VicnuunJ Math. (20161 44 557-^586 DOI 10 l(O7/st0Ol.l-0l^t)17(^y
The Generalized Finite Volume SUSHI Scheme for the Discretization of Richards Equation
Konsbuidn Brenner' • Danielle Hilhorst^ • Huy-Cuong Vu-Dtv*
Received. 24 Decembei 2014 /Accepted 15 lune 2015 / Published online- 2^ October 2015 0 Vieonnn Acndem, of Science «id TechnoloE, |VAST| i„d Spnngei SciencetBnsiness Media Smg.peie
Abstract In diis anicle. we apply die generalized finite volume method SUSHI to die di.s- cretizanoo of Richards equadon. an elliptic-paiabolic equation modeling groundwatei flow where the diffusion term can be anisotropic and heterogeneous This class of locally con- setvauve methods can be applied to a wide tange of unstroctured possibly non-matching polyhedral meshes in aihitrary space dimension As is needed for Richards equation die nme discredzadon is fully implicit. We obtain a convergence result based upon a pnori esti- inates and die application of the Fnichet-Kolmogorov compactness dieorem We implemem die scheme and present numerical le.sLs
Keywords Richards equadon • Finite volume scheme • SUSHI scheme
MaOtemadcs Subject Classlficallon (2010) 35KI5 .15K(i5 63M08 65MI2 76S05
- Danielle Hilhorsl Danielle HilhorM^^maih u-p>.ud fr Konslanim Brenner konManiin brennerle'unice fr Huy^Ciinng Vu-Dii
*dhuscuong@gmail torn
X ™ r F n i " ; r '^-'-^"•""' * <^°"" - <- S.PI.,II-A.„P„|„.
Ubocioiie de MaUieni.li,.es. C r « S el UnivemS d, Pa„,-Sn<l. C h , „ . France Fncnil, ot M.ihemaics n.d Compel., Science. Uni.en.,1, of Science, Ho t hi Mmh. Mcinam
1 Introduction
Let i2 he an open bounded polygonal subset of U" id = 2 or 3) and let 7" be a posiiive constant; we consider the Richards equation in the space-time domain Qj = Q x (0, 7"):
d,9ip) - div (A.(S(p))K(x)V(p + z)) = 0, (1) where p = p{x. t) is the piezometric head. The space coordinates are defined by x = (x, z)
in the case of space dimension 2 and x = (x.y.z) in the case of space dimension 3. The quantity 9ipj is the water storage capacity, also known as the saturation, K(x) is the absolute permeability tensor, and the scalar function kr is the relative permeability.
Next, we perform Kirchhoffs transformation. We define Fis) := f
Jo kAdiT))dT.
and suppose that the function F is invertible. Then we set M = F(p) in Qj and c(u) = '^(Fip)) = S(/j); it turns out that the function c is either qualitatively similar to the function 9 or has a support which is bounded from the left as in Fig. 1 b. We remark dial Kirchhoffs transformation leads to Vu = ktidip))Vp. Equadon (1) becomes
9,C(H) - div(K(x)VK) - divik,iciu))K(.x)Vz) = U. (2) We suppose diat H 6 1V''™(J2) is a given function and consider equation (2) together with the inhomogeneous Duichlet boundary condition
uix.t) = M(X) a.e, on9i2 x (0. 7"), (3) and the inidal condition
M(X, 0) = Mo(x) a,e, ini2. (4) We denote by (P) the problem given by equations (2). (3), and <4). We shall make the
following hypotheses-
, IS a nondecreasing Lipschitz condnuous function c e IV'-^(IR), 0 < c < c
with Lipschitz constant Lc;
kr : R-^ ->• R+ is a nondecreasing Lipschitz continuous function widi Lipschitz constant i.*,, and 0 < k^ < kr where k^ is a positive constant, k^is) > 0 for all s > 0, (Fig. 2)
(a) Typical saturation B ng. 1 Typical salurauon and lis Kirchhoffs transformation
(b) Aprofile of function <:•
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Tbe Rmie Vblume SUSHI Method for Richards Equanon
(M) K is a bounded ftmction from Q to Mrf(R). where M^fK) denotes the set of real died matrices. Moreover fora.e. x m i2. K(x) is a positive definite matrix and diere exisi^two positive constants K and K such that the eigenvalues of K(x) are included in [K. K];
{J(%) UQ e L'^iQ) and w e W'-'^d?).
DefinitiiHi 1 (Weak solution) A ftinction u = u(x.i) is said to be a weak solution of Problem (P) if:
(i) u-Q eL-(0,T.H^(Q));
(li) ciu) e Z.^(0. T: L^(i?)):
(iii) - / I ciu)d,\lfdxdi - I ciuo)yf(:0)dx (5) +J I KVu-V}lfdxdt-i- I I kr(ciu))KVz•V^|,d\dt =0.
roralli/'€Z.^(0. 7;//j(i2))suchthaty'(-. r ) = 0and9,i/f ^ L'^iQj).
The discretization of the Richards equation was performed by means of the finite differ- ence method by Homung [13] and by means of die finite element method by Knabner [15]
Kelanemer [14], and Chounet el al. [3] implemented a mixed finite element method and Frolkovic et al. [ 111 applied a finite volume scheme on die dual mesh of a finite element mesh. We refer to Eymard et al [10] and to Eymard et al. [6] for the smdy of the conver- gence of two slightly different numencai schemes based upon the standard finite volume method.
0 5 0 6 0 7 0 8 0 9 saturation 8
Rg. 2 T>pical purmeabiliiy
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' 1
In Section 2, we introduce die SUSHI scheme, a finite volume scheme using stabiliza- tion and hybrid interfaces which has been proposed by Eymard et al. [7] and define the approxunate Problem (P^.s,). We also present some relevant results which will be use- ftil in die sequel. In Section 3, we prove an a prion estimate on die approximate solution in a discrete norm corresponding to a norm in L~{0. T: //^(J?)). Using these estimates and arguments based on the topological degree, we prove die existence of a solution of Problem {Pg>_s,) in Section 4. In Section 5, we prove estimates on differences of time and space translates. These estimates imply die relative compactness of sequences of approxi- mate solutions by the Frechet-Kolmogorov theorem. We deduce the convergence in L^ of a sequence of approximate solutions to a solution of the continuous problem (P) In Section 6.
For die proofs, we apply mediods inspired upon those of [7] and [8]. In the last section, we describe effective computations in the case of some well-known numerical tests which are often used in literature. We also perform simulations for a realistic model at die end of Section 7
The discretization of Richards equation by means of gradient schemes, which include die SUSHI mediod. has already been proposed by Eymard et al. [9], where they consider Richards equation as a special case of two phase flow; however. Uiey make the extra hypoth- esis that the relative permeability k^ is bounded away from zero, which is not satislled in most geological contexts. Here, we avoid this extra hypodiesis by performing KirchhofTs transformation.
2 The Hybrid Finite Volume Scheme SUSHI
In this section, we construct an approximate solution of Problem (/>) corresponding to a time implicit discretization and a hybrid finite volume scheme. We follow die idea of Eymard et al [7] to construct the fluxes using a .stabilised discrete gradient.
2.1 Space and Time Discretization
U t us nrst define the notion of admissible fimte volume mesh of P and some notations associated with ii.
" . ' S ' ' ' " " ^ ' ^ ' ' - ' •''"""•J""™) I-'l « be a polyhedral open bounded connected subset ol R and 312 = n\S2 its boundary A discretization of K, denolcd by 9). is defined as the triplet & = I-//. ^. :^). where:
. # is a finite familj^of nonemptj^ convex open disjoint subsets of C (the -conBul volumes-) such that !l = UK^_, K.¥or,iny K ,= .il. la3K =-R\l! be the boundary ot XI we denote by |«-| the measure of K and dlK) the diameter of A-.
