𝑛)2
ℎ(𝑡𝑛)]]
+𝑘4𝜑4(𝑘𝐽𝑛,ℎ,0)𝐶ℎ𝜕[
𝐽(𝑡𝑛)3𝑢(𝑡𝑛) + 𝑡𝑛
𝐽(𝑡𝑛)2
ℎ(𝑡𝑛)]
+𝑘
𝑠
∑
𝑖=1
𝑟
∑
𝑙=1
𝜇𝑖,𝑙[𝜑𝑙(𝑘𝐽𝑛,ℎ,0)𝐺𝑛,𝑖,ℎ +𝑘𝜑𝑙+1(𝑘𝐽𝑛,ℎ,0)[𝑃ℎ
ℎ(𝑡𝑛) + 𝐶ℎ𝜕
𝐺𝑛,𝑖 −𝐷ℎ𝜕
𝐽(𝑡𝑛)
𝐺𝑛]
+𝑘2𝜑𝑙+2(𝑘𝐽𝑛,ℎ,0)[𝐶ℎ𝜕[
𝐽(𝑡𝑛)
𝐺𝑛+
ℎ(𝑡𝑛)] − 𝐷ℎ𝜕[
𝐽(𝑡𝑛)2
𝐺𝑛+
𝐽(𝑡𝑛)
ℎ(𝑡𝑛)]]
+𝑘3𝜑𝑙+3(𝑘𝐽𝑛,ℎ,0)𝐶ℎ𝜕[
𝐽(𝑡𝑛)2
𝐺𝑛+
𝐽(𝑡𝑛)
ℎ(𝑡𝑛)]].(12)
whe e
𝐺𝑛,𝑖 =𝛹(𝑢(𝑡𝑛)) + ℎ(𝑡𝑛) − 𝑡𝑛
ℎ(𝑡𝑛) − 𝛹′(𝑢(𝑡𝑛))𝑢(𝑡𝑛) + 𝑐2
𝑖𝑘2
2[𝛹′′(𝑢(𝑡𝑛)) 𝑢(𝑡𝑛)2+
ℎ(𝑡𝑛)].
Now we mus see how o calcula e he e ms on he bounda y aking in o accoun ha ou da a is jus 𝜕𝑢(𝑡) = 𝑔(𝑡),ℎ(𝑡)and 𝑢0in
(1). As s a ed in [17], o 𝑝≥2, i is in p inciple necessa y o eso o nume ical di e en ia ion ei he in space o in ime o bo h
Di ichle and Robin/Neumann bounda y condi ions in o de o app oxima e hose bounda y alues. In o de o a oid ha as a
as possible, we will see ha many imes some simpli ica ions can be pe o med which allow o calcula e he equi ed bounda ies
exac ly in e ms o da a.
Jou nal o Compu a ional and Applied Ma hema ics 453 (2025) 116158
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B. Cano and M.J. Mo e a
3. Fu he simpli ica ions and calcula ion o equi ed bounda ies in e ms o da a
In his sec ion, we will see ha , unde he assump ions
𝑠
∑
𝑖=1
𝜇𝑖,1= 1,
𝑠
∑
𝑖=1
𝜇𝑖,𝑙 = 0 (𝑙= 2,…, 𝑟),(13)
𝑖−1
∑
𝑗=1
𝜆𝑖,𝑗,1=𝑐𝑖,
𝑖−1
∑
𝑗=1
𝜆𝑖,𝑗,𝑙 = 0 (𝑙= 2,…, 𝑟), 𝑖 = 1,…, 𝑠, (14)
some e ms in he gene al exp essions (9),(11),(12) can be simpli ied. In a i s place, he equi ed e ms on he bounda y can
be calcula ed in a much mo e di ec way in e ms o da a, i.e. wi hou eso ing o nume ical di e en ia ion ei he in space no
in ime, excep o he case o Robin/Neumann bounda y condi ions and 𝑝= 3, o which nume ical di e en ia ion o he i s
de i a i e in ime will be equi ed. Secondly, some o he e ms no conce ning he bounda y also simpli y unde hose assump ions
and mo eo e , he e alua ion o 𝐺𝑛,𝑖,ℎ can be pe o med excep o e ms which lead o 𝑂(𝑘𝑝+2)- esidues in 𝑈𝑛+1
ℎand he e o e do no
change ei he he local nei he he global o de . (This las simpli ica ion jus conce ns 𝑝= 1 o bo h he equa ions on he s ages and
he solu ion and 𝑝= 2 o he s ages. I co esponds o he use o
𝐺𝑛,𝑖,ℎ in he o mulas below.) Mo eo e , we will ga he oge he all
e ms in he same 𝜑𝑗so ha K ylo sub ou ines can be di ec ly applied o calcula e a linea combina ion o hose ma ix unc ions
applied o e he co esponding ec o s.
