WELL-BALANCED SCHEMES FOR CONSERVATION LAWS WITH SOURCE TERMS BASED ON A LOCAL DISCONTINUOUS FLUX FORMULATION

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1 WELL-BALANCED SCHEMES FOR CONSERVATION LAWS WITH SOURCE TERMS BASED ON A LOCAL DISCONTINUOUS FLUX FORMULATION K. H. KARLSEN, S. MISHRA, AND N. H. RISEBRO Abstract. We propose ad aalyze a fiite volme scheme of the Godov type for coservatio laws with sorce terms that preserve discrete steady states. The scheme works i the resoat regime as well as for problems with discotios flx. Moreover, a additioal modificatio of the scheme is ot reqired to resolve trasiets, ad soltios of o-liear algebraic eqatios are ot ivolved. Or well-balaced scheme is based o modifyig the flx fctio locally to accot for the sorce term ad to se a merical scheme especially desiged for coservatio laws with discotios flx. De to the difficlty of obtaiig BV estimates, we se the compesated compactess method to prove that the scheme coverges to the iqe etropy soltio as the discretizatio parameter teds to zero. We iclde merical experimets i order to show the featres of the scheme ad how it compares with a wellbalaced scheme from the literatre. 1. Itrodctio I this paper we stdy coservatio laws with sorce terms, ofte referred to as balace laws, a prototype of which is give by { t + f x = Ax, x, t R R +, 1.1 x, 0 = 0 x, x R, where is the scalar kow, f is the flx fctio, ad A is the sorce term. Freqetly the sorce term takes the form 1.2 Ax, = z xb, i which case 1.1 ca be see as a model eqatio for the Sait-Veat shallow water eqatios. We remark that the coefficiet z i 1.2 ca be discotios, which wold correspod to a discotios bottom topography. Formally 1.1 with the sorce 1.2 is eqivalet to U t + AU x = 0, where U =, z ad the matrix A is give by f A = b. 0 0 Date: November 9, Mathematics Sbject Classificatio. 35L65, 74S10, 65M12. Key words ad phrases. Coservatio law, discotios soltio, sorce term, fiite volme scheme, well-balaced scheme, covgerece, compesated compactess method. KHK has bee spported i part by a Otstadig Yog Ivestigators Award from the Research Cocil of Norway. 1

2 2 K. H. KARLSEN, S. MISHRA, AND N. H. RISEBRO The eigevales of the above matrix wave speeds are f ad 0, which ca coicide ad thereby reslt i resoace. If zx = x, the 1.2 redces to t + f x = b, which is the sal case of a atoomos sorce term. If b 1, 1.2 redces to 1.3 t + f x = z x. I fact, 1.3 ca be writte i the followig coservative form, 1.4 t + f zx x = 0, which is a example of a coservatio law with a spatially varyig coefficiet. These eqatios 1.4, i particlar whe the coefficiets are discotios, have bee stdied from a theoretical ad merical poit of view i a large mber of papers, cf. [1, 2, 4, 9, 13, 17, 20, 26, 27] ad refereces therei. The lik betwee 1.1 ad 1.4 will be the basis for the merical scheme itrodced i this paper. Soltios of 1.1 mst be iterpreted i the weak sese, ad so-called etropy coditios are sed to select a iqe weak soltio to the iitial-vale problem; This soltio is referred to as a etropy soltio. Weak ad etropy soltios of 1.2 are well defied whe z L cf. Sectio 2. Wheever b is idepedet of, oe ca iterpret 1.4 i the sese of distribtios, eve for discotios z. Oe of the key isses i desigig merical schemes for 1.1 is the resoltio of steady states. For the cotios problem, at a steady state ū = ūx the flx fctio f ad the sorce term A are balaced, i.e., ū satisfies 1.5 f x = Ax,. More detailed forms of 1.5 ca be derived for 1.2 cf. Sectio 2. The sal strategy of devisig merical methods for 1.1 is to se a Godov type merical flx i a fiite volme method copled with a cetered differecig of the sorce term. It is well kow that this does ot preserve discrete steady states [12]. Aother alterative is provided by the so-called splittig or fractioal steps method, which is based o separatig the pdates for the flx ad the sorce [18]. This method is also deficiet with regard to preservig discrete steady states. Becase of these difficlties, so-called well-balaced schemes have bee proposed. These schemes are desiged specifically for preservig steady states. A variety of well-balaced schemes ca be fod i literatre, see [12, 10, 5, 6] ad the refereces cited therei. For a partial overview, see also the itrodctory part of [14]. Or aim i this paper is to devise a well-balaced scheme for 1.1. The key elemet of or strategy will be a local trasformatio of the balace law 1.1 to a coservatio law with a space-time depedet discotios coefficiet: 1.6 t + fkx, t, x = 0 where f is the flx modified locally by the sorce. Eqatios of this type are by ow mathematically well-stdied withi a proper framework of etropy soltios ad varios types of merical methods have bee devised ad aalyzed for these eqatios cf. the list of refereces give above ad for 1.6 i particlar referece [15]. Ths, or strategy is to employ merical schemes desiged for coservatio laws with discotios coefficiets 1.6 to approximate soltios of 1.1 details are preseted i Sectio 3. This strategy is similar i spirit to that sed i [5, 18].