2. if IS a finite family ofdisjoim subsets of 12 (the -imerfaces"). such that, for all a E <f o IS a nonempty open subset of a hyperplane of R ' and denote b , [<r| its measure. We assume that, forall K € .^tf. diere exi.sts a subset tSx of ,f such dial S/f = u„t i li' IS a family of points of J2 indexed by .W. denoted by .»• = ( „ , ) j , .^. such that lor all K i.^.xg^KtatiK is assumed to be i«-star-shaped, which means that foi all X E «•. the inclusion Ix,, x| c f holds.
For all a e ,y. we denote by x„ the barycenter of <T. For all Jt 6 . # and a <2 Sg vie denote by D,-,, die cone with vetlex x j and basis o. by n,.„ the unit vector normal to a S Sponger
Tie Rnite Viiumc SUSHI Method for Richards Equation
outward lo K and by dfc.a the Euchdean distance between xjc and die hyperplane including cr. For any CTSff, we define . ^ „ = IK e ^ :a e <f/f|. The set of boundary interfaces is denoted by (fe„, and die set of interior interfaces is denoted by f^„,.
We express die finite volume scheme in a weak form. For diat purpose, let us first associate with die mesh the following spaces of discrete unknowns
Xs = {v = (ivK:)Ke.^.(v„)^^s): VK e K, U„ e K] , X9!0= {veXs:v^=OVoE (f„,).
Definition 3 (Time discretization) We divide die ume interval (0. 7) into N uniform time steps of lengdi 5/ = 7"/A', and we define by/„ =n5/where n e [0 N].
Taking into account die Ume discretization leads us to define die following discrete spaces
X% = X^^ {/i = (ft")nen, ..N].h" e Xs>].
^%.o = ^^.n = I'l = ih")„^i,, __ pjf.h" e Xg-.o).
For the sake of simplicity, we restnci our study to die case of constant time steps Never- theless, all results presented below can be easily extended to die case of a non uniform time discretization,
22 Discrete Weak Formulation
We propose here a discrete scheme which is based upon die hybrid finite volume scheme SUSHI. It has been initially proposed for uniformly elliptic problems [7]. Schemes of this type in the case of a parabolic degenerate equation have been recently analyzed in [1]
Remark dial this mediod can also be viewed as a mimetic finite difference or a mixed finite volume mediod (see (1. 2. 5]),
After formally integrating equation (2) on die cell K x (f„_,. f„) for each A- € # and foreach/ie |1 A'l.we obtain
j^iciuix.t„))-ciuix.t„_t)))dx - Y, r fKVunK„dydi
^ / kAciii))KVz-nf;.„dydt = 0. (6) For all A: e ..# and (7 € ,5>-. die diffusive flux - /^ KV» - n^ „dy and die conveciive fiux
J„ kriciu))KVz-nK.„dy are approximated by F,; „{u) and QK „(»). which are defined below by (12) and by (20). respectively
Before nmoducing die numerical scheme, we define die following projection operator n'rttf*- *^^'' ^ ' *"""]•" ^^ ^^* ^^ " " " ™ ' ° ^ ^ ^ '^'^"'"^^ ^y H^'=*^- •»• l*(=^- -'11 loraii A 6 . «• and tr e c'.
Ut Ps-u and P^yuo be the projections of boundary and mitial ftinctions in (1) and (4) respecuvely: we present below a discrete weak problem (Pr^, s, )•
The initial condition is discretized by
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For each n 6 (1 N] and forall K e .4*', find u" such diat u" - P&u 6 X^f^.' satis^-ing
^ | i i : | ( c ( 4 ) - c ( H ^ - ' ) ) i . j r + 5/(M".u)F + 3 f ( « " , i ' ) e = U V r e X s . o , (8) where
{W.V)F = Yl Yl FK.Au})iVK-Vu). (9) and
( t / ; . i ; ) Q : = ^ ^ eA-.^(w)(VK - fo). (10) Let « ^ = « | - /*©« e X%^\ we rewnte die discrete equation (8) as
Y l'^l(c(«K)-c(Hjr'))i>K+5r{H",u}p+5f(/'^H,r)f+fi((M",u)e = 0, (U) The discrete Problem iP^.s,) is given by initial condition (7) and eidier the disable equation (8) or the discrete equation (11),
2.3 The Approximate Flux
The discrete flux FK,^ is expressed in terms of die discrete unknowns. For diis puipose, we apply die SUSHI scheme proposed in [7]. The idea is based upon the identification of the numerical fluxes through the mesh-dependent bilinear form, using die expression of the discrete gradient
Z ] Yl ^'^•"^'"^'^^ii-v^)= I '^&w{x)-K{x)V^vix)dx Vv.weX^a- (12) Ke.xa H^SK •'^
To this purpose, we first define
^/i:"' = j - ^ 1 ^ l'^ICic<r-")K)njf,„ VA: 6 .-^, Vu) e A-® (13) The consistency of formula (13) stems from die following geometrical relation:
J ] klnx.n(Xa-XAr)^ = i/i:|Id "iK^^. (14) where (x„ - x^r)^ is die transpose of Xj, -XK ^ H'^ and Id is the J x rf identity matrix.
Remark I The approximation formula (13) is exact for linear functions. Indeed, for any linear function defined on Q by ^^(x) = G • x with G e W. assuming diat v}„ = v(x^) and WK = V(XK), we obtam W„-WK = (X„ - X^)'"G = (x„ - x^)^Vp: hence. (13) leads to
We also remark that
Y l'^l'>Ar.^= Y «K.ady= I (VI)
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The Finiie Nfalume SUSHI Meihod fw Richaids Equanon
which m e a n s diat die coefficient o f WK in (13) is e q u a l to z e r o . T h u s , a r e c o n s t r u c t i o n of the discrete gradient solely based on ( 1 3 ) c a n n o t lead to a definite discrete bilinear form in the general case. Therefore, w e introduce die stabilized gradient
liK.aW = (W„ -WK- VKW • (x„ - x ^ ) ) . ( 1 5 ) dK.a
We may dien define V ^ m as die p i e c e w i s e constant ftmction e q u a l to V^f, w a.e in the c o n e
VeMF(x) = Vjc„!ii f o r a . e . X^DK.^. ( 1 6 ) Note dial, from die definition ( 1 5 ) . in v i e w of (14) a n d (13). w e d e d u c e dial
2 ^ " RK..,WBKa=0 VKe.^. ( 1 7 )
In order to identify die n u m e r i c a l fluxes Fx.aiw) through relation ( 1 2 ) . w e put die discrete gradient m the form
'JK.^W= X I ("•'"•-"•A:)yA''''- ( 1 8 )
with
I ' ^ ' l ^/d \a'\
Thus,
T : " * : a' • (Sa - XK )nK.„ o d i e r w i s e .
