Nume ical Analysis
Asymp o ically balanced schemes o non-homogeneous
hype bolic sys ems – applica ion o he Shallow Wa e equa ions
Tomás Chacón Rebolloa, An onio Domínguez Delgadob, En ique D. Fe nández Nie ob
aDp o Ecuaciones Di e enciales y Análisis Numé ico, Uni e sidad de Se illa, c/Ta ia s/n, 41012 Se illa, Spain
bDp o Ma emá ica Aplicada I, Uni e sidad de Se illa, A da. Reina Me cedes N. 2, 41012 Se illa, Spain
Recei ed 30 June 2003; accep ed a e e ision 4 No embe 2003
P esen ed by Oli ie Pi onneau
Abs ac
In his wo k we in oduce a class o balanced nume ical schemes, up o second o de , o he solu ion o gene al non-
homogeneous hype bolic sys ems o conse a ion laws. We gi e a gene al echnique o build such schemes. We also p o e ha
hey balance up o second o de a la ge class o s eady solu ions in he whole domain bu some subse whose measu e ends
o ze o as he g id size dec eases o ze o. We inally p esen an applica ion o Shallow Wa e equa ions ha exhibi he good
pe o mances o some o he schemes in oduced. To ci e his a icle: T. Chacón Rebollo e al., C. R. Acad. Sci. Pa is, Se . I
338 (2004).
Schémas asymp o iquemen equilib és pou sys èmes hype boliques non-homogénes – applica ion aux équa ions de
Sain -Venan . Dans ce a ail nous in oduisons une classe de schémas numé iques equilib és au second o d e pou la solu ion
de sys èmes hype boliques de lois de conse a ion. Nous donnons une echnique géné ale pou cons ui e ce ype de schémas.
Nous p ou ons que ces schémas équilib en au second o d e une g ande classe de solu ions s a ionai es, dans ou le domaine,
excep é un sous-ensemble de mesu e qui end e s zé o lo sque la aille de la maille end e s zé o. Nous p ésen ons inalemen
une applica ion aux équa ions de Sain -Venan qui mon e les bonnes pe o mances de quelques-uns des schémas p ésen és.
Pou ci e ce a icle:T. Chacón Rebollo e al., C. R. Acad. Sci. Pa is, Se . I 338 (2004).
Ve sion ançaise ab égée
Dans ce a ail nous in oduisons quelques schémas numé iques equilib és pou la ésolu ion de sys èmes
hype boliques de lois de conse a ion non homogènes.
C’es un ai bien connu, e la gemen ai é dans la li é a u e à ce éga d, que pou qu’un schéma numé ique
ou nisse des solu ions a ec un ni eau de p écision accep able, il doi app oche les équilib es (solu ions
E-mail add esses: [email p o ec ed] (T. Chacón Rebollo), [email p o ec ed] (A. Domínguez Delgado), [email p o ec ed] (E.D. Fe nández
Nie o).
1631-073X/$ – see on ma e
doi:10.1016/j.c ma.2003.11.008
86 T. Chacón Rebollo e al. / C. R. Acad. Sci. Pa is, Se . I 338 (2004) 85–90
s a ionnai es) du sys ème, au moins au deuxième o d e. Au emen , les solu ions ob enues p ésen en de o es
e eu s, é an sou en physiquemen inaccep ables.
Le poin -clé pou cons ui e des schémas equilib és es d’in odui e un décen age du e me sou ce compa ible
a ec le e me de di usion numé ique du schéma (c . [1,5,6]).