3 NUMERICAL SCHEMES FOR BALANCE LAWS 3 The fiite volme scheme devised herei is desiged to preserve discrete steady states ad ca therefore be called well-balaced. The scheme is very simple to implemet ad does ot reqire solvig algebraic eqatios or a addtioal etropy treatmet for discotios steady states. It resolves the trasiets to first order ad the steady states are resolved to machie precisio. The mai featres of the scheme are demostrated throgh a series of merical experimets i Sectio 5. We believe that the approach of sig a local discotios flx formlatio for desigig well-balaced schemes will lead to alterative merical schemes for systems of coservatio laws as well, icldig the shallow water eqatios with bottom topography ad the Eler eqatios for flow i a ozzle. We pla to address the extesio to systems i ftre works. Regardig covergece aalysis of well-balaced schemes, if f 0, it is possible to work withi the stadard BV boded variatio framework, see, e.g., [10, 11]. Whe resoace occrs, i.e., if f = 0 for some, the sitatio becomes more complicated. As with other problems eqatios with discotios flx experiecig resoace pheomea, there is geerally o spatial variatio bod for the coserved variable itself. I order to show compactess of approximate soltios, the so-called siglar mappig approach has bee sed i the last twety years, i particlar for coservatio laws with discotios coefficiets, cf. [1, 2, 9, 13, 17, 20, 26, 27]. More recetly, alterative aalytical tools have bee tilized for discotios flx problems, icldig compesated compactess [15, 16] ad etropy process soltios/kietic soltios [4]. Regardig covergece aalysis for eqatios with sorce terms, very few of the papers deal with the resoat case where BV estimates are ot available. Ideed, we kow oly of [23, 6, 13]. I [23, 13] the siglar mappig techiqe is sed prove the covergece of the Glimm, Godov, ad frot trackig methods, while the athors of [6] prove the covergece of so-called eqilibrim schemes sig the method of kietic soltios. I the preset paper we show that der sitable hypotheses o the flx fctio ad the sorce term, the approximate soltios geerated by or well-balaced scheme coverge to the etropy soltio of 1.2. The covergece proof tilizes the compesated compactess method [24, 25]. Let x deote the approximate soltio geerated by the well-balaced scheme ad let S, Q be a etropy-etropy flx pair. The the mai step i the compesated compactess method is to prove that the etropy prodctio S x t + Q x x is compact i W 1,2 loc. We remark that the W 1,2 loc compactess aalysis is o-trivial; It relies o the properties of the soltio of the Riema problem for a coservatio law with discotios flx. We have orgaized this paper as follows: I Sectio 2 we state or assmptios ad defie the otio of soltios to be sed later o. I Sectio 3 we preset or well-balaced scheme, while the covergece aalysis is give i Sectio 4. I Sectio 5 we report o a series of merical examples ad preset a compariso betwee or scheme ad a well-balaced scheme fod i the literatre. 2. Prelimiaries 2.1. Hypotheses. I this sectio we detail the hypotheses o f, A, 0 ad recall the otio of etropy soltios for coservatio laws. We assme that f satisfies the followig assmptios: A.1 f C 1 loc R. A.2 f has fiitely may poits of extrema.

4 4 K. H. KARLSEN, S. MISHRA, AND N. H. RISEBRO A.3 f is geiely o-liear. More precisely, f 0 for a.e. R. A.4 There exist fiite costats M, C f sch that > M implies f > C f log. Next we state the assmptios o the sorce term A. A.5 We have that Ax, 0 = 0, or Ax, = z x. The mappig Ax, is locally Lipschitz cotios i for all x with a Lipschitz costat C A. Regardig the iitial data, we assme A.6 0 x L R Defiitio of soltios. We defie weak soltios of 1.2 as follows: Defiitio 2.1. Sppose A.1 ad A.5 hold. A fctio L R R + is called a weak soltio of 1.2 if for all ϕ Cc R [0,, ϕ t + fϕ x + Ax, ϕ dxdt + x, 0ϕx, 0 = 0. R + R Next, we cosider the special case A.7 Ax, = z x for some fctio z L R BV R, i which case the followig defiitio of a weak soltio ca be sed: Defiitio 2.2. Sppose A.7 holds. A fctio L R R + is called a weak soltio of 1.1 if for all ϕ Cc R [0,, ϕ t + f zxϕ x dxdt + x, 0ϕx, 0 dx = 0. R + R Observe that this defiitio is meaigfl eve whe z is discotios, which is de to the coservative form of the sorce term A.7. Weak soltios are ot iqely determied by their iitial data ad have to be spplemeted with a etropy coditio to achieve iqeess. A etropy soltio of 1.2 is defied as follows: Defiitio 2.3. Sppose A.5 holds. A fctio L R R + is called a etropy soltio of 1.2 if for all oegative ϕ Cc R [0, ad for all etropy-etropy flx pairs S, Q the followig ieqality holds: 2.1 Sϕ t + Qϕ x + S Ax, ϕ dxdt + S 0 x ϕx, 0 dx 0. R + R R A pair of C 2 fctios S, Q is a etropy-etropy flx pair if S 0, Q = S f. It is sfficiet to establish 2.1 for the Kržkov etropy-etropy flx pairs 2.2 S c := c, Q c = sig c f fc, c R, where the sig fctio sig satisfies sig 0 = 0. It is well-kow fact that there exists a iqe etropy soltio of 1.1 if 0 L R ad Ax, is Lipshitz cotios i x ad locally Lipschitz cotios i. Moreover, if 0 BV R, the belogs to BV R 0, T for ay T > 0. Let s ow tr to the siglar case A.7. For the sake of havig a wellposed problem at or disposal, we shall reiforce A.7 by itrodcig the followig piecewise smoothess assmptio: R R

