MONOTONE DIFFERENCE SCHEMES STABILIZED BY DISCRETE MOLLIFICATION FOR STRONGLY DEGENERATE PARABOLIC EQUATIONS

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1 MONOTONE DIFFERENCE SCHEMES STABILIZED BY DISCRETE MOLLIFICATION FOR STRONGLY DEGENERATE PARABOLIC EQUATIONS CARLOS D. ACOSTA A, RAIMUND BÜRGERB, AND CARLOS E. MEJÍAC Abstract. The discrete mollificatio method is a covoltio-based filterig procedre sitable for the reglarizatio of ill-posed problems ad for the stabilizatio of explicit schemes for the soltio of PDEs. This method is applied to the discretizatio of the diffsive terms of a kow first-order mootoe fiite differece scheme [S. Eve ad K.H. Karlse, SIAM J Nmer Aal ] for iitial vale problems of strogly degeerate parabolic eqatios i oe space dimesio. It is proved that the mollified scheme is mootoe, ad coverges to the iqe etropy soltio of the iitial vale problem, der a CFL stability coditio which permits to se time steps that are larger tha with the -mollified basic scheme. Several merical experimets illstrate the performace, ad gais i CPU time, for the mollified scheme. Applicatios to iitial-bodary vale problems are iclded. 1. Itrodctio 1.1. Scope. This paper is cocered with fiite differece methods for the followig iitial vale problem for a degeerate parabolic eqatio: t + f x = A xx, x, t Π T := R, T, T >, 1.1 where the itegrated diffsio coefficiet A is defied by x, = x, x R, 1.2 A = as ds, a, a L R L 1 R. 1.3 The fctio a is allowed to vaish o -itervals of positive legth, o which 1.1 degeerates to a firstorder scalar coservatio law. Therefore, 1.1 is called strogly degeerate. It is well kow that soltios of 1.1, 1.2 are, i geeral, discotios eve if is smooth, ad eed to be defied as weak soltios alog with a etropy coditio to select the physically relevat soltio, the etropy soltio. Applicatios of degeerate parabolic eqatios iclde two-phase flow i poros media, traffic flow, ad sedimetatiocosolidatio processes. Eve ad Karlse [1] itrodced a explicit mootoe differece scheme for the approximatio of etropy soltios of 1.1, 1.2 based o the first-order accrate, mootoe Egqist-Osher merical flx [2] for the covective part combied with a coservative discretizatio of the degeerate diffsio term. If x ad t deote the spatial meshwidth ad the time step, respectively, they proved covergece of the scheme to a etropy soltio as x, t provided that the followig CFL stability coditio is satisfied: λ f + 2µ a 1, λ := t/ x, µ := t/ x Similar coditios appear for explicit fiite differece schemes approximatig smooth soltios of strictly parabolic covectio-diffsio eqatios. For these eqatios, the so-called method of discrete mollificatio [3, 4] cosists i sig certai covex combiatios of fiite differece stecils rather tha a sigle oe. This Date: December 3, 29. A Uiversidad Nacioal de Colombia, Departmet of Mathematics ad Statistics, Maizales, Colombia. cdacostam@al.ed.co. B CI 2 MA ad Departameto de Igeiería Matemática, Facltad de Ciecias Físicas y Matemáticas, Uiversidad de Cocepció, Casilla 16-C, Cocepció, Chile. rbrger@ig-mat.dec.cl. C Uiversidad Nacioal de Colombia, Departmet of Mathematics, Medellí, Colombia. cemeia@al.ed.co. 1