Dans ce a ail nous p ésen ons d’abo d une echnique géné ale pou cons ui e des e mes sou ces décen és,
de açon à ga an i que les équilib es son calculés au moins au second o d e (schéma (2)–(7)). L’idée consis e
à ajou e un e me co ec eu au sys ème équi alen . Ceci s’applique a des sys èmes hype boliques de lois de
conse a ionsnon-homogènesquelconques,à des schémas de ype Flux-di e encee Flux-spli ing e à une g ande
classe de solu ions s a ionnai es (Théo ème3.3). Essen iellemen , nous démon onsque no e schéma équilib e les
é a s s a ionnai es à l’o d e 2 dans les zones du domaine où les i esses ca ac é is iques ga den un signe cons an
(même zé o). Asymp o iquemen (lo sque la aille de la maille ends e s zé o), on équilib e ces é a s pa ou
sau su un ensemble de mesu e nulle. L’in é ê de ce ésul a es qu’on n’a pas besoin de connaî e les solu ions
s a ionnai es aupa a an .
Nousdonnonségalemen descondi ionssu isan espou queno eschémaéquilib ede açonexac eune solu ion
s a ionnai e connue.
Finalemen , la echnique p oposée es appliquée aux équa ions de Sain -Venan : nous p oposons une
disc é isa ion conc è e du e me sou ce, e mon ons quelques ésul a s numé iques assez sa is aisan s pou des
es s signi ica i s (Sec ion 4).
1. In oduc ion
In his pape we s udy he nume ical disc e iza ion o non-homogeneous hype bolic sys ems o conse a ion
laws. I is well known ha in o de ha he scheme is accu a e enough, i should sol e some equilib ia o he
sys em up o second o de . The e is a wide li e a u e which ea s his p oblem. Be múdez–Vázquez in [1] de ine
he C-p ope y; which asks ha he scheme calcula es a gi en s a iona y solu ion o a gi en sys em a g id nodes,
exac ly o up o second o de . In [3] an ex ension o a lux-spli ing scheme o non-homogeneous 1D SWE is
de i ed. In [2], based on he ideas o [3], a sys ema ic echnique o build nume ical schemes which e i y he
C-p ope y is in oduced. In [5] G eenbe g–Le oux in oduce he concep o a well-balanced scheme. In his
pape , o non-homogeneous scala equa ions, he au ho s in oduce a nume ical scheme based on he esolu ion
o egula ized Riemann p oblems, which p ese e he balance be ween in e nal o ces and he sou ce e m o he
hype bolic conse a ion law. The nume ical scheme balances he equilib ia o non-homogeneousequa ions. In he
case o such a sys em, in [6], Jin in oduces he in e ace-me hod. He p o es ha he p oposed scheme balances
up o second o de all s a iona y solu ions o he sys em o Shallow Wa e equa ions, a cell in e aces. In [7]
Pe hame–Simeoni p o e an ex ension o he heo em o Lax–Wend o o non-homogeneous scala conse a ion
laws. The ema kable ac , in ou con ex , is ha he heo em holds,p o ided he scheme balances ce ain s a iona y
solu ions.
In he p esen wo k we in oduce, o gene al non-homogeneoushype bolic sys ems, nume ical schemes which
balance i ually all s a iona y solu ions up o second o de .
We conside a gene al o mula ion o nume ical schemes, unde which can be w i en a wide class o known
and new schemes. We p esen an applica ion o Shallow Wa e equa ions ha exhibi s he good pe o mances o
he schemes in oduced.
2. Mo i a ion
In his sec ion, we gi e he gene al s uc u e o he schemes ha we p opose.
We conside a non-homogeneoushype bolic sys em,
∂W
∂ +∂F
∂x =G. (1)
T. Chacón Rebollo e al. / C. R. Acad. Sci. Pa is, Se . I 338 (2004) 85–90 87
The s uc u e o schemes in conse a i e o m o he hype bolic sys em is ob ained by in eg a ing he di e en ial
sys em in he con ol olume, in he 1D case, (xi−1/2,xi+1/2),whe exi=ix. Then, o non-homogeneous
hype bolic sys ems wi h an explici disc e iza ion in ime, i  is he ime s ep, we ha e
Wn+1
i−Wn
i
 +
φn
i+1/2−φn
i−1/2
x =Gn
C,i,(2)
whe e Gn
C,i is a cen e ed (second o de )app oxima iono 1
x xi+1/2
xi−1/2G(x, Wn
h)dx,Wn
hbeing a unc ion buil om
{Wn
i}i. The nume ical lux unc ion, φi+1/2is an app oxima ion o Fa x=xi+1/2.