5 NUMERICAL SCHEMES FOR BALANCE LAWS 5 A.8 Coditio A.7 holds ad zx is piecewise C 1 with a fiite mber of jmp discotiity poits located at x 0 < x 1 < < x M, M 0. Uder this coditio we will employ a otio of etropy soltio take from [14]. Defiitio 2.4. Sppose A.8 holds. A fctio is said to be a etropy soltio of 1.3 if for all c R ad for all 0 ϕ Cc R [0,, c ϕ t + sig c f fcϕ x dxdt R R sig c z xϕ dxdt R D R + M + zx m + zx m ϕx m, t dt + 0 c ϕ dx 0, R + R m=0 where D = {x 0, x 1,..., x M }. For varios existece ad iqeess reslts for etropy soltios i the sese of Defiitio 2.3 we refer to [13] Covergece framework. We will se the compesated compactess method [24, 25] to prove covergece of or approximate soltios. For simplicity, we will se the Yog measre idepedet versio of this method [7, 19]. The followig lemma, which cotais the compesated compactess method as we rely o it herei, is take from [7, 19]. Lemma 2.1. Let { ε } ε>0 be a seqece of fctios sch that i ε C for all ε > 0; ii The two seqeces 2.4 {S 1 ε t + Q 1 ε x } ε>0 ad {S 2 ε t + Q 2 ε x } ε>0 are i a compact sbset of W 1,2 loc R R +, where for all c R. S 1 = c, S 2 = f fc, Q 2 = Q 1 = f fc, c f ξ 2 dξ, The there is a sbseqece of { ε } covergig a.e. to a fctio L R R +. We will also eed the followig techical reslt [22]. Lemma 2.2. Let Ω R d be a ope set. If 1 < q < 2 < r <, the { } { } compact set of W 1,q loc Ω boded set of W 1,r loc Ω { } compact set of W 1,2 loc Ω.

6 6 K. H. KARLSEN, S. MISHRA, AND N. H. RISEBRO 3. The merical scheme I this sectio we defie or the well-balaced scheme for 1.1 ad 1.2, which is a Godov type fiite volme scheme. Let t ad x be the time step ad mesh size respectively. Fix T > 0 ad set C = M expct, where C is a costat to be determied later. Now set M f = max f. [ C,C ] We assme that the time step ad the mesh size satisfy the followig CFL-coditio: 2λM f 1, λ = t x. Let t = t, ad x j = j x for = 0, 1, 2,... ad j =..., 1, 1/2, 0, 1/2, 1,.... Let deote the iterval [x j 1/2, x ad I the iterval [t, t +1. Set 1 j x, t = 1 Ij x1 I t, where 1 Ω deotes the characteristic fctio of the set Ω. The iitial data is defied by 0 j = 1 x x x j 1/2 x, 0 dx. Fixig a time level t o which or approximate soltio j is give, we describe ext how to costrct the approximate soltio +1 j at the sbseqet time level t +1. Let x ad B x be defied as x = j j 1 Ij x, B x = j B j 1 Ij x ad B j B j 1 = q x x j 1, x j, Ax, x, where q x is some approximate itegratio sch that q x a, b, hx b a hx dx, as x 0, for boded fctios h. For t [t, t +1 we solve the followig coservatio law with a discotios coefficiet: 3.1 t + f B x x = 0, x, t = x, ad the defie +1 j as +1 j = 1 y, t +1 dy. x By the CFL-coditio, waves emergig from the Riema problems at x = x will ot iteract, ad therefore we obtai j = j λ F F j 1/2, where F is the flx across x = x.

7 NUMERICAL SCHEMES FOR BALANCE LAWS 7 The flx F is determied by solvig the Riema problem for a coservatio law with discotios flx: { t + f Bj 3.3 = 0, x, 0 = x j, x < x, t + f Bj+1 = 0, x, 0 = x j+1, x x.. The Riema problems 3.3 ca be exactly solved [9, 3, 21]. These formlas provide simple expressios for the Godov flx to be sed across each iterface. For example, if f has oly oe miimm for = θ ad o maxima, the we have the followig formla for the iterface flx: F = max { f max { j, θ } + Bj, f mi { θ, } } j+1 + B j+1 For other flx fctios, similar explicit formlas ca be obtaied. For the covergece aalysis, we defie a approximate soltio x for t t ad x R by settig x x, t = x, t t t < t +1, x R, where is the etropy soltio of 3.1, cf. Defiitio 2.4 We are goig to show i the sbseqet sectio that 3.2 is a well-balaced scheme preserves discrete steady states exactly ad that x coverges as x 0 to a etropy soltio of Covergece Aalysis I this sectio, we will carry ot the covergece aalysis for the well-balaced Godov type scheme 3.2. First, we show that the scheme preserves discrete steady states. A discrete steady state for or scheme is defied by 4.1 f j+1 f j = B j+1 B j, j Z, where we observe that B j+1 B j xj+1 x j A y, y dy; Uder the form 1.2 we se the formla Bj+1 Bj = z x z x j + b j + z x j+1 z x + b j+1. The well-balacig properties of the scheme are defied below. Lemma 4.1. Let { } j be a seqece sch that ad j Z f j+1 f j = B j+1 B j, j Z, 4.2 f j f j+1 < 0 = f j > 0 ad f j+1 < 0, j Z. The +1 j = j, j Z, where { +1 } j is compted by the scheme 3.2. j Z