2 2 ACOSTA, BÜRGER, AND MEJÍA device leads to a cosistet merical method with a ew CFL coditio that allows to se larger time steps, i.e. it stabilizes the give method. I or case, the applicatio of discrete mollificatio to the scheme itrodced i [1] leads to a mollified scheme whose CFL stability coditio is give by λ f + 2µε η a 1, where ε η := C η 1 w, 1, 1.5 where C η ad w are a parameter ad the cetral weight, respectively, of the discrete mollificatio operator that deped o the width η of the mollificatio stecil. Sice ε η, 1, for give x the coditio 1.5 admits to employ vales of t that are p to several times larger tha for the stadard, -mollified versio of the scheme with the restrictio 1.4. This accelerates the give scheme bt, as or merical experimets show, itrodces a at most moderate additioal error. It is the prpose of this paper to demostrate that discrete mollificatio ca also be applied to differece methods for the iitial vale problem for a strogly degeerate parabolic eqatio 1.1, 1.2. We prove that the ew method is mootoe ad coverges to the iqe etropy soltio of 1.1, 1.2 der the less restrictive CFL coditio 1.5. The performace of the method is illstrated i several merical examples Related work. Discrete mollificatio [5, 6] is a versatile covoltio-based filterig procedre for the reglarizatio of ill-posed problems [6, 7, 8, 9] ad the stabilizatio of explicit schemes for the soltio of PDEs. I [3] ad [4] the mollificatio method was itrodced as a stabilizer for merical schemes for strictly parabolic covectio-diffsio eqatios ad o-liear scalar coservatio laws, respectively. I [1] it is show that a particlar discrete approximatio of the secod derivative of a smooth fctio, based o discrete mollificatio, stabilizes operator splittig methods [11] for the merical soltio of covectiodiffsio problems. The method of [1] is applied herei to strogly degeerate parabolic eqatios. O the other had, mootoe schemes for first-order coservatio laws correspodig to A were first aalyzed i [12, 13]. Their attractive featre is the covergece to a etropy soltio, which remais valid for the applicatio to strogly degeerate parabolic eqatios. This was first exploited by Eve ad Karlse i [1]. Related aalyses iclde implicit mootoe schemes for degeerate parabolic eqatios [14], problems with bodary coditios [15], mltidimesioal degeerate parabolic eqatios [16], eqatios with discotios coefficiets [17, 18, 19], ad problems of parameter idetificatio [2] this list is far from beig complete. Of corse, the robstess of mootoe schemes, i particlar the covergece to the etropy soltio, comes at the well-kow price of the geeric limitatio to first-order accracy Otlie of the paper. The remaider of the paper is orgaized as follows. I Sectio 2 we preset prelimiary material, icldig a defiitio of a etropy soltio of 1.1, 1.2 i Sectio 2.1, a descriptio of the mollified basic scheme from [1] i Sectio 2.2, a precise statemet of the assmptios derlyig the covergece aalysis Sectio 2.3, ad a otlie of the discrete mollificatio operator ad its properties Sectio 2.4. Sectio 3 deals with the mollified scheme, which is motivated i Sectio 3.1, ad whose covergece to a etopy soltio of 1.1, 1.2 is show i Sectio 3.2. Based o stadard compactess argmets, we prove that der the CFL coditio 1.5 the mollified scheme is coservative ad mootoe, ad prodces iformly boded approximate soltios that satisfy the L 1 Lipschitz cotiity i time property. Moreover, der a additioal limitatio of the choice of the mollificatio stecil the approximatios of the itegrated diffsio coefficiet have the appropriate reglarity properties. Sice the scheme is mootoe, it satisfies a discrete etropy ieqality, ad by a Lax-Wedroff-type argmet we prove that it coverges to a etropy soltio of 1.1, 1.2 as t, x. The mollified scheme is frther spported by merical experimets preseted i Sectio 4, which are motivated by three differet applicative models that also iclde bodary coditios. Some coclsios are collected i Sectio Prelimiaries 2.1. Defiitio of a etropy soltio. We recall here the defiitio of etropy soltios of 1.1, 1.2 from [1]. Defiitio 2.1. A boded measrable fctio is said to be a etropy soltio of 1.1, 1.2 if it satisfies

3 MOLLIFIED SCHEMES FOR STRONGLY DEGENERATE PARABOLIC EQUATIONS 3 1 BV Π T ad A C 1,1/2 Π T. 2 For all o-egative test fctios φ C Π T with φ t=t = ad ay c R, the followig etropy Kržkov-type ieqality is satisfied: { c φ t + sg c f fc φ x + } A Ac φxx dt dx Π T c φx, dx. R It is well kow that etropy soltios of 1.1, 1.2 i the sese of Defiitio 2.1 are L 1 -cotractive, i.e., if 1 ad 2 are two etropy soltios correspodig to the respective iitial data 1 ad 2, the 1, t 2, t for t T. I particlar, etropy soltios of 1.1, 1.2 are iqe. This is, i fact, valid for iitial vale problems of more geeral strogly degeerate parabolic eqatios, also i several space dimesios. See [21] for details The mollified scheme basic scheme. We select a mesh size x > ad a time step t > sch that there exists a iteger N with N t = T. Let deote the vale of the differece approximatio at x, t for Z ad = 1,..., N. We the discretize the iitial datm by := 1 x +1/2 x 1/2 x x dx, Z, 2.2 ad calclate the soltio vales at time level t +1 from those at time t by the explicit marchig formla +1 = λ + F EO 1, + µ 2 A, Z, =, 1, 2,..., N 1, 2.3 where we defie the stadard differece operators + V 1 = V = V 1/2 := V V 1, V := V +1 V 1 /2, 2 V := V +1 2V + V 1, ad se the Egqist-Osher [2] merical flx give by F EO, +1 = f + + f +1, where f + := f + max { f s, } ds, f := mi { f s, } ds. Uder the CFL coditio 1.4 the scheme 2.3 is mootoe, therefore first-order accrate, ad coverges to the iqe etropy soltio [1]. The ew scheme will be based o replacig the term µ 2 A i 2.3 by a differet expressio ivolvig discrete mollificatio. We will therefore refer to 2.3 as the basic scheme Assmptios. With the otatio related to the discretizatio at had, we state, similarly to [15], as a frther assmptio that B := { BV R M1, M 2 > : x = cost. for x [ M 1, M 1 ], TVA x M 2 }. 2.4 This meas, i particlar, that there exists a costat M 3 sch that 2 A m M3 x iformly i x. m Z 2.4. The discrete mollificatio operator. The mollificatio method [5, 6] is based o replacig the discrete fctio y = {y } Z, which ca, for example, cosist of evalatios or cell averages of a real fctio y = yx give at eqidistat grid poits x = x + x, x >, Z, by its mollified versio J η y, where J η is the so-called mollificatio operator defied by [J η y] := η w i y i, Z, 2.5 i= η