3.1. 𝑝= 1
We no ice ha , in his case, o he s ages, jus he e m 𝜕𝑢(𝑡𝑛) = 𝑔(𝑡𝑛)on he bounda y is equi ed (see (9)). Mo eo e , aking
in o accoun ha
𝐺𝑛,𝑗,ℎ =𝛹(𝐾𝑛,𝑗,ℎ) + 𝑃ℎℎ(𝑡𝑛) − 𝑡𝑛𝑃ℎ
ℎ(𝑡𝑛) − diag(𝛹′(𝑈𝑛
ℎ))𝐾𝑛,𝑗,ℎ +𝑂(𝑘2),(15)
and he i s pa o (14), he e ms in 𝜑1(𝑐𝑖𝑘𝐽𝑛,ℎ,0)wi hou bounda ies can be simpli ied o
𝑘𝜑1(𝑐𝑖𝑘𝐽𝑛,ℎ,0)[𝑐𝑖𝑃ℎℎ(𝑡𝑛) + ∑𝜆𝑖,𝑗,1
𝐺𝑛,𝑗,ℎ],
whe e
𝐺𝑛,𝑗,ℎ =𝛹(𝐾𝑛,𝑗,ℎ) − diag(𝛹′(𝑈𝑛
ℎ))𝐾𝑛,𝑗,ℎ.
On he o he hand, he e m in 𝜑2(𝑐𝑖𝑘𝐽𝑛,ℎ,0), conside ing he second pa o (14) is
𝑘𝜑2(𝑐𝑖𝑘𝐽𝑛,ℎ,0)[∑𝜆𝑖,𝑗,2
𝐺𝑛,𝑗,ℎ +𝑐2
𝑖𝑘𝑃ℎ
ℎ(𝑡𝑛)].
Summing up,
𝐾𝑛,𝑖,ℎ =𝑒𝑐𝑖𝑘𝐽𝑛,ℎ,0𝑈𝑛
ℎ+𝑘𝜑1(𝑐𝑖𝑘𝐽𝑛,ℎ,0)[𝑐𝑖[𝑃ℎℎ(𝑡𝑛) + 𝐶ℎ𝜕𝑢(𝑡𝑛)] + ∑𝜆𝑖,𝑗,1
𝐺𝑛,𝑗,ℎ]
+𝑘𝜑2(𝑐𝑖𝑘𝐽𝑛,ℎ,0)[∑𝜆𝑖,𝑗,2
𝐺𝑛,𝑗,ℎ +𝑐2
𝑖𝑘𝑃ℎ
ℎ(𝑡𝑛)]
+𝑘
𝑟
∑
𝑙=3
𝜑𝑙(𝑐𝑖𝑘𝐽𝑛,ℎ,0)∑𝜆𝑖,𝑗,𝑙
𝐺𝑛,𝑗,ℎ.
As o 𝑈𝑛+1
ℎin (9), mul iplying 𝑘𝜑1(𝑘𝐽𝑛,ℎ,0)𝐶ℎ, jus 𝜕𝑢(𝑡𝑛) = 𝑔(𝑡𝑛) u ns up again; on he o he hand, he e m in 𝐷ℎ𝜕
𝐽(𝑡𝑛)𝑢(𝑡𝑛)can
be w i en as
−𝑘𝜑1(𝑘𝐽𝑛,ℎ,0)𝐷ℎ𝜕[𝑢(𝑡𝑛) − 𝛹(𝑢(𝑡𝑛)) + 𝛹′(𝑢(𝑡𝑛))𝑢(𝑡𝑛) − ℎ(𝑡𝑛)].(16)
This e m can be exac ly calcula ed when conside ing Di ichle bounda y condi ions and, wi h Robin/Neumann bounda y condi ions,
i can be app oxima ed h ough he nume ical app oxima ion a he bounda y gi en by he space disc e iza ion o (1) i sel , bu
wi hou eso ing o nume ical di e en ia ion. On he o he hand, no ice ha , by using he i s pa o (13), some e ms cancel and
he e m mul iplying 𝑘2𝜑2(𝑘𝐽𝑛,ℎ,0)𝐶ℎis jus 𝜕 𝑢(𝑡𝑛). The e o e, ha bounda y can be calcula ed exac ly in e ms o da a as 𝑔(𝑡𝑛). As
o he e ms in 𝑘2𝜑𝑙+1(𝑘𝐽𝑛,ℎ,0)𝐶ℎwi h 𝑙≥2, no ice ha hey anish in (9) because o he second pa o (13). Mo eo e , conside ing
(15) and he i s pa o (13), he e ms in 𝜑1(𝑘𝐽𝑛,ℎ,0)wi hou bounda ies can be simpli ied o
𝑘𝜑1(𝑘𝐽𝑛,ℎ,0)[𝑃ℎℎ(𝑡𝑛) + ∑𝜇𝑖,1
𝐺𝑛,𝑖,ℎ].
In a simila way, bu using now also he second pa o (13), he e ms in 𝜑2(𝑘𝐽𝑛,ℎ,0)and 𝜑𝑙(𝑘𝐽𝑛,ℎ,0)wi hou bounda ies, wi h 𝑙≥3,
can be simpli ied o
𝑘𝜑2(𝑘𝐽𝑛,ℎ,0)[∑𝜇𝑖,2
𝐺𝑛,𝑖,ℎ +𝑘𝑃ℎ
ℎ(𝑡𝑛)]
𝑘𝜑𝑙(𝑘𝐽𝑛,ℎ,0)[∑𝜇𝑖,𝑙
𝐺𝑛,𝑖,ℎ], 𝑙 ≥3.