8 8 K. H. KARLSEN, S. MISHRA, AND N. H. RISEBRO Proof. Sice j satisfies both the Rakie-Hgoiot coditio ad the etropy coditio 4.2 across each iterface x, we have that F = f j + B j = f j+1 + B j+1, j Z. Therefore, F = F j 1/2 ad the lemma is proved. Remark 4.1. Lemma 4.1 says that a discrete steady state 4.1 satisfyig coditio 4.2 is preserved by the well-balaced scheme 3.2. The additioal costrait 4.2 is a etropy coditio essetially excldig der-compressive waves. Next, we will show that the approximate soltios are boded i L. Lemma 4.2. There exists a costat σ, idepedet of x, sch that x x, t Me σt, M := 0 L R. Proof. Let M, C A, C f be the costats defied i A.4 ad A.5. Withot loss of geerality we ca assme that 0 j < M. Now either f > C f log or f < C f log for > M. We assme that f > C f log, the other case beig similar. To start we assme idctively that j < Meσt =: M. Ay Riema problem arisig at t = t will ivolve left ad right flx fctios whose differece is boded by xc A λ 1 tc A =:. Frthermore ay left ad right states are smaller tha M. Therefore, ay states i the soltio will be less tha ū, where ū solves Therefore f ū = f M +. C f log ū log M, or ū M e /C f = Me σt +C A /λc f t. Settig σ = C A /λc f shows that +1 j < Me σt+1. Showig the correspodig lower bods is similar. We will apply the compesated compactess Lemma 2.1 to the seqece { x }. To this ed, we will eed to establish certai etropy dissipatio estimates to be able to verify the W 1,2 loc compactess of the etropy prodctio associated with { x }. To prove the dissipatio estimates we will adapt a approach from [16] developed for coservatio laws with discotios coefficiets. As x locally is the soltio of Riema problems for coservatio laws with discotios coefficiets, we start by recallig some reslts relatig to the problem { t + g k l, x = 0, x, 0 = l x < 0, t + g k r, x = 0, x, 0 = r x > 0, where k l,r ad l,r are give costats. For the momet, we jst assme that g is a cotiosly differetiable fctio. The Rakie-Hgoiot coditio tells s that the vales l,r = lim x 0 x, t, satisfy g 0 := g k l, l = g k r, r.

9 NUMERICAL SCHEMES FOR BALANCE LAWS 9 I geeral, this does ot determie l,r iqely, ad we eed a additioal coditio. We se here the so-called miimal jmp etropy coditio, which states that amog the possible choices we select l ad r sch that l r is miimal. This choice has the followig coseqeces see [9]: { g kl, g k l, l for all [ l, r], or l r = g k r, g k r, r for all [ l, r], 4.3 r l = { g kl, g k l, l for all v [ r, l ], or g k r, g k r, r for all [ r, l ]. Lemma 4.3 [16]. If the vales l ad r are chose accordig to the miimal jmp etropy coditio, the for ay costat c Q r r, c Q l l, c g k r, c g k l, c, where Q l ad Q r deote the Kržkov etropy flxes Q l v, c = sig v c g k l, v g k l, c, Q r v, c = sig v c g k r, v g k r, c. Next we cotie with the proofs of the etropy dissipatio estimates. We shall se the otatio [α]x, t = lim h 0 αx + h, t αx h, t, for ay qatity α = αx, t. Fix a etropy-etropy flx pair S, Q. The the etropy dissipatio of x associated with S, Q is defied to be 4.4 Eϕ = S x ϕ t + Q x ϕ x dxdt, ϕ Cc R R +, Π X,T where we let Π X,T deote the set [ X, X] [0, T ] ad where ϕ C Π X,T. Withot loss of geerality, we ca assme that X ad T are sch that X = x J+1/2 ad T = t N for some itegers J ad N. By a itegratio by parts ad the local Riema soltio strctre of the approximatio x, we ca split Eϕ as 4.5 Eϕ = I 1 ϕ + I 2 ϕ + I 3 ϕ + I 4 ϕ + I 5 ϕ, where I 1 ϕ = I 2 ϕ = I 4 ϕ = X X N 1 S x x, t ϕx, t t=t t=0 dx, X =0 X J N 1 [ ] S x x, t S x x, t + ϕx, t dx, t +1 j= J =0 t [ ] Q, Q,+ ϕx j, t dt,

10 10 K. H. KARLSEN, S. MISHRA, AND N. H. RISEBRO I 5 ϕ = I 3 ϕ = N 1 =0 J t +1 t N 1 j= J =0 Q j=j j 1/2 ϕx j, t j= J σ t +1 t dt, [σ [S ] + [Q]] ϕx + σt, t dt, where the smmatio over σ exteds to all shocks with speeds σ i the soltio of the Riema problem at the iterface x. We have also sed the otatio,± = lim x x, t. x x ± We have the followig lemma o the variatio across each time level. Lemma 4.4. Let x ad j be geerated by the well-balaced scheme 3.2. There exists a costat C = CX, T idepedet of x sch that N 1 J 4.6 x x, t 2 j dx C. =1 j= J Proof. I what follows we se the etropy-etropy flx pair S = 1 2 2, Q = f. Withot loss of geerality, we assme that the merical soltio x has compact spport, so that we ca se the test fctio ϕ = 1, which implies Eϕ = 0. Sice x is boded, 4.7 I 1 ϕ C 1 X, T ad I 5 ϕ C 5 X, T. Next we estimate I 2. Writig for x, t we fid I 2 ϕ = S S j dx,j = j dx,j = j dx = 1 2,j,j 2 j dx,,j as the secod term i the third lie above is zero sice 4.8 j = 1 x x dx. j j dx Regardig the term I 3 ϕ, we se the fact that x is the exact soltio of a Riema problem. Ths, by the etropy coditio, σ [S x ] + [Q x ] 0,