We conside upwind schemes, ha ea he con ec ion dominance by in oducing some nume ical di usion.
Fo hese schemes, he nume ical lux unc ion has he s uc u e
φi+1/2=FC(Wi,W
i+1)−1
2D(Wi,W
i+1)(Wi+1−Wi), (3)
whe e by FC(Wi,W
i+1), we deno e a cen e ed (second o de ) app oxima ion o F(W) in x=xi+1/2,and
D(Wi,W
i+1)is a i s -o de app oxima ion o he di usion ma ix D(
Wi+1/2),whe e
Wi+1/2is an in e media e
s a e be ween Wiand Wi+1.
These schemes can be in e p e ed as second o de app oxima ionsin he space o a pa abolic equi alen sys em.
I we also ake a cen e ed app oxima ion o he sou ce e m, hen his equi alen sys em is
∂W
∂ +∂
∂xF(W)−ν∂
∂xD(W) ∂
∂xW=G(x, W), (4)
whe e νis equal o hal o he space s ep (ν=x/2).
No ice ha i Wis a s a iona y solu ion o (1) hen, i e i ies ∂
∂xF(W)=G(x, W).ThenWis no a s a iona y
solu ion o (4), as he eis a i s o de e o due o he upwindingo he lux. I is well known ha his is a sou ce o
la ge e o s in he nume ical solu ion. These la ge e o s disappea when he scheme has a high p ecision in space
(c . [1,5]). We shall de elop he e schemes wi h second o de accu acy in space. Fo his pu pose, we p opose o
add a co ec ing e m ( ha we deno e by VST, Viscosi y Sou ce Te m) o he equi alen sys em:
∂W
∂ +∂
∂xF(x,W)−ν∂
∂xD(x, W )∂W
∂x =G(x, W ) +VST.(5)
The e m VST, o s a iona y solu ions o (1) mus e i y −ν∂
∂x(D(W)∂W
∂x )=VST.
To gi e he de ini ion o VST we conside a smoo h s a iona y solu ion Wo (1). By A(W), we deno e he
Jacobian ma ix o F.I A(W) is non singula , hen
A(W)∂W
∂x =G(x, W ) ⇒−ν∂
∂xD(W)∂W
∂x =−ν∂
∂xD(W)A−1(W)G(x, W).
Wi h his de ini ion o VST, we can w i e he sys em (5) (wi h a conse a i e s uc u e) as
∂W
∂ +∂
∂xF(W)−νD(W)∂W
∂x −A−1G(x, W )=G(x, W). (6)
We can conclude hen ha o build balanced nume ical schemes, up o second o de , o he hype bolic sys em (1),
i is enough o use app oxima ionso o de wo in space o (6). This mo i a es he de ini ion o φi+1/2, as a second
o de app oxima ion o F(W)−νD(W)(∂W
∂x −A−1G(x, W )) in x=xi+1/2. Conc e ely, we de ine
φi+1/2=FC(Wi,W
i+1)−νD(Wi,W
i+1)Wi+1−Wi
x −
A−1(Wi,W
i+1)GD(xi,xi+1,W
i,W
i+1),(7)
whe e GDand 
A−1a e i s -o de app oxima ions o Gand A−1(
Wi+1/2), espec i ely.
88 T. Chacón Rebollo e al. / C. R. Acad. Sci. Pa is, Se . I 338 (2004) 85–90
The e is s ill he di icul y o de ining VST when Ais singula . I by λj,j=1,...,N, we deno e he eigen alues
o A, we shall assume ha he e exis s Ncon inuous su aces γj⊂RN,j=1,...,N, such ha λjis in e ible
and C2in RN γj.