Summing up, unde assump ions (13),𝑈𝑛+1
ℎin (9) can be simpli ied o
𝑈𝑛+1
ℎ=𝑒𝑘𝐽𝑛,ℎ,0𝑈𝑛
ℎ+𝑘𝜑1(𝑘𝐽𝑛,ℎ,0)[𝑃ℎℎ(𝑡𝑛) + ∑𝜇𝑖,1
𝐺𝑛,𝑖,ℎ +𝐶ℎ𝜕𝑢(𝑡𝑛) − 𝐷ℎ𝜕[𝑢(𝑡𝑛) − 𝛹(𝑢(𝑡𝑛)) + 𝛹′(𝑢(𝑡𝑛))𝑢(𝑡𝑛) − ℎ(𝑡𝑛)]]
Jou nal o Compu a ional and Applied Ma hema ics 453 (2025) 116158
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B. Cano and M.J. Mo e a
+𝑘𝜑2(𝑘𝐽𝑛,ℎ,0)[∑𝜇𝑖,2
𝐺𝑛,𝑖,ℎ +𝑘[𝑃ℎ
ℎ(𝑡𝑛) + 𝐶ℎ𝜕 𝑢(𝑡𝑛)]]+𝑘
𝑟
∑
𝑙=3
𝜑𝑙(𝑘𝐽𝑛,ℎ,0)∑𝜇𝑖,𝑙
𝐺𝑛,𝑖,ℎ.
3.2. 𝑝= 2
Wi h simila a gumen s as hose o 𝑝= 1, he s ages o 𝑝= 2 in (11) can be simpli ied o he ollowing o mulas:
𝐾𝑛,𝑖,ℎ =𝑒𝑐𝑖𝑘𝐽𝑛,ℎ,0𝑈𝑛
ℎ
+𝑘𝜑1(𝑐𝑖𝑘𝐽𝑛,ℎ,0)[∑𝜆𝑖,𝑗,1
𝐺𝑛,𝑗,ℎ +𝑐𝑖[𝑃ℎℎ(𝑡𝑛) + 𝐶ℎ𝜕𝑢(𝑡𝑛) − 𝐷ℎ𝜕[𝑢(𝑡𝑛) − 𝛹(𝑢(𝑡𝑛)) + 𝛹′(𝑢(𝑡𝑛))𝑢(𝑡𝑛) − ℎ(𝑡𝑛)]]]
+𝑘𝜑2(𝑐𝑖𝑘𝐽𝑛,ℎ,0)[∑𝜆𝑖,𝑗,2
𝐺𝑛,𝑗,ℎ +𝑐2
𝑖𝑘[𝑃ℎ
ℎ(𝑡𝑛) + 𝐶ℎ𝜕 𝑢(𝑡𝑛)]]+𝑘
𝑟
∑
𝑙=3
𝜑𝑙(𝑐𝑖𝑘𝐽𝑛,ℎ,0)∑𝜆𝑖,𝑗,𝑙
𝐺𝑛,𝑗,ℎ.
As o 𝑈𝑛+1
ℎin (11), now he e m mul iplying 𝑘2𝜑2(𝑘𝐽𝑛,ℎ,0)𝐶ℎis again 𝜕 𝑢(𝑡𝑛). As o 𝑘2𝜑2(𝑘𝐽𝑛,ℎ,0)𝐷ℎ, we ha e
−𝜕[
𝐽(𝑡𝑛)2𝑢(𝑡𝑛) +
𝐽(𝑡𝑛)[𝛹(𝑢(𝑡𝑛)) + ℎ(𝑡𝑛) − 𝛹′(𝑢(𝑡𝑛))𝑢(𝑡𝑛)]]
= −𝜕
𝐽(𝑡𝑛)𝑢(𝑡𝑛)=−𝜕[𝑢(𝑡𝑛) −
ℎ(𝑡𝑛)],
whe e, o he las equali y, we ha e jus conside ed he di e en ia ion o Eq. (1). On he o he hand, he e m mul iplying in
𝑘3𝜑3(𝑘𝐽𝑛,ℎ,0)𝐶ℎis
𝜕[
𝐽(𝑡𝑛)2𝑢(𝑡𝑛) +
𝐽(𝑡𝑛)[𝛹(𝑢(𝑡𝑛)) + ℎ(𝑡𝑛) − 𝛹′(𝑢(𝑡𝑛))𝑢(𝑡𝑛)] +
ℎ(𝑡𝑛)]
=𝜕[
𝐽(𝑡𝑛)𝑢(𝑡𝑛) +
ℎ(𝑡𝑛)] = 𝜕 𝑢(𝑡𝑛),
and he e m in 𝜑3(𝑘𝐽𝑛,ℎ,0)𝐷ℎ anishes because ∑𝜇𝑖,2= 0. In a simila way, he e ms in 𝜑𝑙(𝑘𝐽𝑛,ℎ,0)𝐶ℎand 𝜑𝑙(𝑘𝐽𝑛,ℎ,0)𝐷ℎ o 𝑙≥4
anish because o he second pa o (13). Conside ing his, 𝑈𝑛+1
ℎin (11) can be calcula ed as
𝑈𝑛+1
ℎ=𝑒𝑘𝐽𝑛,ℎ,0𝑈𝑛
ℎ+𝑘𝜑1(𝑘𝐽𝑛,ℎ,0)[∑𝜇𝑖,1
𝐺𝑛,𝑖,ℎ +𝐶ℎ𝜕𝑢(𝑡𝑛) − 𝐷ℎ𝜕[𝑢(𝑡𝑛) − 𝛹(𝑢(𝑡𝑛)) + 𝛹′(𝑢(𝑡𝑛))𝑢(𝑡𝑛) − ℎ(𝑡𝑛)] ]
+𝑘𝜑2(𝑘𝐽𝑛,ℎ,0)[∑𝜇𝑖,2
𝐺𝑛,𝑖,ℎ +𝑘[𝑃ℎ
ℎ(𝑡𝑛) + 𝐶ℎ𝜕 𝑢(𝑡𝑛) − 𝐷ℎ𝜕[𝑢(𝑡𝑛) −
ℎ(𝑡𝑛)]]]
+𝑘𝜑3(𝑘𝐽𝑛,ℎ,0)[∑𝜇𝑖,3
𝐺𝑛,𝑖,ℎ +𝑘2𝐶ℎ𝜕 𝑢(𝑡𝑛)]
+𝑘
𝑟
∑
𝑙=4
𝜑𝑙(𝑘𝐽𝑛,ℎ,0)∑𝜇𝑖,𝑙
𝐺𝑛,𝑖,ℎ,
whe e
𝐺𝑛,𝑗,ℎ =𝛹(𝐾𝑛,𝑗,ℎ) − diag(𝛹′(𝑈𝑛
ℎ))𝐾𝑛,𝑗,ℎ +𝑃ℎℎ(𝑡𝑛,𝑗 ) − 𝑐𝑗𝑘𝑃ℎ
ℎ(𝑡𝑛).