11 NUMERICAL SCHEMES FOR BALANCE LAWS 11 ad coseqetly I 3 ϕ 0. For ay covex C 2 fctio S, we have sig Lemma 4.3 ad a approximatio argmet see [16] for details Q,+ Q, B j+1 Bj xj+1 x j C A x, A y, y dy where C A is specified i A.5. If A.7 holds, the we ca bod the last term by the total variatio of z i the iterval I. I both cases we have I 4 ϕ C, where C = CX, T is a positive costat idepedet of x. This fiishes the proof of the lemma. Estimate 4.6 ca be coverted ito a estimate of the variatio of the approximate soltios, which is the cotet of the sbseqet lemma. Lemma 4.5. Let j be defied by the well-balaced scheme 3.2. There exists a costat C = CX, T idepedet of x sch that N 1 J x,,+ + j, + j+1,+ C, =0 j= J where J ad N are sch that T = t N ad x J+1/2 = X. Proof. Eqipped with 4.6, the proof follows alog the lies of [8] see estimate 7.13 o page 67 i that paper. Now we prove the W 1,2 loc compactess of the etropy prodctio associated with the approximate soltios. Lemma 4.6. Let x be geerated by the well-balaced scheme 3.2 ad let S i, Q i, i = 1, 2, be the etropy-etropy flx pairs defied i Lemma 2.1. Defie the fctioal Ê by Êϕ = S i x ϕ t + Q i x ϕ x dxdt, i = 1, 2. Π X,T {Ê} 1,2 The the seqece is compact i Wloc R R +. x>0 Proof. First we ote that by the L bods o x, we have that Êϕ C ϕ W 1, Π X,T, so that Ê is boded i W 1,r for ay r 2, ]. By sig the bods 4.7 ad Lemma 4.4 we coclde I 1 ϕ I 5 ϕ I 3 ϕ C 1 ϕ L Π X,T. To estimate the I 2 -term we split it as follows: I 2 ϕ = I 2,1 ϕ + I 2,2 ϕ,

12 I 2,1 ϕ =,j 12 K. H. KARLSEN, S. MISHRA, AND N. H. RISEBRO where S S j ϕ j dx, S S j ϕ j ϕx, t dx, I 2,2 ϕ =,j where ϕ j = ϕx j, t. Next S S j = S j 1 j + 2 S θ j 2 j, for some itermediate vale θj x. The itegral of the first term above over the iterval x j 1/2, x is zero. Therefore we ca write I 2,1 ϕ = 1 ϕ j S θ 2 j 2 j dx,j C ϕ L Π X,T, by Lemma 4.4. Let each term i I 2,2 be deoted by, 2,2. The, 2,2 ϕ = S S j ϕ j ϕx, t dx ϕ C 0,α Π X,T xα S S j dx, where α 0, 1 is to be chose later. The, by a weighted Yog s ieqality,, 2,2 ϕ x2α+1 ϕ C 0,α Π X,T xα x δ + ϕ C 0,α Π X,T x δ S S 2 j dx. Now, smmig over j,, ad sig Lemma 4.4 as whe bodig I 2,1 we arrive at I 2,2 ϕ ϕ C 0,α Π X,T x 2α 1 δ + C x δ C ϕ C 0,α Π X,T xδ x 2α δ Therefore, if we have that α > δ, I 2,2 ϕ C ϕ C 0,α Π X,T xδ. Next we estimate the term I 4 for the etropy-etropy flx pairs defied i 2.4. We have, by the properties of the etropy soltio of the Riema problem for coservatio laws with discotios flx, [Q 1 ] = f = f,, f,+ + Bj f,+ B j+1 + Bj+1 Bj

13 NUMERICAL SCHEMES FOR BALANCE LAWS 13 = B j+1 B j xj+1 x j A y, y dy C A x, from which the desired estimate follows: 4.9 I 4 ϕ t [Q 1 ] C ϕ L. j, Next we cosider the etropy flx Q 2. We have, [Q 2 ] =,+ f ξ 2 dξ By sig the L bods o x ad the fact that f C for boded we see that + [Q 2 ] f ξ dξ =: q, where we have itrodced the otatio { } { } = mi,,,+ + = max,,,+. If f does ot chage sig i the iterval, + we ca estimate the jmp i Q 2 as we estimated the jmp i Q 1, cocldig that 4.10 q xj+1 x j A y, y dy C A x. Let s assme that < θ < + ad that f θ = 0. The, by sig a Taylor expasio abot ξ = θ, + + q = f ξ dξ C ξ θ dξ θ + = C ξ θ dξ C 2 θ ξ θ dξ + θ 2 + θ 2 C + 2 = C,+, 2. So by combiig this estimate ad 4.10 with 4.9 we coclde that [Q i ] 2 C x +,+,, i = 1, 2, ad hece I 4 ϕ C t ϕ L Π X,T [Q i ] j, C ϕ L Π X,T T + x j, 2,+, CX, T ϕ L Π X,T.

14 14 K. H. KARLSEN, S. MISHRA, AND N. H. RISEBRO {Ê} Now we ca se stadard argmets see [15, 16] to coclde that x>0 compact i W 1,q for some 1 < q < 2. Hece by Mrat s Lemma 2.2, we obtai the W 1,2 compactess of the approximate soltios. To prove that ay limit of { x } x>0 is a etropy soltio, we shall eed the two scceedig lemmas. Lemma 4.7. Cosider the Riema problem { t + f + B l x = 0, x, 0 = l, x < 0, t + f + B r x = 0, x, 0 = r, x 0, where B l,r are costats. Let deote the limits For each fixed c R, = lim x, t. x 0 Q c Q c + sig ũ c B l B r, where ũ = or ũ = + ad Q c deotes the Kržkov etropy flx, cf Proof. Assme first that sig c = sig + c. The Q c Q c + = sig c f f + = sig c B l B r. Hece the lemma holds i this case. Next assme that sig c sig + c. If + <, the we have that + < c <. By the miimal jmp etropy coditio, i this case 4.3, either fc f or fc f +. Assme ow that the first of these ieqalities hold. The, sig i additio the Rakie-Hgoiot coditio, Q c Q c + = f fc + f + fc f + fc If fc f +, the we fid similarly that Q c Q c + f fc f f + = f + + B r fc + Br = f + B l fc + Br f c + B l fc + B r = sig c B l B r. = f + B l f + + B r Bl B r = sig + c B l B r. This cocldes the case where + < c <. The aalysis i the case where < c < + is similar. To prove that a limit fctio of { x } x is a etropy soltio, we shall eed the followig techical lemma. is