^ V , u , ( x ) . K ( x ) V ^ „ ( x W x = J2 E 1 F ' < » V - « ' i - l ( i V - n;.l ie<fif o'e^i-
with a. a' e £K and
^T = Y yi'" • A^'y^""' Aj = f o'-^^, Jo, „
The local matrices A"/ are s y m m e t r i c and positive, and die identification of dte numerical fiuxes using relation (12) leads to die expression:
^ A - . o ( w ' ) = Y AY'iwK -w^-). ( 1 9 ,
Next, we consider die conveciive flux. T o tins purpose w e first define gK ^ = j K V - -
^K.ndy. Then die convective flux is defined as - o -
QK.aiu-') = -kAdu-'fc .))gK.^ 'iK^.W.a e < ? x - ( 2 0 ) where ii';^-„ satisfies die upwind-sort formula
» , . . = («•« i f s * - . < 0 .
t U'o otherwise. * " ' ' Moreover, in view of die definition of gKo. w e remark that
gK.<, = -gL.a V(7 e (?„,,..,#•„ = [K.L). ( 2 2 )
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^ K. Brenner etdlf 2.4 Tbe Properties of the Scheme
Next, we introduce some extra notations related to die mesh. Let 5? be a discretization of fl - ^ in die sense of Definition 2. The size of the discretization 5* is defined by
1^ — sup diK). i and die regularity of die mesh by:
M = n i a x f max ^ , max '-^) We will suppose in die sequel that /@ < 1,
Definition 4 U t 5^ be a discretization of i2 in die sense of Definition 2, and let fir be the time step defined in Defimtion 3. For v 6 XQ, we define the semi-norm
i»ij,=i;^E^ !;<".-».)=•
ForalWi = | * " | „ | | „, e x g , we define die semi-noim
l''li'i=i:«'l''"|-J,-
Let //.#(fi) c L-(i?) be the set of piecewise constant ftinctions on the control volumes of the mesh . # . For all » E X3, we denote by D.^v £ H.„ia) die piecewise function from 12 to K defined by n,.,i,(xl = vg for almost every x e JC, for all *• e .^
Let H ; , ( C X (0, D ) c i^O? X (0, T)) be the sel of piecewise constam function, on die space-dme control volumes. We denote by n ' i : ; { « - . t J ( e r ) the mapping
n ' i - ' l x . ' I = 4 forall ( x . O e j r x(i„_,,t„]. (23) We also define 11%: X%-^ L'iQr)'' by
V|i,(x. /) = Van" for all Ix.t) e K x (t,_i, t„). (24) Next, following n j . for all » s Xg, we define die following related notm
lin-«'«llL,..,,= E E l ' l - ' ^ - . f ^ ) ' . (25)
with If, = 1,,^ , - , . , , , , . I. O , „ = | „ , _ „,, ,f . ^ ^ , , ^ ,. ,^ ^ „ , d . # „ = jr. A result stated in [7] gives die lelation
r i ["lis, Vu e Xs,.!,. (26) II n . .
Lett™, 1 (Poincare like inequality) Let S be a discreilzalion of n in the sense ofDefni- Ihen there e.xms C, o,i/y depending on d. Q and n such that
Iin-i"'lli.^(j3l < C i | | n , ^ i , | | , , 2 . ^ Vt;eA-0, (27) li-fene |[n #!.||, 2, # IS defined by i25)
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The Finite Volume SUSHI Melliod for Richards Equation
Proof InviewofLemma5.4in[7], foreachp> 1, tiiere exists ^ > p only dependmg on/?
and diere exists a positive constant C only depending on d and /? such diat ||n.#L'|| j;«(j3i <
C\\T\_av\\i_p _/^ for all V e X Q . We remark diat for all 9 > p, then lin^i^lk^lQ, < |i2|<''-'""''||n_#i'||i„i3,.
We set p = 2 and C\ = |i2|<'»-->^^C to conclude the proof. D Definition 5 Let @ be a discretization of i? in die sense of Definition 2. and let St he die
time step defined in Definition 3. We define die i^-norm of the discrete gradient by l|V9.(x)||J,,„,, = E E ^ ^ [ ' - ^ - . " l ' VnEX^.
and
\Kl'<'-mi:iQ,yi=j:!-E E * ^ l ' « . * " l ' ™e;(S.
where V^,„ and V@ is defined by (13)-(16),
a 2 Let & be a discretization ofQ in the sense of Definition 2 and suppose that there e.xists a positive constant ji such that p& < p for all @, let St be the time step defined in Defimtion 3.
(i) Then there exist positive constants C2 and Ci only depending on p and d such that Ci\v\\^ < l|Vsi;(x)||^,,„,, < Cml^ Vi. e x ^ . (ii) Moreover, we have
C:|/.|^,^ < \\V%h{x. <)\\\^.,Q^y, < C3J/,|^., W, e X%
Proof We refer to Lemma 4,2 in [7] for the proof of (i) In view of die definition of die serai-norm in die space X^ and die i^-norm of the discrete gradient, we deduce (ii). D Lemma 3 Let ^ be a discretization ofi2 in the sense of Definition 2 and suppose that there exists a positive constant p such that pa < p for all 9; there exists a positive
{v.v)r^a\v\-^^. (28) Proof In view of Hypothesis (.yf,) and Lemma 2. we also obtain
(u. u)f = / K(x)(Ve^v(x))-dx
2 t C , i i i | = , .
Setting Of = KC2 permits lo complete the proof • Definition 6 Let 5^ be a discretization of Q in die sense of Definition 2 and let ,5/ be die
time step defined in DefiniUon 3. U l u% e X^ be a solution of Problem (P^^,s,). We say diat n*^H^(x. /) IS an approximate solution of Problem (P)
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We now state a weak compacmess result for die discrete gradient.
Lemma 4 Let ^ be a family of discretizations ofn ia the sense of Definition 2 and suppose that there crisis a posiiive constant p such that PS < p for all & e ,^. Let {h%) be
a family of unknowns such that ^-^
(1) h% 6 X%„forallS< tl .^ and for all Si E (0. I).-
(dl there exists C > 0 such thai | / i | | ^ j < C for all S e » and for all St 6 (0,1),- (iii) them easts h n L^IQj) such that n'^h%lti. D converges to h iveakly in L^IQy) a, Then h e (,'(0, T. H„1(C)) and V%h% conyerges to Vh weaUy In L\QT)' as I3 and
Proof We extend the fiincdons n * , * | and V%h% by zero outside SI. In view of (ii) of Lemma 2. diere exists a funcdon .X" f: LH«'' X (0. T))' such diat up to a subsequence V^* J, weakly converges to J T in LHU' X (0. D ) ' ' as (5,. it -y 0. We show below dial J T = VA. Let,, e C^lW x (0, D ) ' ' be given; we consider the term defined by
K''=f / V | * | ( x , i ) . y ( x . , ) r f x d i . In view of the definition of Vg in (13)-(16), we infer dial r » = Tf + 7 f , where
'i"' = E " E E I'lK-*!)"*,.*-;: •"•titfi^ — f" ftfUt.Ddxdt,
and
iV
' j " = E E E *«•"*" "I.n • / / il>lx.l) dtidt.
n^l K::.lKo^Sr Jt„-llDs„
Which by (17) yields 1/
?J* = E E T, "'-''•""K-it- f f {v(tl.D-tl>'e)dsdl.
Applying Cauchy-Schwarz inequality, we deduce that
<^3"'' s ( E " E E ^ ( « X „ / . " :
•'[EE E
V=i KuMa^Se
We deduce from formulas (4.11) and (4.12) in [7] die inequality
;^y-'^Ej^.:.-n,A
^ Sprioger
\a\dK St^f L (<P ~ 9>"K)dxdtA .
TTH: Finlie \falunie SUSM MethcxI For Richards Equation
which in turn implies that
^ ( . . . ) ^ , 2 | L ( . : - . i ) . , 2 „ ^ ^ ^ 5 : ^ | l , f , : , _ . i , .
We remark diat ! ; „ , , , " g i = ' ""i *»• £ . " . , " £ « . . * £ , « , ; ^ ( * : - ftjl^ = l * | l > < C, which yields
T."!!'E~r^(Pk-,it")'i2ii-ypV-)c. oo)
n=l Ki-ytocSr
By die regularity properdes of die funcdon tp. diere exists C , only depending on t, such
* • ' I /il'.i / D , .<«"«•') - fVdfdi] < C , J t ( » S l & i , which implies dial
H^lliX,."""-"""'""'"! ^"^'•^'i'-
S''^'^'' Eoe^s ^ ^ ^ = l^^l, 11 follows that
E E E
,.i,t-i>.^.'"™-"»^(/.-il.'''"''"'^''''^') =^™c^'s- 13"Ftom (29), (30) and (311, we deduce dial lim,,. „ ^ „ r f = 0. Next, wc compare T'^ to T«
defined by - "t n
'^^-Y^'Y Y\°\^K-h\)ny,„ < ^-nh,fl"^ = J - r f^dydt.