We no ice ha again a e m like (16) u ns up, which can ei he be calcula ed exac ly o app oxima ed wi hou eso ing o
nume ical di e en ia ion. As o he o he e ms on he bounda y, hey can be calcula ed in e ms o da a wi h bo h Di ichle
and Robin/Neumann bounda y condi ions since 𝜕 𝑢(𝑡) = 𝑔(𝑡)and 𝜕 𝑢(𝑡) = 𝑔(𝑡).
3.3. 𝑝= 3
Simila ly o he calcula ion o 𝑈𝑛+1
ℎwi h 𝑝= 2, bu using (14), he s ages in (12) can be simpli ied o
𝐾𝑛,𝑖,ℎ =𝑒𝑐𝑖𝑘𝐽𝑛,ℎ,0𝑈𝑛
ℎ+𝑘𝜑1(𝑐𝑖𝑘𝐽𝑛,ℎ,0)[∑𝜆𝑖,𝑗,1
𝐺𝑛,𝑗,ℎ +𝑐𝑖[𝐶ℎ𝜕𝑢(𝑡𝑛) − 𝐷ℎ𝜕[𝑢(𝑡𝑛) − 𝛹(𝑢(𝑡𝑛)) + 𝛹′(𝑢(𝑡𝑛))𝑢(𝑡𝑛) − ℎ(𝑡𝑛)]]]
+𝑘𝜑2(𝑐𝑖𝑘𝐽𝑛,ℎ,0)[∑𝜆𝑖,𝑗,2
𝐺𝑛,𝑗,ℎ +𝑐2
𝑖𝑘[𝑃ℎ
ℎ(𝑡𝑛) + 𝐶ℎ𝜕 𝑢(𝑡𝑛) − 𝐷ℎ𝜕[𝑢(𝑡𝑛) −
ℎ(𝑡𝑛)]]]
+𝑘𝜑3(𝑐𝑖𝑘𝐽𝑛,ℎ,0)[∑𝜆𝑖,𝑗,3
𝐺𝑛,𝑗,ℎ +𝑐3
𝑖𝑘2𝐶ℎ𝜕 𝑢(𝑡𝑛)]
+𝑘
𝑟
∑
𝑙=4
𝜑𝑙(𝑐𝑖𝑘𝐽𝑛,ℎ,0)∑𝜆𝑖,𝑗,𝑙
𝐺𝑛,𝑗,ℎ.(17)
As o he e ms conce ning bounda ies o calcula e 𝑈𝑛+1
ℎin (12), using he le pa o (13) and (14), he e m in 𝜑2(𝑘𝐽𝑛,ℎ,0)𝐶ℎ𝜕can
be simpli ied o
𝑘2𝜑2(𝑘𝐽𝑛,ℎ,0)𝐶ℎ𝜕[𝑢(𝑡𝑛) + 𝑘2
2(∑𝜇𝑖,1𝑐2
𝑖)[𝛹′′(𝑢(𝑡𝑛)) 𝑢(𝑡𝑛)2+
ℎ(𝑡𝑛)]],
which can again be calcula ed exac ly in e ms o da a wi h Di ichle bounda y condi ions and can be app oxima ed wi h
Robin/Neumann ones conside ing he e o om he app oxima ion i sel a he bounda y and eso ing o nume ical di e en ia ion
jus o he i s ime de i a i e 𝑢.