15 NUMERICAL SCHEMES FOR BALANCE LAWS 15 Lemma 4.8 [14]. Let Ω R d be a boded ope set, g L 1 Ω, ad sppose that {g ν } ν>0 is a seqece sch that g ν g a.e. i Ω as ν 0. The there exists a set Θ, which is at most cotable, sch that for ay c R \ Θ, sig g ν c sig g c a.e. i Ω as ν 0. Let c Θ ad defie E c = { x Ω gx = c }. There exists seqeces {c ν } ad {c ν } sch that c ν c ad c ν c as ν 0, ad c ν ad c ν are i R \ Θ, ad sig gc c ν sig gx c a.e. x Ω \ E c, sig gc c ν sig gx c a.e. x Ω \ E c. Now we are i a positio to state or mai covergece theorem. Theorem 4.1. Sppose coditios A.1-A.6 hold. Let x be the approximate soltios geerated by the well-balaced Godov type scheme 3.2. The there exists a limit fctio L R R + sch that x x 0 i L p loc R R + for p < ad a.e. i R R +. Frthermore, is a etropy soltio of 1.2. Proof. The claimed covergece of { x } x>0 to a limit fctio is a straightforward coseqece of Lemmas 4.6 ad 2.1. Let s ow prove that the limit fctio is a etropy soltio i the sese of Defiitio 2.4. Fixig ay costat c R, we cosider i what follows the Kržkov etropy-etropy flx pairs S c, Q c defied i 2.2. By 4.4 ad 4.5, S c x ϕ t + Q c x ϕ x + S c x A x, x ϕ dxdt R R S c x x, 0 ϕx, 0 dx I 2 ϕ + I 3 ϕ + I 4 ϕ + I 5 ϕ, R for ϕ Cc R [0,. The term I 3 ϕ is o-egative, while I 5 ϕ is zero if we choose J sfficietly large. We are left with I 2 ad I 4. By covexity of S c ad 4.8, I 2 ϕ 0. Regardig the term I 4 ϕ, we se Lemma 4.7 ad obtai I 4 ϕ =,j,j t +1 t ] [Q c, Q c,+ ϕx, t dt t +1 t xj+1 sig ũ c A x, x x, t dx ϕx, t dt, x j where ũ eqals either, or,+ accordig to Lemma 4.7. Defiig the piecewise costat fctios ũ x, û x ad ϕ x as ũ x x, t = j, ũ 1 x, t, û x x, t = x1 I t,

16 16 K. H. KARLSEN, S. MISHRA, AND N. H. RISEBRO ϕ x x, t = j ϕ x, t 1 I x, we have I 4 ϕ sig ũ x c A x, û x ϕ x dxdt. Π X,T From their defiitios ad sig Lemma 4.5, it follows that, N 1 J 2 ũ x û x L2 [0,T ] X,X C t x,,+ Similarly, =0 j= J + j, N 1 x û x L 2 [0,T ] X,X C t x Hece, it follows that J =0 j= J 2 + j+1,+ 2 = 0 t lim x 0 ũ x = lim x 0 û x = lim x = i L 1 locr R +. x j, j+1,+ = 0 t We also have that ϕ x ϕ for all x, t as x 0. Hece, sig Lemma 4.8, lim I 4ϕ S Ax, ϕ dxdt, x 0 Π X,T for ay c R\Θ. Comparig this with the last term i the doble itegral i 4.11 ad repeatig the argmet from [14] to exted the set of permissible c s to R, we coclde that the limit is a etropy soltio i the sese of Defiitio 2.3 Fially, we cosider briefly the siglar case A.8. Combig the above chai of argmets with those fod i [14] we ca prove Theorem 4.2. Sppose coditios A.1-A.4, A.6, ad A.7 hold. Let x be the approximate soltios geerated by the well-balaced Godov type scheme 3.2. The there exists a limit fctio L R R + sch that x x 0 i L p loc R R + for p < ad a.e. i R R +. Frthermore, is a etropy soltio of 1.3 i the sese of Defiitio Nmerical Experimets I this sectio we report several merical experimets with or scheme 3.2, ad compare it with the well-balaced scheme of [6] as well as with a stadard cetered sorce scheme. The well-balaced scheme of [6], which is formlated for the case 1.2, ca be writte as 5.1 v +1 j = v j λ F vj, v,+ F v, j 1/2, v j, where F is a cosistet ad mootoe merical flx fctio. Moreover, v,± j±1/2 solve the algebraic eqatios D v,± j±1/2 z j = D v j±1 zj±1,

17 NUMERICAL SCHEMES FOR BALANCE LAWS 17 where we defie D ext. A steady state is a soltio of 1.5. For the simpler case of 1.2, we have a more explicit represetatio of the flx-sorce balace. If b 0, we ca formally write 1.5 as 5.2 D zx = C, D = 0 f s bs ds, for some costat C. It is clear that 5.2 costittes a o-liear algebraic eqatio from which a steady state ca be calclated. I the more specific case 1.2, eqatio 5.2 takes o the form f z = C. I or comptatios, we take F i 5.1 to be the stadard Godov flx. We will refer to 3.2 as the AWBS scheme ad 5.1 as the BPV scheme. The stadard cetered sorce scheme defied as w +1 j = w j λ F w j, w j+1 F w j 1, w j 1 2 bw j z j+1 z j, where F is ay cosistet ad mootoe merical flx fctio. We will refer the above scheme as the CS scheme. Example 1. We start with a experimet that ivolves a o-liear flx ad a o-trivial bottom topography. Cosider 1.1, 1.2 with { f = 1 cosπx, 4.5 < x < 5.5, 2 2, b = zx = 0, Otherwise. I aalogy with the shallow water eqatios, we refer to z as the bottom topography. The topography z is cotios i this case. This example is take from [6]. We compte o the domai [0, 10] with iitial data 0 0 ad bodary data x=0 = 2 to eforce the steady state. It is easy to see that the steady state is give by x = 2 + z. The steady state is reached oce the shock frot has passed the domai. The reslts at the steady state are show i Figre AWBS: BPV:... CS : BT: o o o o o o o o o o o Figre 1. Discrete steady states compted by the three schemes for Example 1. The reslts of the AWBS ad BPV schemes are idistigishable. BT refers to the bottom topography. Note that the AWBS ad BPV schemes resolve the steady state accrately eve at this corse mesh resoltio, whereas the CS scheme does ot resolve the steady state well. The errors at steady state are show i Table 1.