We have diat
c-f-^fi^JEs-E Es^i'.;-*!)^]
M E * E E I'ft.w-Ki'l
V..I x , < l - « « . /
< Ihli,, TdtS2\cill.
x^ -e -r
which leads to lim,,,„_„ | r f - r f ) = 0. On die odier hand, since
' ' * = " £ ' • • E E I ' l ' x " , . , • < = - / ' / nVi*(x.i)div,.(x.tWxdt.
it follows diat lim,„,.,_„ 7;» = - / ; 4 , „ , . „div^,., Ddtidt. Tlius. dte function .#= E
* J^ !m r ^Hl I T "ft'°'" '" "" " '"• ^'- ^'"" * = ° ™"'* "•"« *""= ft"
« e i (0. r . //„ (121), and die uniqueness of die Bmit implies dial the whole family v " / i »
weakly converges in l - ( K ' ' x ( 0 , n ) ' ' t o Vft as ; 5 , , J i - . o . * g
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3 A Priori Estimates
Lemma S Let & be a discretization of ii in the sense of Definition 2. and lei St be a time step in the interval (0, 7") in the sense of Definition 3. Let u% € X^ be the solution af Problem^ {PQ.SI). Let ii% = u%- P^u. There exists a positive constant C5 only depending -.
onK.kr.T.Q.aaswellason llclit.'^dK). ll«ok~C.fi)> llwllt^^tfl)'""'ll"llB"~(0)-t«cArtof
l"Slx|^C5, (32) V^\\%^C,. (33)
Proof Setting u = ii« in the scheme (11) and summing over n e [I, ...,JV} implies
Y Y i^i(^(4)-c(«r'))K-(p®")jf)
N N fj
+ Y^'{u".ii")F + Y^'(P^"^«")F + Y^'i"''-"")Q=0^ (34) which can be rewritten as
A| - A2 -I- B| -I- .S2 -I- C = 0. P5) where
^••=EEi'fi('«)-<(4-'))"i,
' 1 = 1 K^.^
B, =YSt{u".u")F.
N
B2 = Y^'(''^"-»")F.
N
C = E * ( " " . "">!!•
n = l
Next, we define
B J = c ( i i ' i ) » i - y 'c(r)rfT
Since r is nondecreasing, ,t follows that SJ > 0 for all „ e (, A'l and if ,= rf Moreover, we have diat • ' ^"" « £ - • « •
B J - B r ' . (c(.'i) - clu",-')) „ J -^ jT^^ ( , ( „ j ^ i , _ ^ . , ^ j ^ ^^^
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The Fmiie VfaJume SUSHI Method [or Richards Equanon
where die last term is negative. Thus.
N
^' ^ E E I'fIK - er") = E 1*^19^ - E Iflel-
Note diat 0 ^ is positive and 0 ° , can be written as
©Sr - jT ' (ciu\) - ciT))dx < 2||«ok-.i7,l|ck=.,R,.
It Implies that
- A | < 2 | I 2 | | | „ „ | | i . , „ , | | c | | t , . „ , . ,35, In view of die hypodieses ( . ^ ) and (J«5), we obtain
I'i2l<2|fi|l|i:il/..|„,lkk,.,ai. (37) We deduce from die coercivity propeny in Lemma 3 diat
«i2oE^'l""lx
X s - (38)Applying die first Holders inequahty and dien Young's inequality, we deduce that diets exists a posidve constant Q such dial for all e, > 0
\B2\ < E « ' y « ' » ' ' » « l l t ' l O , ' l ' i ? t i " l l l = , „ , ' i5i;C,lrF||n||i,,,„,-^&f;^,;Fllv^,;.|lJ^^^^^
- jiyQryii,-,,;,,„,„, + ^ c , « ' E * l " " l i , - 139)
^ As for die term C, we deduce from | g , , | < |„ | y and Young's inequality diat for all
l"^! = E " E E fc(«""«-.))sx,(.il-i;;)l
n=l k^-ltn^Es
= E»' E E {Trfi^) /CM|,;I -„-;,)
- 1 K.„,iSr^ ) \ \ III:, ' ' )
' i E * ' E E \-'\ik.rrK.y'-lJ2st J: X; ^T^iri-g-u',)'
-.-1 k,,r,,e, - . . I ,^^,t?. ''"'
' 5 ; ^ i " * ' ^ + y ^ E * i " " i i , - (40)
We deduce from (35) dial
fi] ~~Ai -i-A2-B2-C.
^ Springer
fCBieonerctaL
Bi < - A i + [Ai|-l-|B2|-HC|. (41) We gather the inequalities (36)-(41). Then in view of Definidon 4 of the space-dme norm
l"Sl',i, = E " . | i ' I » " l i 3 , . we deduce that
("-" ' , ^ " " g ) l 4 l i i . S 2|I2|ll<:|k-,El(ll«olll»(ni-l-||i;ili-i,j,l
Choosing ei = a/l2CiK) and €2 = tt/l2K) permits to complete die proof of Lemma 5, D
4 Existence of a Discrete Solution
Let )z e [0.1 i and ii"-^ e Xg,; we consider die following extended pttiblem. Find u"->' e Jf S" such that for all n e ;f ;^ „
- E l*^! ('•<«x"> - <•(""«"')) "K + Sllu"-". v)r -y pSilii'-'. ii)„ = 0. (42) Ke.n:
It can be shown by a similar proof as that of Lemma -"i that the solution of die extended problem (42) satisfies
''\ti""\l,<liC,<Ca. (43) where a only depends on -R.K.T.Q.^ as well as on | | e | | t « , „ . | | „ „ | | i „ „ , , | | i t . , „ |
•mdllnllivi^in,.
Theorem 1 (Exi.iience of a disctete solmion) The discrete problem tP.j „) possesses at least one .solution.
Pmof The extended problem (42) can be written as die abstract system of nonlinear equadons
« ( « " " , » " - ' , W ) = 0 , (44) where H is a conlinuous mapping from X3 K Xs, >t [0. 1) to Xs,. Indeed, setting u , =
1. nt = 0. for all Z. / If, !,„ = 0 for all o e <r, we obtain the equadon
" « ( ' • ( " « " ) • < • ( " " « • " ' ) . 4 " ' K ' ' ) , e ^ , . « ) = 0 foraU Ki.,».
and seding i-g = 0 for all K e . ^ . y , , ) and „,. = 0 for all o' ^ o. we deduce the equation
" " ( ^ ( " " I j . , , . , ( ( i C " ) „ < - , ) ^ ^ ^ -p\=a forallir e,«;„i.
Setting r = 2 / 5 . , we deduce from (43) that die system (44) has no .soluuon on die boundary ot die ball fi, of radius r for /i e [0, I ].
Before pursuing the proof, we recall results due to [4. Theorem 3.1).
fl Spnnger
11 r'
The Fiiilie \o\ame SUSHI IHediod for Richank Equation
Proposition 1 Ut M = [(f. Q. y ) with Q an open bounded set ofW. f e CiQ) and y t / O ^ J i and let d : M ^ J. be the topological degree. Vten d has the following propenies.
( d l ) d{id.a.y) = \fory&Q.
(d2) dif a. y) == d(f. 1 2 , . y) +d{f S22. y) whenever Qi and Qi are disjoint open subsets of i2 such that y ^ fii2 \ ( i 2 , U i 2 i ) ) .
(d3) dibit. •). Q. y{t)) is independent of t whene\-er /i : [0. 1] x I ? -*- E " and y : [0, 1] -" W are continuous and y(t) i fit. dQ) for every t £ [0, 1].
(d4) dif Q.y)^Q implies f-\y):^&.