We p opose o de ine he ma ix 
A−1(U, V ) as 
A−1(U, V ) =X(
W(U,V))
Λ−1(U, V )X−1(
W(U,V)) whe e

W(U,V) is an in e media e s a e be ween Uand V,byXwe deno e he ma ix de ined by he eigen ec o s o A
and

Λ−1(U, V ) =Diag
λ−1
1(U, V ), . . . , 
λ−1
N(U, V )wi h 
λ−1
j=λ−1
j
Wi L[U,V]⊂RN γj,
0o he wise,
whe e L[U,V]is he segmen in RN ha connec s Uand V.
This de ini ion o 
A−1s a es ha when some eigen alueo A anishes, hen no upwindingo he sou ce e m in
cha ac e is ics a iables mus be pe o med. Fo scala equa ions, we may p o e ha his de ini ion o 
A−1ensu es
he s abili y o he scheme.
This choice allows us o balance he scheme, e en in some cases when Ais singula .
3. Balance p ope ies
In his sec ion we p o e ha he scheme (2)–(7) balances up o second o de he sys em (1) o a wide class
o s a iona y solu ions. We also p o e ha unde some na u al hypo heses upon ma ix D, he de ined scheme
balances sys em (1) o all s a iona y solu ions in all [0,L]bu in a se whose measu e ends o ze o as x →0.
We will call hem ‘asymp o ically balanced schemes’.
Fi s ly, we obse e ha ma ix D(W) is only a Lipschi z unc ion, wi h non-con inuous de i a i e a he poin s
whe e i s eigen alues anish. A possible choice o he ma ix is D(W) =|A(W)|; o example his happens o he
scheme o Roe. Thus, we canno hope ha he in oduced scheme balances all s a iona y solu ions, wi h o de wo,
in all he domain [0,L]. We in oduce he ollowing de ini ions,
De ini ion 3.1 (Balanced schemes).We say ha he nume ical scheme (2)–(7) is balanced on a smoo h s a iona y
solu ion W(x) o he hype bolic sys em (1) i he associa ed consis ency e o has second o de accu acy.
De ini ion 3.2 (Asimp o ically balanced schemes).We say ha he scheme (2)–(7) is asymp o ically balanced on
a smoo h s a iona y solu ion W(x) o he hype bolic sys em (1) i he e exis s an inc easing sequence {Kn}no
compac subse s o [0,L]such ha
(1) µ([0,L]nKn)=0, whe e by µwe deno e Lebesgue measu e in R.
(2) Fo all n he e exis s a δn>0 such ha i 0 <x<δ
n, hen he scheme balances he sys em in Kn.
Assuming ha A−1and Dp esen singula i ies only on cu es in RN, we ob ain he ollowing esul :
Theo em 3.3. We conside W:[0,L]→RNa s a iona y solu ion o class C2o he sys em (1). Then,
(a) I A(W) does no ha e singula poin s, he scheme (2)–(7) balances sys em (1) on Wup o second o de .
(b) I he se o poin s xwhe e A(W(x)) is singula has measu e ze o, hen he scheme (2)–(7) asymp o ically
balances sys em (1) on W.
(c) I Dhas he same eigen ec o s as Aand hei eigen alues anish a he same poin s, hen he scheme (2)–(7)
asymp o ically balances sys em (1) on W.
The p oo o his heo em is based on he ac ha in egions o [0,L]whe e Ais in e sible, a smoo h solu ion
o he o iginal sys em (1) is also a solu ion o he equi alen sys em (6). Poin (c) applies o s eady solu ions W o
T. Chacón Rebollo e al. / C. R. Acad. Sci. Pa is, Se . I 338 (2004) 85–90 89
which he se o poin whe e A(W) is singula has non-ze o measu e. The comple e p oo o he heo em can be
ound in [4].
The in e es o poin (c) is ha i p o ides a wide class o schemes ha balance all (smoo h) s eady solu ions o
sys em (1).
We also ha e he ollowing esul ha yields su icien condi ions o he exac calcula ion o s a iona y solu ions.
Theo em 3.4. Le Wbe a s a iona y solu ion o (1). Assume ha FC,GC, and GDe alua ed on W e i y
FC(Wi,W
i+1)−FC(Wi−1,W
i)
x =GC,i,Wi+1−Wi
x =
A−1(Wi,W
i+1)GD(xi,xi+1,W
i,W
i+1).