18 18 K. H. KARLSEN, S. MISHRA, AND N. H. RISEBRO L L 1 CS AWBS BPV Table 1. Errors at the steady state for the CS, AWBS, ad BPV schemes with x = 0.1 i Example 1. Next we focs o the trasiets ad show the cotor plots of the reslts with the AWBS ad BPV schemes i Figre 2. Figre 2. Cotor plots i x, t plae of the soltios compted by the AWBS ad BPV schemes with x = 0.1 i Example 1. Observe that althogh both the AWBS ad BPV schemes coverge to the same steady state, their behavior i resolvig trasiets is very differet. I particlar, whe the right-movig shock is comig i from the bodary ad has yet to reach the o-trivial dip i the bottom topography, the BPV scheme prodces a travelig hmp which appears to be o-physical. O the other had, as soo as the shock hits the dip, the BPV scheme closely resembles the soltio compted by the AWBS scheme. To check whether the hmp is a merical artifact ad disappears as x 0, we display the soltios compted by the BPV scheme x = 0.1, 0.01 i Figre 3. From Figre 3 it is clear that while oe part of the hmp seems to disappear i the limit, the part to the right seems to remai ad is i fact amplified as the mesh is refied. Hece, the BPV scheme appears to coverge to a differet soltio tha the AWBS scheme i the trasiet phase of the flow. Example 2. We cosider the same problem as i Example 1, except that we replace the bottom topography zx by the the discotios fctio { cosπx 5 < x < 6, zx = 0, Otherwise. This topography is similar to the oe give i [6] bt with the opposite sig. The steady states are show i Figre 4 ad the trasiets are show i Figre 5. I this case, both the steady states as well as the trasiet soltios give by the AWBS ad BPV schemes are differet. The steady state of the AWBS scheme

19 NUMERICAL SCHEMES FOR BALANCE LAWS t = 3 BPVDelta x =0.1: BPVDelta x=0.01: Figre 3. The two soltios compted by the BPV scheme with x = 0.1, 0.01 at time t = 3 for Example AWBS: BPV: CS: o o o o o o o BT: t = Figre 4. The discrete steady states compted by the AWBS ad BPV schemes with x = 0.1 for Example 2. Figre 5. Cotor plots i x, t plae of the soltios compted by the AWBS ad BPV schemes with x = 0.1 i Example 2.

20 20 K. H. KARLSEN, S. MISHRA, AND N. H. RISEBRO is the etropy satisfyig steady state as it satisfies the relatio x = 2 + zx, whereas the steady state give by the BPV does ot satisfy this relatio ad hece is ot the etropy soltio. As i the previos experimet, the BPV scheme geerates travelig waves almost istataeosly de to the effect of topography ad coverges to the the wrog steady state. Despite the discotiities i the bottom topography zx, the AWBS scheme resolves the steady state almost to machie precisio ad correctly resolves the trasiet. Example 3. I or third example we specify the relevat fctios as follows: f = 2 3 3, b =, 0 x = 1, { cosπx, 4.5 < x < 5.5, zx = 0, otherwise. I this case, D = 2 ad soltios to the steady state eqatio 5.2 may ot exist or may be mlti-valed. Extra care is reqired to defie the BPV scheme see [6] for details, whereas the AWBS scheme is well-defied. I fact, it is very easy to implemet as the flx fctio is mootoe i this case. The trasiet ad steady state soltios compted by the AWBS scheme are show i Figre AWBS; t = AwBS: BT: t = Figre 6. Soltios compted by the AWBS scheme at t = 3, 10 with x = 0.1 for Example 3. BT refers to bottom topography. We observe that the resoltio of the steady state, which i this example is give by x = 1 + zx, is resolved to almost machie precisio. Additioally, the trasiets are resolved very well o this coarse mesh as well. Example 4. I or fial example we specify the data as follows: f = 1 2 2, Ax, = si2πx 2, 0 x = 0, 0, t = 1. I this example the strctre of the sorce term is very geeral ad well-balacig based o the o-liear trasformatio D ad 5.2 is ot possible, whereas the AWBS scheme is well defied. We have compted soltios o the domai [0, 5] ad determied the steady state by solvig the ODE 2 x = Ax,