N e x t ^ w e denote by diHi:u"-\ p). B,.Q) the topological degree of die application Hi: u"^ . p) widi respect to die ball B , and die n g h t - h a n d side 0. For M = 0, die system / / ( - . u" , 0) = 0 reduces to a linear system widi a positive definite matrix. A p p l y i n e property (d I) in Proposition I, w e obtain
diH{-.u"-\0).Br.Q) = 1.
Then, in view of die h o m o t o p y invariance of the topological degree (properly (d3) in Proposition I ) , w e have that
d(Hi: u"-'.p). Br.p) = d(Hi: u " - ' . 0 ) . Br. 0) f o r a l l p e (0. 1].
As a result, in the case w h e r e p= 1. w e have that diM{-.u"-\ l ) . a , . | ) = 1.
Thus, by the properly (d4) m Proposition 1, die system / / ( • . « " - ' . | ) is invertible T h e n t h e r e e x i s t s « ' ' s u c h l h a t / / ( . M ' - i . l ) = O s o d m t H * ^ = ( « " ) „ e l , ^ ) i s a solution of die
discrete problem iPi^j,). r-,
S Estimates on Space and Time IVanslales
In this section, w e perform estimates on time and space translates of die discrete saturaUon.
S.1 E s t i m a t e s o n T i m e T r a n s l a t e s
Let fsl denote die smallest integer larger or equal to j . We state widiout proof two technical lemmas deduced from [ 10|, which will be useful for proving the estimate on time translates.
Lemma 6 Ut T > 0. r > 0. St > 0. and N be a positive integer .such that t e (0 D as well as SI = T/N. Let iy")„^M be a family of non-negative real values Then
/ Y y"^' ^^Yy"
L e m m a 7 Z^r 7 > 0, r > 0. < > 0. 5r > 0, and N he a positive integer such tha, f € [0, r ) . r e (0. T) as well as S, = T/N. Le, iy- , „ , , , he a family af non-negative real values. Then
E l-"'+"'"l<f,<r^,,"
"=[(/*'!
e Springei
I
^ K, Brennw etri.*
Lemma S Let u | be a solution of Problem (Ps,_s,). There exists a positive constant Cf only depending on p such that for all v e (0, T), there holds
Proof Let p, = f ^ l and q, = \-^'], we obtain
[ ' X { - { n ^ « | ( x , t + r))-c {U'j^u%ix. 0))^dxd,
= j r El^l('^K')--(4))''^f- (45)
Since c is monotone and Lipschitz condnuous, we deduce diat
r E K^K'K')-'K))'<"s["L, T. m(•:(.;•)-e(I,;))(.» - , ; , „ .
Xf^-« " Ks-rr We substitute i; = itfi - ut' e x@,o in die scheme (8) to obtain
E I f I (<• K) - c (u-f')) (uj' - « | ) = -St(„". u'- - u*)p - Si(„", u" - »«)e.
(47) At first, we consider die first term on the nght-hand side of (47). Applying Holder's inequahty yields
[(»", U': -U")y\< Wl\u")p\ + 1(1,", ««.),!
S *r[|V,„"||^!|„,,||Vs,nl'i||t,,„|,
+ ^ l l ' » " " l l i > ( O l ' l l ' 3 » " l l l ! , o j ' , (48) Since 2ab < a^ -I- i,^ and in view of Lemma 2, one has
l l ' » « " l l i i , o ) ' I|v»«"l[i2„,,, + lV3»''|li,,„,,||Va«"||^il„,j
< i||V3„-"|lii,„|„ + jllVs,««||2.,„,, + ||V3„"[|J,,„,,
^Yl-'-lL + fl^'Mj.-i-CjKil,.
Next, we consider the second term in the tight-hand side of (47); in view of (40) widi llii'.uf -,t*)Q\ < l(»",»«)(,|-H(«".„«')e|
< d | C | l ; ' x - ^ j ^ k - l i ^ + | y | „ « | J , . (49)
© Springer
•nte Finiie Wjlume SUSHI Metliod TOT Ricbartts Equation
Taking die sum of (47) widi respect to n from 9, -!- I to p, and substimting (48)-(49) yields
E l * ^ U f < < > - ^ ( ' ' ' ^ » ( < - 4 ) (50)
< « [ f; !:s±i|i,-.,-i,+ ± a±i,„.,j^, f-,c,i„-,=,+.it2e=,].
In view of estimates (46M50) and Lemmas 6,7, and 5. (46) becomes f ' E tl!\{c(u'g)-clii'i)fdl
. ' i r ^ E ' . ( ^ I " " i ; , + ' ^ l « " l | , +C3,„-,i, +.,^P=)
< r L r ^ ((2C3 -I- 1 )C5 -^ ^ 7 - | i 2 | V ) We obtain
I I ( ' • ( n ' i 4 ( x . / -l-r)) - c ( n ' ; 4 ( x . , ) ) ) = r f x r f , s C,r.
where C, =. t , r ( ( 2 C 3 -f DCs -F J r i r a i t ' ) . n We tematk diaf there also holds diat
/ E l « ^ l ( ' ' ( < ) - r t " J > ) ' " " < < ' L , | K | r . (51)
• ^ ' ~ ' Xe.A- 5,2 Estimates on Space TVanslates
In diis section, we prove an estimate m die t - norm of differences of space translates of die discrete salutation. At first, we state without proof die following result from [ 1 ].
Leimna 9 Let^Jtbea discretization of SI m the sense ofDefintttan 2 and let r, > I) be such yiSdg. Idt.. < l / , / „ r » H , r e ,S;.|, , i / , „ e . C = {K.L). There exi,t q >2and (-8 > 0 only depending ond. 12 and ij such that
||n,»i«(x-|-y) - n.«.u,(xU,:,j,.| < C,|y|'||n.«.u,|| ._,,.
when-p = i j ^ . „, 5 x%. Ui = 0 outside QT aiid\\ I,,, ,., ts defined b,-125) . J ' "'"'•PPly • ' » . " " " " " ' ' l ^ " " " ' •- n'i,« J . We fiist extend n'y.'J. by Ooutside Qy and extend n „P,j,u by the boundary value outside QT. Figure 3 shows how we proceed
© Spni
I
ig.3 Extension of funcliomi
K. BrenneraaL
Lemma 10 Ut 9 be a discretization of S2 in the .'tense of Definition 2 and let q > 0 be such that n^dK.„/dL.a < l/ijforalla € ^iny. where .^„ = \K. L\. There exist q > 2 and C9 only depending on d. Q and i) such that
lk(n>g(« + y.'))-c(n>S(».'))lk=,s., < c,i,nin^i"ll,,j„.,
Vt e (/„_i.(„].Vy elR''. (32) PriMf From die Lipschitz continuity of c, we have
lk(n'i..4(x-i-y.il)-e(nfv.«(x,,))|^,,,„,<i, |n'i,„»(^ + ,,„_ni„S(x.,i||,,„,,,
We remark that n<;,„S(x -^ y,,) - n i „ S ( x , , ) = n ' ; , i | ( x -1-y,,) - n%ii%l,t.l).
Applying Lemma 9, we deduce that thete exists p > 0 such diat
IIn';,»S(x -1-,.,) - ni„S(x.,)II,,,,,, < Qiyr IIny,u' | ,^ ^ . (53)
The inequality (52) follows from (53) by setting C, =C,L, • Theorem 2 Let .^ be u family of discretizations ofn in Ihe sense ofDeftnilion 2 such that
there crisis p > pg for all S e .9 and let St 6 (0. 1). The family (c (O* « | ) ) , ,
»/o,,prav,,»o,c .,„,,,„,,,<,„, is relalively compact in LHQT)- In panicular. 'then, exists a subsequence of \c (n iiJ)], ,i,/„c4 „ denote again by ( c ( n ' ; , i , | ) | , and a function I c e t ; ""'" " " " ' ' • ' " ' " » " ""•""'" " ™ * ' " » '" ' • ' e ^ ) ">"» -r^i' •'"
Proof In view of Lemma 10. mtegrating in time, we obtain
ll'(n'>4(x + y,,))-c(n'i„S(x.,))f,.,,,^,„^,, <cliyP'fjtin^ii-lli,,..,,
TTieRiiiteVbljme SUSHI Method for Richards B
which by die inequality (26) and L e m m a 5 yields
Ik(n'i»S(x + y,,)) - c (ni4(x. t))||,.„.„„„ i v/J7c,Lv|'.