On he o he hand, he e m in 𝜑2(𝑘𝐽𝑛,ℎ,0)𝐷ℎ𝜕can be simpli ied o
−𝑘2𝜑2(𝑘𝐽𝑛,ℎ,0)𝐷ℎ𝜕[
𝐽(𝑡𝑛)2𝑢(𝑡𝑛) +
𝐽(𝑡𝑛)[𝛹(𝑢(𝑡𝑛)) − 𝛹′(𝑢(𝑡𝑛))𝑢(𝑡𝑛) + ℎ(𝑡𝑛)]]
Jou nal o Compu a ional and Applied Ma hema ics 453 (2025) 116158
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B. Cano and M.J. Mo e a
= −𝑘2𝜑2(𝑘𝐽𝑛,ℎ,0)𝐷ℎ𝜕
𝐽(𝑡𝑛)𝑢(𝑡𝑛)=−𝑘2𝜑2(𝑘𝐽𝑛,ℎ,0)𝐷ℎ𝜕[𝑢(𝑡𝑛) −
ℎ(𝑡𝑛)],
which is exac ly calculable in e ms o da a o bo h Di ichle and Robin/Neumann bounda y condi ions.
As o he e ms in 𝜑3(𝑘𝐽𝑛,ℎ,0)𝐶ℎ𝜕, hey can be w i en as
𝑘2𝜑3(𝑘𝐽𝑛,ℎ,0)𝐶ℎ𝜕[𝑘2
2(∑𝜇𝑖,2𝑐2
𝑖)[𝛹′′(𝑢(𝑡𝑛)) 𝑢(𝑡𝑛)2+
ℎ(𝑡𝑛)]
+𝑘[
𝐽(𝑡𝑛)2𝑢(𝑡𝑛) +
𝐽(𝑡𝑛)[𝛹(𝑢(𝑡𝑛)) − 𝛹′(𝑢(𝑡𝑛))𝑢(𝑡𝑛) + ℎ(𝑡𝑛)] +
ℎ(𝑡𝑛)]]
=𝑘2𝜑3(𝑘𝐽𝑛,ℎ,0)𝐶ℎ𝜕[𝑘2
2(∑𝜇𝑖,2𝑐2
𝑖)[𝛹′′(𝑢(𝑡𝑛)) 𝑢(𝑡𝑛)2+
ℎ(𝑡𝑛)] + 𝑘[
𝐽(𝑡𝑛)𝑢(𝑡𝑛) +
ℎ(𝑡𝑛)]]
=𝑘2𝜑3(𝑘𝐽𝑛,ℎ,0)𝐶ℎ𝜕[𝑘2
2(∑𝜇𝑖,2𝑐2
𝑖)[𝛹′′(𝑢(𝑡𝑛)) 𝑢(𝑡𝑛)2+
ℎ(𝑡𝑛)] + 𝑘𝑢(𝑡𝑛)].
Simila ly, he e m in 𝜑3(𝑘𝐽𝑛,ℎ,0)𝐷ℎ𝜕can be w i en as
−𝑘3𝜑3(𝑘𝐽𝑛,ℎ,0)𝐷ℎ𝜕[
𝐽(𝑡𝑛)3𝑢(𝑡𝑛) +
𝐽(𝑡𝑛)2[𝛹(𝑢(𝑡𝑛)) + ℎ(𝑡𝑛) − 𝛹′(𝑢(𝑡𝑛))𝑢(𝑡𝑛)] +
𝐽(𝑡𝑛)
ℎ(𝑡𝑛)]
= −𝑘3𝜑3(𝑘𝐽𝑛,ℎ,0)𝐷ℎ𝜕[
𝐽(𝑡𝑛)[
𝐽(𝑡𝑛)𝑢(𝑡𝑛) +
ℎ(𝑡𝑛)]]
= −𝑘3𝜑3(𝑘𝐽𝑛,ℎ,0)𝐷ℎ𝜕
𝐽(𝑡𝑛)𝑢(𝑡𝑛)
= −𝑘3𝜑3(𝑘𝐽𝑛,ℎ,0)𝐷ℎ𝜕[…
𝑢(𝑡𝑛) −
ℎ(𝑡𝑛) − 𝛹′′(𝑢(𝑡𝑛)) 𝑢(𝑡𝑛)2],
whe e he las equali y comes om di e en ia ing (1) h ee imes.
In a simila way, i can be deduced ha he e m in 𝜑4(𝑘𝐽𝑛,ℎ,0)𝐶ℎis
𝑘4𝜑4(𝑘𝐽𝑛,ℎ,0)𝐶ℎ𝜕[1
2(∑𝜇𝑖,3𝑐2
𝑖)[𝛹′′(𝑢(𝑡𝑛)) 𝑢(𝑡𝑛)2+
ℎ(𝑡𝑛)]+ …
𝑢(𝑡𝑛) −
ℎ(𝑡𝑛) − 𝛹′′(𝑢(𝑡𝑛)) 𝑢(𝑡𝑛)2]
,
bu ha in 𝜑4(𝑘𝐽𝑛,ℎ,0)𝐷ℎ𝜕 anishes, as well as he possible e ms 𝜑𝑙(𝑘𝐽𝑛,ℎ,0)𝐶ℎ𝜕and 𝜑𝑙(𝑘𝐽𝑛,ℎ,0)𝐷ℎ𝜕 o 𝑙≥5.