21 NUMERICAL SCHEMES FOR BALANCE LAWS 21 by a high-order Rge-Ktta method. The steady state ad trasiet soltios are show i Figre AWBS: RK4 :o o o o o o o Figre 7. Soltios compted by the AWBS scheme at t = 20 with x = 0.02 for Example 3. The steady state is resolved qite well with the AWBS scheme. It is ot immediately clear how to modify other well-balaced schemes like the BPV scheme so that they ca be applied to preset problem. We remark that i this particlar example it was easy to determie the soltio of the ODE for the steady state as it tred ot to be a smooth fctio. I geeral, however, it may ot be possible to obtai etropy satisfyig steady states from ODE solvers withot first bildig the etropy coditio ito the solver, whereas AWBS scheme by costrctio atomatically captres the etropy soltio. Refereces [1] Adimrthi, J.Jaffre ad G. D.Veerappa Gowda. Godov type methods for scalar coservatio laws with flx fctio discotios i the space variable. SIAM J. Nmer. Aal., 42 1: , [2] Adimrthi, Siddhartha Mishra ad G.D.Veerappa Gowda. Optimal etropy soltios for coservatio laws with discotios flx fctios. J. Hyperbolic Differ. Eq., 2 4:1 56, [3] Adimrthi, Siddhartha Mishra ad G.D.Veerappa Gowda. Coservatio laws with flx fctio discotios i the space variable - III, The geeral case. Preprit, [4] F. Bachma ad J. Vovelle. Existece ad iqeess of etropy soltio of scalar coservatio laws with a flx fctio ivolvig discotios coefficiets. Comm. Partial Differetial Eqatios, 311-3: , [5] D. S. Bale, R. J. Leveqe, S. Mitra, ad J. A. Rossmaith. A wave propagatio method for coservatio laws ad balace laws with spatially varyig flx fctios. SIAM J. Sci. Compt., 243: electroic, [6] R. Botchorishvili, B. Perthame, ad A. Vasser. Eqilibrim schemes for scalar coservatio laws with stiff sorces. Math. Comp., 72241: electroic, [7] G.-Q. Che. Compactess methods ad oliear hyperbolic coservatio laws. I Some crret topics o oliear coservatio laws, volme 15 of AMS/IP Std. Adv. Math., pages Amer. Math. Soc., Providece, RI, [8] R. J. DiPera. Covergece of approximate soltios to coservatio laws. Arch. Ratioal Mech. Aal., 821:27 70, [9] T. Gimse ad N. H. Risebro. Soltio of the cachy problem for a coservatio law with a discotios flx fctio. SIAM Joral o Mathematical Aalysis, 233: , [10] L. Gosse ad A.-Y. Lerox. U schéma-éqilibre adapté ax lois de coservatio scalaires o-homogèes. C. R. Acad. Sci. Paris Sér. I Math., 3235: , 1996.

22 22 K. H. KARLSEN, S. MISHRA, AND N. H. RISEBRO [11] L. Gosse. Localizatio effects ad measre sorce terms i merical schemes for balace laws. Math. Comp., 71238: electroic, [12] J. M. Greeberg ad A. Y. Lerox. A well-balaced scheme for the merical processig of sorce terms i hyperbolic eqatios. SIAM J. Nmer. Aal., 331:1 16, [13] K. H. Karlse, N. H. Risebro, ad J. D. Towers. L 1 stability for etropy soltios of oliear degeerate parabolic covectio-diffsio eqatios with discotios coefficiets. Skr. K. Nor. Videsk. Selsk., 3:1 49, [14] K. H. Karlse, N. H. Risebro, ad J. D. Towers. Frot trackig for scalar balace eqatios. J. Hyperbolic Differ. Eq., 11: , [15] K. H. Karlse ad J. D. Towers. Covergece of the Lax-Friedrichs scheme ad stability for coservatio laws with a discotios space-time depedet flx. Chiese A. Math. Ser. B, 253: , [16] K. H. Karlse, S. Mishra, ad N. H. Risebro. Covergece of fiite volme schemes for triaglar systems of coservatio laws. Sbmitted, [17] C. Kligeberg ad N. H. Risebro. Covex coservatio laws with discotios coefficiets. Existece, iqeess ad asymptotic behavior. Comm. Partial Differetial Eqatios, : , [18] R. J. LeVeqe. Fiite volme methods for hyperbolic problems. Cambridge Texts i Applied Mathematics. Cambridge Uiversity Press, Cambridge, [19] Y. L. Hyperbolic coservatio laws ad the compesated compactess method, volme 128 of Chapma & Hall/CRC Moographs ad Srveys i Pre ad Applied Mathematics. Chapma & Hall/CRC, Boca Rato, FL, [20] S. Mishra. Covergece of pwid fiite differece schemes for a scalar coservatio law with idefiite discotiities i the flx fctio. SIAM J. Nmer. Aal., 432: electroic, [21] S. Mishra. Aalysis ad Nmerical approximatio of coservatio laws with discotios coefficiets PhD Thesis, Idia Istitte of Sciece, Bagalore, [22] F. Mrat. L ijectio d côe positif de H 1 das W 1, q est compacte por tot q < 2. J. Math. Pres Appl. 9, 603: , [23] A. Nossair. Riema problem with oliear resoace effects ad well-balaced Godov scheme for shallow flid flow past a obstacle. SIAM J. Nmer. Aal., 391:52 72 electroic, [24] L. Tartar. Compesated compactess ad applicatios to partial differetial eqatios. I Noliear aalysis ad mechaics: Heriot-Watt Symposim, Vol. IV, pages Pitma, Bosto, Mass., [25] L. Tartar. The compesated compactess method applied to systems of coservatio laws. I Systems of oliear partial differetial eqatios Oxford, 1982, volme 111 of NATO Adv. Sci. Ist. Ser. C Math. Phys. Sci., pages Reidel, Dordrecht, [26] J. D. Towers. Covergece of a differece scheme for coservatio laws with a discotios flx. SIAM J. Nmer. Aal., 382: electroic, [27] J. D. Towers. A differece scheme for coservatio laws with a discotios flx: the ocovex case. SIAM J. Nmer. Aal., 394: electroic, 2001.

23 NUMERICAL SCHEMES FOR BALANCE LAWS 23 Keeth Hvistedahl Karlse Cetre of Mathematics for Applicatios CMA Uiversity of Oslo P.O. Box 1053, Blider N 0316 Oslo, Norway address: keethk@math.io.o URL: Siddhartha Mishra Cetre of Mathematics for Applicatios CMA Uiversity of Oslo P.O. Box 1053, Blider N 0316 Oslo, Norway address: siddharm@cma.io.o Nils Herik Risebro Cetre of Mathematics for Applicatios CMA Uiversity of Oslo P.O. Box 1053, Blider N 0316 Oslo, Norway address: ilshr@math.io.o