We c o m b i n e this result widi L e m m a 8 to obtain
l|c(n!^4('^+y-' + r))-c(n*;,4(x.r))|,,,^,^,„^„
< k W 4 ( x - F y , , - H r ) ) - c ( n V ^ ( x - K y , r ) ) | , , , ^ , ^ , ^ ^ , ,
+ lk(nM(x + y,/))-.(nV4(x.o)|L.,^..,„_,„
SCiodTrl'^^-Hyl").
where C m = m a x ( v ^ C 9 . -JOi).
Moreover, we recall that \\c{n'^u%) [ | ^ , , ^ . , , „ ^ ^ , < \n\T\\c\\l^^^^. A p p l y i n g t h e Frechet-Kolmogorov c o m p a c t n e s s theorem, w e d e d u c e that d i e s e q u e n c e [c{n\ii%) ] is relatively compact in LHQT). T h u s , there exist a ^ e L^Qj) and a s u b s e q u e n c e of F ( n . * " s t ) l ^hi'^h converges to i3 strongly in L-iQr) as /^t and fir tend to zero D
Lemma 11 U t ^ be a family of discretizations of Q in the sen.ie of Defmition 2 andlet St e (0, ]).Let{n'^u^)^^^ be a sequent e of appro.vimale solutions of Problem iPs's,) such that {r]%u%] converges to ii weakly in LHQT) and {c (n^'^u%)] be a sequence of approximale saturations which converges lo a limn i? strongly in L-{Qj) and a.e. m Qj as Icj and St tend to zero. Then i? = £•(»}
Proof We deduce from the monotonicity o f t that for all 0 e L-iQr)
Because of die weak convergence of n^^aU% and the strong convergence of c (n%y^).
respectively, the expression above tends to / J " / ( c ( 0 ) - ,3 ){0 - u )dxdt. Let S > O a n d set (t>^ii-i-S(&-c(u));weohmi^
S / ( ( ( » - h 5 ( i ? - f ( H ) ) ) - i ? ) ( t f - c - ( , 7 ) ) ^ x r f / > 0 Jo Jn -
We divide the inequality above by 5 and let , 5 ^ 0 . T h i s implies diat
- / / iciii) - &)-dxdi > 0 , JO Jn
s o d i a i r d i ) = 1? a.e, in QT.
^ S p n .
i^ K. BreiiMretjL' 6 Conveigence
Theoi^m 3 Ut .^ be a family of discretizations of Q in the sense af Definition 2 such that there exists n > ps-forall 9 ^.^.Let&t e (0. D and let {ii%),j,^ be afamilvof solutions of Problem iPg;s,). Then
(i) Theree.xistafi,nctionu e L'iH) and a .subsequence of \n^'^u%\. which we denote again by {n^^'ffii^}. which converges to ij weakly in L-{QT)asl5«. St ->• 0;
(ii) There exists a subsequence of [c {H^'^ii^^)} which converges to ciu) strongly in LHQT)ashj:St ^Q.
Moreover. H is a weak solution of Problem (P), H - H e Z.-(0. 7"; H^ (i2)) and Vgug converges to Vu weakly in L'iQr)'' as h^.St -* 0.
Proof Estimate (32) togedier with the discrete Poincare inequality (27) imply that the sequences [n^;^H^^| and {V^»^} are bounded Thus, there exist H in L^iQr) and a sub- sequence of {n^;^(r^}, which we denote again by {U%u%]. which converges weakly to ii mL-iQrn^ls'-Si ^0.
Lei H^ = « ^ - P.yu. It is easy to see that D ^ P,j.S converges to u sti-ongly in L^iS2}
aslcy ^ 0. We deduce that
n%r,% ^ i i = U-u in LHQT) as Is-. St - * 0,
It follows from Lemma 4 that u e L'-«). T; //J (J2)) and that V%u% converges weakly to Vi( m L-iQr)'' as hj,. St ->• 0. Moreover, vl^Pc^i't converges lo VH strongly in L-{i2) as l'_^ -* 0 [10, Lemma 4.4] Thus, we deduce that
V^J.ri% -t- V^Pryf, _ v,7 + V« in L-iQr) as 1^. 6t -^ 0, or else
V ^ U ^ ^ ^ V M in L-(QT)^sil&.at^O.
and diat H - M e i.-(0, T"; //J (12)).
By Theorern 2, there exist a function t? e L\QT) and a subsequence of {c (n%u%)\
such that c {n'^u%) converges to tf strongly in L^-iQr) as /g, and St tend lo zero. Also applying Lemma 11 we deduce diat i? = £•(«).
Next, we prove that u is a weak solution of Problem iP). We first introduce the function
<^'^{^lts C-(Q X [0. 7"]). ^ = 0 on 9f2 X [0. T]. ^i-. T) = 0|. (54) Let V ' e * a n d s e t . = /'5<iA(x.,„_,)in(8) Taking t h e s u m o n . = (I,, .,/V}. we obtain n + Ty + TQ^O. with
„ - l •'«
T-g - - ^ , 5 ^ ^ ^ *'(r("5rJ)S^^((fr(xj^./«-i)-ti'(x„./„_,)). (55)
1 = 1 Aferf-txetfA^
fl Springer
•nie Finiie ^fa^ume SUSHI Method for Richards E
Time Evolution T e r m
lelp" = C ( H ^ ) and q" = ^(xic.t„). A d d i n g and s u b t r a c d n g Xljcg ^ l / C l p ^ ^ ^ in dte expression of TV, w e d e d u c e that
rT = YY li^ip"^"-'-Y Y \f^\p"-v-' - Y I ' ^ I P V + E I^I^^"'
n=l Ke-rf- „ = | / - e ^ j f g ^ f,^ ^
= YY iKiP-r-'-Y Y I^IPV- Y \f^\py+ Y i^iz-v
(1=1 K^-^ „=l Ke.Yf K^.M K^.fi N
= ~YY \K\p"iq"-q''-')- Y I^IPV+ Y I/^IPV.
As a result, w c d e d u c e that Tj = A\ - A2 — A-t, where
^\^ Y t-^M«^5i'(x^,(,v).
Kf2.va
^2= Y l'^k(«^)(fr(xjf.O).
^^ = Y Y ['^lc(""^)('/'(XA-.f„)-iA(x;f,/„_,)).
Since i/r(x.(;^) = ^{x.T) = 0, the first term/I, vanishes.
Next, we add and subtract •Zx^.tf !K ' • ( " ^ ' / ' ( X . 0)^X to die terni A. and compare A^
with / „ C(«O(X))I;KX, 0)t/x. This yields
A2-j^c(uaix))>l,ix.0)dx^ ^ |A-|c(4)Vr(xj,,0)rfx- E /"c04);Kx.0)r/x + E ciu'*.)fix.O)dx- ciuoix))>}fix.O)dx
E h-i"'K)i'l'(XK-Q)-f(x.O))dx
^ H I ('^(''K)-t-(«o(x)))v(x,0)^x.