Summing up, unde he assump ions in (14),𝑈𝑛+1
ℎin (12) can be w i en as
𝑈𝑛+1
ℎ=𝑒𝑘𝐽𝑛,ℎ,0𝑈𝑛
ℎ+𝑘𝜑1(𝑘𝐽𝑛,ℎ,0)[∑𝜇𝑖,1
𝐺𝑛,𝑖,ℎ +𝐶ℎ𝜕𝑢(𝑡𝑛) − 𝐷ℎ𝜕[𝑢(𝑡𝑛) − 𝛹(𝑢(𝑡𝑛)) + 𝛹′(𝑢(𝑡𝑛))𝑢(𝑡𝑛) − ℎ(𝑡𝑛)]]
+𝑘𝜑2(𝑘𝐽𝑛,ℎ,0)[∑𝜇𝑖,2
𝐺𝑛,𝑖,ℎ +𝑘[𝑃ℎ
ℎ(𝑡𝑛) + 𝐶ℎ𝜕[𝑢(𝑡𝑛) + 𝑘2
2(∑𝜇𝑖,1𝑐2
𝑖)[𝛹′′(𝑢(𝑡𝑛)) 𝑢(𝑡𝑛)2+
ℎ(𝑡𝑛)]]−𝐷ℎ𝜕[𝑢(𝑡𝑛) −
ℎ(𝑡𝑛)]]]
+𝑘𝜑3(𝑘𝐽𝑛,ℎ,0)[∑𝜇𝑖,3
𝐺𝑛,𝑖,ℎ +𝑘2[𝐶ℎ𝜕[𝑘
2(∑𝜇𝑖,2𝑐2
𝑖)[𝛹′′(𝑢(𝑡𝑛)) 𝑢(𝑡𝑛)2+
ℎ(𝑡𝑛)] + 𝑢(𝑡𝑛)]
−𝐷ℎ𝜕[…
𝑢(𝑡𝑛) −
ℎ(𝑡𝑛) − 𝛹′(𝑢(𝑡𝑛)) 𝑢(𝑡𝑛)2]]]
+𝑘𝜑4(𝑘𝐽𝑛,ℎ,0)[∑𝜇𝑖,4
𝐺𝑛,𝑖,ℎ +𝑘3𝐶ℎ𝜕[…
𝑢(𝑡𝑛)+(1
2∑𝜇𝑖,3𝑐2
𝑖− 1)[𝛹′′(𝑢(𝑡𝑛)) 𝑢(𝑡𝑛)2+
ℎ(𝑡𝑛)]]]
+𝑘
𝑟
∑
𝑙=5
𝜑𝑙(𝑘𝐽𝑛,ℎ,0)∑𝜇𝑖,𝑙
𝐺𝑛,𝑖,ℎ (18)
Again all he e ms a he bounda y in his exp ession can be exac ly calcula ed in e ms o da a wi h Di ichle bounda y condi ions
and in an app oxima ed way wi h Robin/Neumann bounda y ones aking in o accoun he app oxima ed alues o he space
disc e iza ion o (1) a he bounda y and he app oxima ion o 𝑢 h ough nume ical di e en ia ion in ime.
3.3.1. Concluding ema ks
Rema k 3.1. We no ice ha , in any case, no nume ical di e en ia ion in space is equi ed o app oxima e he equi ed bounda y
alues, so ha no weak CFL condi ion is equi ed o p o e he classical o de o he me hod, as i was in p inciple necessa y in he
mo e gene al case [17].
Rema k 3.2. Al hough, o he sake o b e i y, we do no show he calcula ions he e, o 𝑝= 4, nume ical di e en ia ion in space
would be equi ed o app oxima e bounda y alues wi h bo h Di ichle and Robin/Neumann bounda y condi ions, e en unde
assump ions (13)–(14).
4. Condi ions unde which he simpli ying assump ions a e sa is ied
In his sec ion, we will see why assump ions (13) and (14) a e nea ly always sa is ied o me hods o classical o de ≤4. Fo ha ,
we will conside he classical o de condi ions on he coe icien s o he Rosenb ock me hod (2) and, om hem, we will jus i y when
(13) and (14) a e gua an eed. Classical o de condi ions ill o de ou we e de i ed in [13], al hough jus assuming ha 𝑠= 2 and
ha ∑𝑏𝑖(𝑧) = 𝜑1(𝑧). In he gene al case, jus conside ing Taylo expansions o (5)–(6) on he imes epsize, he ac ha 𝐺′
𝑛(𝑈𝑛)=0
because o (7), and compa ing wi h he Taylo expansion o he exac solu ion o (4), he o de condi ions o Table 1 u n up.
Then, we ha e he ollowing esul :
Jou nal o Compu a ional and Applied Ma hema ics 453 (2025) 116158
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Table 1
Classical o de condi ions o exponen ial Rosenb ock me hods.
O de Condi ions
1∑𝑏𝑖(0) = 1
2∑𝑏′
𝑖(0) = 1
2
3∑𝑏′′
𝑖(0) = 1
3∑𝑏𝑖(0)𝑐2
𝑖=1
3
4∑𝑏′′′
𝑖(0) = 1
4∑𝑏′
𝑖(0)𝑐2
𝑖=1
12 ∑𝑏𝑖(0)𝑐𝑖𝑎′
𝑖𝑗 (0) = 1
8∑𝑏𝑖(0)𝑐3
𝑖=1
4
Theo em 4.1. I 𝑞deno es he classical o de o an exponen ial Rosenb ock me hod (1≤𝑞≤4) and 𝑟in (2) sa is ies 𝑟≤𝑞, hen (13) is
sa is ied.