Then, he scheme de ined by (2)–(7) balances in an exac way he sys em (1) o he s a iona y solu ion a g id
poin s. This happens independen ly o he choice o he upwinding ma ix D(U,V ).
4. Applica ion o Shallow Wa e equa ions
In his sec ion we apply he p oposed scheme o Shallow Wa e equa ions, which model he beha iou o wa e
lows in channels. The unknowns a e h, he heigh o he wa e , and q he discha ge. The physical lux unc ionand
he sou ce e m o a iable dep h, a e de ined by
F(W)=q
q2
h+1
2gh2,G(x,W)=0
ghH(x) whe e W=h
q,
gis he g a i y cons an and H(x)=¯
h−zb(x) is he dep h o he channel, abou a ixed e e ence le el.
The di e en elemen s ha de e mine scheme (2)–(7) ha we use a e de ined as ollows:
FC(Wi,W
i+1)=F(W
i+α)+F(W
i+1−α)/2,whe e Wi+α=(1−α)Wi+αWi+1wi h α∈[0,1].
The i s componen o GDand GCis equal o ze o, in bo h cases, and he second componen s a e
GD(xi,xi+1,W
i,W
i+1)2=ghi+1/2Hi+1−Hi
x ,
[GC,i]2=g
2αhi+α/2+(1−α)hi+(1−α)/2Hi+1−Hi
x +αhi−α/2+(1−α)hi+(α−1)/2Hi−Hi−1
x .
Ma ix Dhas he same eigen ec o s as ma ix Aand i s eigen alues a e gi en by dj(Wi,W
i+1)=|λj(Wi+1/2)|.
Wi h his choice, a leas o he linea homogeneouscase, he scheme is L2s able.
Using Theo em 3.3 we p o e ha he scheme is asymp o ically balanced o all smoo h s a iona y solu ions.
Mo eo e , he scheme exac ly calcula es wa e a es , (h, q) =(H, 0). I is enough o obse e ha he condi ions
o he Theo em 3.4 a e e i ied.
Nume ical es . We es ou scheme o wo s a iona y solu ions, in a domain o leng h L=25 m and bo om
unc ion de ined by he maximum be ween ze o and he ‘bump’ zb(x) =0.2−0.05(x −10)2. Fo his es he
discha ge mus be cons an , so his es is in e es ing in o de o obse e he pe o mance o he scheme nea he
bo om bump.
We use a CFL condi ion equal o 0.8, a s ep disc e iza ion equal o x =0.25 and as ini ial condi ion wa e
a es wi h cons an ee su ace a 0.5. As bounda y condi ions, we conside wo di e en cases: o Tes 1.a we
impose h=0.66 m downs eam when he low is subc i ical and q=1.53 m2/s ups eam. Tes 1.b is ob ained by
imposing h=0.33 downs eam and q=0.18 m2/s ups eam. In bo h cases we conside ha he scheme con e ges
o a s a iona y solu ion when he ela i e e o be ween wo consecu i e app oxima ions in ime is less han 10−5.

90 T. Chacón Rebollo e al. / C. R. Acad. Sci. Pa is, Se . I 338 (2004) 85–90
Fig. 1. Tes 1.a. Fig. 2. Tes 1.b.
In Figs. 1 and 2 we compa e he calcula ed discha ge o he Su ace G adien Me hod (SGM) (c . [8]) and he
p oposedschemewi h α=1/8.Inbo hcases hep oposedschemep o idessomeimp o emen .Thisispa icula ly
ele an o Tes 1.b, as in his case he e is a hyd aulic jump (discon inui y o he ee su ace) nea x=12.
Acknowledgemen s
This wo k has been pa ially inanced by he Spanish Go e nmen Resea ch p ojec s REN 2000-1162-C02-01,
REN 2000-1168-C02-01and CYTMAR MAR97-1055-C02-02.
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