Applying hypodiesis (J«^), we deduce diat
|A2-y^(-(«„(x)),(r(x,0)^x < |U-|lt-(R, E f ifi^K.O)->lrlx.O)\dx +1-C Y ] K"5:-«o('')llv('(x.O)|(/x, (56) Since (fr e C-(r? x |0, T]), there exists a positive constant cf. which only depends on lfr,T and i2. such that
Y y . l ^ ( ' " f - 0 * - ^ ( x . O ) | < J 2 c f / a .
fl Spnnger
jfBte^
K. BTcnnen
J
We conclude that die first term on die nght-hand side of (56) converges to zero as % tends to zero. By die definition of u% in (7), die second term on die right-hand side o|',- (56) tends to zero as /g tends to zero. FinaUy, Aj -+ / „ c(uo(x))V'(x. 0)dx as Ig,, St tend' to zero.
Next, we add and subduct Y;_^^^ Y.Ke.^ !,'"_, JK ciu"^)h,-^dxdt to die difference A 3 - / I c{u{x,t))a,irdxdl.
Jo Jn to deduce that
" ' " i ['»''• •))»!* <''<'l = J2T, f':i''"K)Hrlttx.l,)-irlxg.l,-)))dx - E E /"' f cl-tiWilrdxdt
+ E E / / clu"g)3,tl,dttdl
»=l K&-ni-^'"-:-'1 -J j ciaiti. t))3,ilr dtidl
= E / / '("'i)(9,V'(XK, t) - Siiilx, l))d%dt + E E / / K"«r)-t;(«(",')))S.*»,t)t(itift.
Thus
! / ' ' • / • I
U 3 - / / clulx.t))3,iird\dt\
\ Ja IQ j
- E E / I MiiVWHtfOtK.D-Stiiltt.DWttdt 1=1 Ks-m '"-'-11-
+ E E / / l'(«!-)-<^('i(t,/))[|S,i(.(x,l)|<(iK/t. (57) n=l ke.^-""-!^!-
For aU X 6 ^ and for all /f e - # , we have
laitKxK.tl-SilKx,!)] < C j % ,
whete q * is a positive constant. Since c is bounded, the first term on die nght-hand side of (57) ttnds to zero as ( , , i , tend to zero. Since c{n%u%) strongly conveige!
to rt») ; " / - - ( e 7 ) . die second tetm lends to zero as / s . s, tend to zero. Thus, A, -.
lo Ji7tl"(x,'))3ii/'tfxdf as/s^,5/tendtozero. Weconcludediat
'"' " " i X " " " - ' " ' ' * • " " " - X r t " » ' « ) * ( x , 0) tfx as / , , J, ^ 0. (58)
fl Sponger
TTie Fmile \btuilie SUSHI MeUiod for Richards Equadon
Next, we consider die difliision term / > (55). Adding and subtractiiig /o la K ( x ) V | « | ( x . I) - V ^ ( i . Dd-xdt yields
TF- I K(x)Vi/(x, I ) . V,Kx, Ddttdl Jt> Js2
= J2" KVs«" . VsPsiH-n, l,-i)dx - j I K(x)V|iiS(x. I ) . ViKi. Ddtidt n=l ^^ JO J12
+ K ( x ) V | « | ( x . / ) . Vil.(x.Ddidt - f t Kmvaix.D-ViHn Ddtidt JO J£2 Jo Ji2
= E j ( / ''(x)V»i,"(x.t)-(Vi,Ps„li(x,t._|)-Vi>(x.t))<;x</l
+j j K(x) ( V | i , | ( x . I) - Vii(x, /)) - ViA(x. Ddxdt. (59) Since V|i,J(x. t) weakly converges to Vn in t ^ t C r ) , die second term on die right-hand
side of (59) tends to zero as 1^, 5t tend to zero.
We denote by fy die first term on the right-hand side of (59). We have dial
l^fl = E y / K(x)Vs,»"(x.l).(Vs/>y^(x./._|)-v^(x,i))dx</t
< J^l y«'»""<''-')ll,:!|0,jltV»Ps,Kx.t._i)-V,(,(x.t)|lt!,„,j.
togedier with
l|Vii.f»i(.(x.t._i)-V*(x,t)||^^,„,, < l | V s , p , , K x . t . _ i ) - v , | , ( x . i . _ , ) | l t , , „ , , , -H|Vi(,(x, t,_i) - V,Kx. i)l[i«,„|i (60) In view of die regularity of i/,. diere holds
l|Vi/.(x. / . _ , ) - Vii,(x, ( l | l i , | „ , i < C*Sl.
where cf is a
In view of [7. Lemma 4.4], we have ^V^P^ilrlx.t„-i) - Vl|r|x.t„-i)\\^^^^^J < c * where C, 'S a posidve constant. As aresult. die tetm ||Vs,Ps^i/,(x,t„_i)-V|fr(x.t)||^^ „,^
tends to 0 as fs., St tend to zero In view of Lemma 2 and estimate (33). die fu^t term on die nght-hand side of (59) tends to zero as /g. St tend to zero. We conclude dial
TF^ f f K(xlVii(x.i).
Jo JS2
Q Springer
^ K.Breiuiereiil.
Convection Term
Finally, we prove diat the convection term TQ tends to —^jf/o*:r(c((i))K(x)V; • Vijfix. I) dxdt as 1^. St -^ 0. For this purpose, we intnxjuce the following two terms
^C = Y^' Y Y ''riciul))gf:^il,ix„.t„^t).
n=i K-i./yoeffK
. - . / - i - l ) .
'^Q = Y^'Y Y ^>(c«.))sA-„i/-(x,-..
We show below diat lim/5.j,^o I^G - (7"p - T"^)! = 0.
''g-iT^-T^)
= Y^' Y I ^ *'W(/"^„))s^o(V'(xjc,/„-i)-^(x„./„_,))
-Y^' Y Z!''^'"'"*''X)>SK''(^(''«-'"-i)-V'(x„.'«-!))
n=\ K^.^^aSt^K
= Y^- Y Z^ 5'^<'<V'(x/:./„-.)-V(Xa.^-i)(tr(f(«5:„))-itf(c«))),(62)
1=1 K^.A'oe.?^
We denote by tg the term on the right-hand side of (62). Since \gK a\ < 'K\a|, using die Cauchy-Schwai^ inequality, we deduce from die Lipschitz continuity of the functions *, and c diat
'^e'' s Y^' Y Y l'^l'''^.^('/'(xA--''.-i)-'/'(x„.(„_i))^
^ E ^ ' E E P-i'^riciu",J)-kAciul))f
n - i K^.y/i^i?y °
- E ' " E E '^l^^"l^t^<'<'f-'i-l)-v!'(x„,i„-|}|-
Ke-^a^gn (63)
DeliniMon 4 and estimate (33) implies that the second let™ on the right-hand side of (63) IS bounded. In view of die regularity ptopetties of (», we deduce dial IT,,)' < ctll. As a result, we have — i w
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i i i r - • '
The Finite Volume SUSHI Method for Richaids B
Next, w e c o n s i d e r t e r m T^. B e c a u s e of die regularity of ^ . it is e a s y to see ttiat [TQ-TQ) ^ O a s / ® - » O w h e r e
N
^ = Y^' Y ^-(^K)) E j^'^z^^Kody
n=l /fe.*? H&SK
= Y^' Y ''rMu'i;)) f <iMK(x)Vzn^.t))dx
n = l K^.^ • ' * '
= j j kricin%u%))±viKix)Vzyr)dxdt.
Since c ( n ^ u | ) converges to dH) strongly m L^iQr), w e h a v e diat
^ e ^ / / i r ( f ( H ) ) d i v ( K ( x ) V ; i / f ) r f x d ( as I^.St ->• 0, (65)
Next, w e c o n s i d e r term Tg.
^Q = Y^' Y ^'•{c(u'-f.))ifixK,rn-i) Y l^'^-^^Kody
= E ^ ' E * ^ ( ' - ( « A - l > l ^ ( x x - ' « - i ) / ' d " v ( K ( x ) V c ) r f x
/ / kric{n''^u%))diyiK(x)Vz)n^'Ps^dxdt.