P oo . We i s ly no ice ha , conside ing (2), he i s column o condi ions in Table 1 is equi alen o
𝑟
∑
𝑙=1
1
𝑙!
𝑠
∑
𝑖=1
𝜇𝑖,𝑙 = 1,
𝑟
∑
𝑙=1
1
(𝑙+ 1)!
𝑠
∑
𝑖=1
𝜇𝑖,𝑙 =1
2,
𝑟
∑
𝑙=1
1
(𝑙+ 2)!
𝑠
∑
𝑖=1
𝜇𝑖,𝑙 =1
6,
𝑟
∑
𝑙=1
1
(𝑙+ 3)!
𝑠
∑
𝑖=1
𝜇𝑖,𝑙 =1
24 .(19)
The e o e, when 𝑞= 1, jus he i s equa ion mus hold and, when 𝑟= 1,(13) ollows di ec ly.
When 𝑞= 2, he i s wo equa ions mus hold. When 𝑟= 1, bo h equa ions a e he same and again (13) ollows immedia ely.
When 𝑟= 2, we ha e a linea sys em o wo equa ions in he wo unknowns ∑𝜇𝑖,1and ∑𝜇𝑖,2. The ma ix associa ed o ha sys em
is clea ly nonsingula . Because o ha , he e is a unique solu ion o ha sys em, which ob iously co esponds o ∑𝜇𝑖,1= 1 and
∑𝜇𝑖,2= 0.
When 𝑞= 3, he i s h ee equa ions mus hold. When 𝑟≤2, he i s wo o hem lead o (13) wi h he same a gumen s han
be o e and he co esponding solu ion happen o also sa is y he hi d equa ion. When 𝑟= 3, we ha e a linea sys em o h ee
equa ions and h ee unknowns, which ma ix is again non-singula and he unique solu ion o he sys em is ∑𝜇𝑖,1= 1,∑𝜇𝑖,2= 0
and ∑𝜇𝑖,3= 0.
Fo 𝑞= 4, a simila a gumen leads o he esul . □
Le us see now unde which condi ions (14) is gua an eed.
Theo em 4.2.
(i) I 𝑟= 1 o 𝜆𝑖,𝑗,𝑙 = 0 o 𝑙≥2,(14) is always sa is ied.
(ii) I 𝑟= 2 o 𝜆𝑖,𝑗,𝑙 = 0 o 𝑙≥3,𝑠= 2 and 𝑞= 4,(14) is sa is ied.
P oo . (i) comes di ec ly om (3), which can be w i en like his conside ing (2)
𝑟
∑
𝑙=1
1
𝑙!
𝑖−1
∑
𝑗=1
𝜆𝑖,𝑗,𝑙 =𝑐𝑖, 𝑖 = 1,…, 𝑠. (20)
In o de o p o e (ii), we no ice ha , o 𝑞= 4 and 𝑠= 2, he las wo condi ions in he las ow o Table 1 ead
𝑏2(0)𝑐2𝑎′
21(0) = 1
8, 𝑏2(0)𝑐3
2=1
4.
Conside ing (2) again, he la e can also be w i en as
𝑐2
2(
𝑟
∑
𝑙=1
𝜇2,𝑙
1
𝑙!)(
𝑟
∑
𝑙=1
𝜆2,1,𝑙
1
(𝑙+ 1)! )=1
8, 𝑐3
2
𝑟
∑
𝑙=1
𝜇2,𝑙
1
𝑙!=1
4,
which imply ha
𝑟
∑
𝑙=1
𝜆2,1,𝑙
1
(𝑙+ 1)! =𝑐2
2.
Taking he e 𝑟= 2 as well as in (20) wi h 𝑖= 2, a uniquely sol able linea sys ems o wo equa ions and wo unknowns u n up,
which lead o 𝜆2,1,1=𝑐2and 𝜆2,1,2= 0.□
Jou nal o Compu a ional and Applied Ma hema ics 453 (2025) 116158
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B. Cano and M.J. Mo e a
Rema k 4.3. We no ice ha he condi ions which gua an ee ha he simpli ying assump ions a e sa is ied mainly conce n he
maximum alue 𝑟 o he index 𝑙in he 𝜑𝑙- unc ions, which mus be small enough wi h espec o he classical o de which wan s
o be achie ed. This is no a se ious d awback since, o he sake o simplici y, in he cons uc ion o me hods, 𝑟is aken as small
as possible. Only in case (ii) o Theo em 4.2 he e is also a es ic ion o o de 𝑞= 4 on he numbe o s ages i 𝜑2is u ning up in
he coe icien s 𝑎𝑖𝑗 . Howe e , his is no es ic i e ei he since, as s a ed in he in oduc ion, wi h Rosenb ock me hods, e y ew
s ages a e equi ed o ge a desi ed accu acy. In pa icula , classical o de 4can be ob ained wi h jus 2s ages.