Jo Jn '
where U^^P^^ix. t) = i / r ( x ^ , ( „ _ , ) for all ix,t)eKx [ ( „ _ , , ; „ ) . Since f ( n * ; ( , i / ^ ) c o n - verges to ciu) strongly in L^iQr) and since U^Ps^-^ converges lo ^|) strongly m LHQT)~
we deduce that
^Q-* j j * r ( c ( « ) ) d i v ( K ( x ) V j ) i ^ t / x r f ; as l^^.St - > 0. (66)
We remark that d i v ( K ( x ) V z V ) = d i v { K ( x ) V z ) , ^ -|- K ( x ) V ; V V . . It follows from ( 6 4 ) - (66) dial
TQ-^ - \ / kric{ii))Kix)Vz • Vi/dxdi Ja Jn
From ( 5 5 ) . ( 5 8 ) . (61), and (67), w e deduce that H satisfies die w e a k form (5) for lest funcUons (fr e * , Finally, w e deduce from die density of the set f in die set
* = j(fr e L\0. T: H^(Q)). a,^ ^ L^iQr). (fr(-, 7") = o | , (68)
diat a is a w e a k solution of the continuous problem ( P ) in die sense of Definition 1 D
flSprir
K, Brenner etii;'
1
7 Numerical Tests
7,1 The Honmng-Messing Problem
The Homung-Messmg problem is a statidanj test (cf. for instance [10]). We consider a horizontal flow in a homogeneous ground 12 = [0. IJ^ and set T = 1. Its chamcterisdcs are given bv
Slf) -- tslir) --
j n'/2 - 2arctan2(i^) if i^ < 0.
I tr /2 otherwise.
| 2 / ( l + « ' If i / , < 0 . ^ , ^ ,^
12 odienvise. "(x) = Id.
An analytical solution is given by
Pix, z. t) = ' ~s/2
- t a n f
if J < 0.
otherwise.
where s ~x-z-t.The problem after Kirchhoffs transformation is given by Problem (2) widi (Fig. 4)
ciu) = 9ip) = 7 r V 2 - 2 a r c t a n - f - - ^ j if/j < 0,
^ /2 otherwise,
Mile of (he spate dnmam where 4 9348 and lully .saturated elsewhere
fi Springer
The Finite \falunie SUSHI Method for Richards Equanon
l U i l e l Number or m
Mesh UniTomi Umfonn Unifonn Uniform Adaptive
i sieps A', mesh si7e / g . number of unknowns N,^. ihe er neiT(fjuj). and the experimental order of convergence eo
0.025 0 143
2.40 6.09 1 53 3.76- 5 62 1-32-
10-- 10-^
10--' 10--'
! 0 - ' 10--'
1-60 4-13 2.90- 1 83- 3.67- 2 19
•10- 10- 10- 10- 10- 10-
I -F pIx. z.
2plx.z.l) if p < 0.
otherwise. (69)
We apply the SUSHI scheme using an adapdve mesh driven by die vanal ration. We prescribe die Neumann boundary condition deduced from (69) on die hue v = 0 and an inhomogeneous Dmchlel boundary condition elsewheie. We use an initially square mesh, which is such dial each square can be decomposed again into four smaller square elements. Wheieas die suindard finite volume scheme is not suited to handle such a non- eonformmg adaptive mesh, the SUSHI scheme is compadble with these non-confonninE volume elements.
We introduce die tclaove e m r in LHQT) between die exact and die numencai .soluuon eiT(«) = - " S l . i l [|f.lig.
I"™«llt.,5„ • as well as the experimental order of convergence
roc I... log(cn-(ii,)/eiT(",ti)>
log(»s-,/''»,-i)
"here It, ts die solmion coiresponding to the space discietization 9;. Table 1 shows the erior using a umforni square mesh with various mesh sizes and dme steps m die four first lines Note dial die scheme is only tlrst-order accurate with respect to time: dierefore in order to obtiun second-order convergence we choose St proportional to h-„. We also compare die enor for die appraximaie saturation using a unifonn mesh and an'adaptive mesh with a siinilar number of unknowns. In bodi cases, about 300 unknowns (line 2-line 5) and l^oo unknowns (line 3-line 61. the adaptive mesh compared to the fixed one provides sli-hilv better results for die saturation r(«). The obsersed compuuitional gain in relative error is tadier small (about 10-20 %). which is due to die fact dial die area of high gradients of c is comparauvely large
0 Springer
K. Brenner etal 7.2 The HaverkaiDp Problem
We considerthe case ofasandgroundrepresented by the space domain^ = (0.2)x(0.40) on the time interval [0,600], The parameters are given by [12] (Fig. 5)
I -1- \ap\P if p <0.
otherwise.
where 0, = 0,287, 9, = 0.075, a = 0.0271, /i = 3.96. K, = 9,44c -5, A = 0.0524, and y = 4 74. From 8 and AT, we have tabulated suitable values for the functions c and K^.
We have taken here the initial condition p = -61,5, a homogeneous Neumann boundary condition for J: = Oandj: = 2. the Dirichlel boundary condition/) = -61.5 for; = Oand p = -20.7 for;: = 40.
p
K(P)
\ P
. *.
( a ) Profiles of saturation e{p) and permeability t , ( p ) in the Haverkamp problen
( b ) The ftinctions u[p) and t , ( u ) m the HaveAainp problem Fig. 3 Parameiers in Ihe Hiiverkamp problem
^ Spnnger
TTie Fmite \blume SUSHI t^eUiixl for Richards Equa
Wc use an adaptive mesh and die tune step Ji = I to peifonn die test. Figure 6a rep- lesents die pressun; profile at various dmes. In diis test, no analydcal solufion is known.
Therefotc, we compare our numerical soludon widi diat of Pierre Sochala [16. Fig. 2.6. p.
35] which is obtained by means of a finite element mediod. Our results are quite similar to his. Figure 6b shows die dme evoludon of die mesh at different times conespouduig to the pressure profiles in Fig. 6a.
t=120s
N I I I I I I I I I I I I
I I I I I 11 I I I I |-T^
Fig, i Time e«.ladon of Oi. pressure p and ihi «laplive mesh ia die Haveitamp pmbkn
© Springer
Acknowledgments We thank Professor Pascal Omnes as well as the referees for a careful rereading of our.
manuscnpi which has led lo many improvements This work was supponed by the ITN Mane Cune Network.
FIRST dnd Fondauon lacques Hadamard.
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neous porous media and Richards equation, ZAMM 94.560-585 (2014)
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12 Haverkamp. R.. Vauclin. M., Touma. J , Wierenga. P. Vachaud. G,: A companson of numencai simulation models for one-dimensionalmfiltradon. Soil Sci Soc Am.J 41.285-294(1977) 13 Homung, U . Numensche Simulation von gesaitigi-ungesaltigl Wasserfiilssen in por^sen Medien
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14 Kelanemer, Y- Transferts Couplifs de Ma,sse el de Chaleur dans les Milieux Poreux; Modehsauon et Elude Num^ntjue- Ph D Thesis. University Pans-Sud (1994)
'^ l!"''Ji^"«'''rr' ^ " " ' ^ ' ' ' ^ ' " = " ' ' ' • " " ' ^ ' • " ^ " f s^'T^'c'l-unsaiuniied now through porous media In-Deufl- imi) ^ * ^ ' ' ' ' ^ ' ^ ^ ^'"''' ^^'^"'''•"•" Computing, Prog Sci Compui. vol, 7. pp. 83-93 16, Sochalju P Miihodes Numenques pour les EcoulemenLs Souierrains et Couplage avec le RuissellemenL
Ph-D Thesis, Ecole Nal.onal des Poms el Chaussifes [200B)
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