Rema k 4.4. We also ema k ha he simpli ying assump ions (13)–(14) a e equi alen o he simpli ying assump ions in [12]
𝑠
∑
𝑖=1
𝑏𝑖(𝑧) = 𝜑1(𝑧),
𝑖−1
∑
𝑗=1
𝑎𝑖𝑗 (𝑧) = 𝑐𝑖𝜑1(𝑐𝑖𝑧),1≤𝑖≤𝑠,
unde which i was assu ed ha equilib ia o au onomous p oblems we e p ese ed and which allowed o simpli y s i o de
condi ions in ha pape . Mo e pa icula ly, s i o de 2was assu ed unde hose assump ions when in eg a ing ha ype o
p oblems when conside ing anishing bounda y condi ions. Because o ha , assump ions (13)–(14) a e sa is ied by mainly all al eady
cons uc ed me hods. As dis inc , in his pape , we jus i y h ough Theo ems 4.1 and 4.2 ha hose assump ions a e assu ed o be
sa is ied in many cases, wi hou he need o eso o s i o de condi ions o any kind o he p ese a ion o equilib ia.
5. Recommended me hods depending on he desi ed accu acy
In his sec ion, conside ing he esul s on he p e ious ones, we will sugges wha we hink is he bes choice up o he momen
o exponen ial Rosenb ock me hods o achie e he pa icula o de s o accu acy 𝑞= 2,3and 4.
5.1. 𝑞= 2
Rosenb ock-Eule me hod, which jus has one s age and co esponds o 𝑏1(𝑧) = 𝜑1(𝑧), is well-known o ha e classical o de 2.
By looking a Table 1, we can see ha no only he condi ions o classical o de 2a e sa is ied, bu also one o he condi ions o
classical o de 3. Mo eo e , he o he condi ion o achie e he la e accu acy canno be sa is ied wi h any me hod which jus has
one s age since, in such a case, because o (3),𝑐1= 0. The e o e, his seems o be an unbea able second-o de me hod. Wha is mo e,
i happens o sa is y (13) and (14). (In ac , ha could also be deduced di ec ly om Theo ems 4.1 and 4.2.) Because o ha , he
echnique in [17] wi h 𝑝= 2 can be applied o achie e local o de 3wi hou eso ing o nume ical di e en ia ion o calcula e he
equi ed bounda y alues. (We no ice ha Eule -Rosenb ock me hod has s i o de 2acco ding o [12], bu jus shows local o de
2when implemen ed h ough he s anda d me hod o lines when he bounda y condi ion 𝑔(𝑡)is ime-dependen ).
Nume ical esul s in [17] show he big ad an age in compu a ional ime o using he modi ied Rosenb ock-Eule me hod agains
Rosenb ock-Eule wi h he s anda d me hod o lines. The simpli ica ions o ha pa icula simple modi ied Rosenb ock-Eule
me hod we e al eady done in [17], whe e i was obse ed ha he di e ence wi h he s anda d me hod o lines jus consis ed
on adding a e m o he o m 𝑘3𝜑3(𝑘𝐽𝑛,ℎ,0)𝐶ℎ𝜕 𝑢(𝑡𝑛), when calcula ing 𝑈𝑛+1
ℎ om 𝑈𝑛
ℎ.
5.2. 𝑞= 3
As s a ed be o e, i is impossible o ge an exponen ial Rosenb ock me hod o classical o de 3wi h jus one s age. Because o
ha , we look o one wi h wo s ages. T ying o be as e icien as possible, we ake 𝑐2= 1 so ha possible e alua ions a 𝑡=𝑡𝑛+𝑐2𝑘
can also be used a he nex s ep. The simple unc ion 𝑎21(𝑧)o he o m (2) sa is ying (3) is hen 𝑎2,1(𝑧) = 𝜑1(𝑧). I we now look o
unc ions 𝑏1(𝑧)and 𝑏2(𝑧)sa is ying he ou necessa y condi ions in Table 1, we can see ha we can achie e ha jus by conside ing
𝑟= 1. Al hough a linea sys em o ou equa ions wi h wo unknowns is ob ained, h ee o hem a e equi alen and al oge he lead
o
𝑏1(𝑧) = 2
3𝜑1(𝑧), 𝑏2(𝑧) = 1
3𝜑1(𝑧).
This me hod again sa is ies condi ions (13) and (14), as i was also assu ed h ough Theo ems 4.1 and 4.2. By conside ing 𝑟= 2, a
one-pa ame e amily o me hods u n up, which co espond o
𝜇1,1=2
3+𝜇2,2
2, 𝜇1,2= −𝜇2,2, 𝜇2,1=1
3−𝜇2,2
2.(21)
We no ice ha wi h all hese me hods, he i s equa ion in he las ow o Table 1 is sa is ied. As o he hi d and ou h equa ion
in he same ow, hey a e ne e sa is ied. Howe e , he second equa ion in ha ow is jus sa is ied o 𝜇2,2= 1, and ha leads o
𝑏1(𝑧) = 7
6𝜑1(𝑧) − 𝜑2(𝑧), 𝑏2(𝑧)=−1
6𝜑1(𝑧) + 𝜑2(𝑧).
We may he e o e expec ha his leads o he smalles local e o s inside he amily (21).
All p e ious me hods ha e s i o de 2bu no s i o de 3acco ding o [12] since
𝑏1(𝑧) + 𝑏2(𝑧) = 𝜑1(𝑧), 𝑎21(𝑧) = 𝑐2𝜑1(𝑐2𝑧), 𝑏2(𝑧)𝑐2
2≠2𝜑3(𝑧).
Howe e , he echnique in [17] can be applied o a oid o de educ ion. Again he simpli ying assump ions (13) and (14) a e
sa is ied and he e o e, he ad an ages o he simpli ied o mulas in Sec ion 3can be used.