4 4 ACOSTA, BÜRGER, AND MEJÍA η i = i = 1 i = 2 i = 3 i = 4 i = 5 i = 6 i = 7 i = 8 ζ e e e e e e e e e e e e e e e e e e e e e e e e e Table 1. Discrete mollificatio weights w i give by 2.7 with p = 3, alog with the vale of ζ defied i 3.15 see Lemma 3.5. where η N is the spport parameter idicatig the width of the mollificatio stecil ad the so-called weights w i satisfy η w i = w i, w i w i 1, i = 1,..., η; w i = The weights w i are obtaied by merical itegratio of the trcated Gassia kerel { A p δ 1 exp t 2 /δ 2 for t pδ, p 1 κ pδ t := where A p := exp s 2 ds, otherwise, p ad δ ad p are positive parameters. This kerel satisfies κ pδ, κ pδ C pδ, pδ, κ pδ = otside [ pδ, pδ], ad R κ pδ = 1. The we defie ξ 1/2 := 1/2 x for Z ad compte the weights by w i := ξi+1/2 i= η ξ i 1/2 κ pδ s ds, i = η,..., η. 2.7 Usally p = 3 is take ad δ, whose role is to determie the shape of the kerel s Gassia bell, is cosidered as reglarizatio parameter, ad it is estimated by meas of methods like Geeralized Cross Validatio GCV [6, 8]. I ay case, i this work the mai relatioship betwee δ ad η is give by δ = η + 1/2 x/p. This choice geerates weights w η,..., w η, that are idepedet of x. The resltig vales of w i for several vales of η ad p = 3 are give i Table 1. We coclde this sectio with some approximatio ad stability reslts. Lemma 2.1. The discrete mollificatio operator ca be writte i the forms η η [J η y] = y + ψ ψ 1 = y ρ i y i+1/2 + ρ i y +i 1/2, 2.8 where we defie ρ k := ψ := i=1 η ρ k y +k y k+1 = k=1 η 1 k= η+1 η w i, k = η,..., η; Q k = Q k := i=k i=1 Q k y +k+1/2, η i=k+1 ρ i, k =,..., η We assme that g is a sfficietly smooth real fctio, set y = gx, ad employ the Taylor expasio y +i = y + i x g x i x2 g x i x3 g x i x4 g 4 ξ,i,

5 MOLLIFIED SCHEMES FOR STRONGLY DEGENERATE PARABOLIC EQUATIONS 5 where ξ,i is a real mber betwee x ad x +i. The, defiig 1 η C η := i 2 w i, we ca write [J η y] = η i= η i= η w i y +i = y + x2 g x + x4 2C η 24 η i 4 w i g 4 ξ,i. 2.1 Theorem 2.1. Let g C 4 R with g 4 boded o R, ad set y = gx. If the data {y ε } Z satisfy y ε y ε for all Z, i= η the [Jη y ε ] [J η y] ε for all Z. Additioally, for each compact set K = [a, b] there exists a costat C = CK sch that [J ηy] gx x2 g x 2C η C x4 for all Z Moreover, the followig ieqalities hold for all Z, where C is a differet costat i each ieqality: [J η y] gx C x 2, [J η y] xg x C x 3, + [J η y] xg x C x 2, + [J η y] x 2 g x C x 4. Details of the proofs of Lemma 2.1 ad of the third part of Theorem 2.1 ca be fod i [3], while 2.11 is a way of rewritig Mollified scheme 3.1. Motivatio of the mollified scheme. The ew scheme is based o the cosistecy reslt for discrete mollificatio 2.11, which implies the approximatio g x = 2C η x 2 [J η g] gx + O x 2 as x. 3.1 Assme ow for the momet that A is a smooth fctio of x. The we have 1 x 2 2 A x = A xx x=x + O x 2, 2C η x 2 [J η A] A x = A xx x=x + O x 2, so we obtai the followig ew cosistet scheme if we replace the expressio µ 2 A i the right-had side of 2.3 by 2µC η [J η A] A : +1 = λ + F EO 1, + 2µCη [J η A ] A. 3.2 As we will see, the ew scheme 3.2 has a more favorable CFL coditio tha 2.3. For strictly parabolic covectio-diffsio problems where A is ideed smooth, 3.2 represets a obviosly cosistet scheme see [1]. For the preset settig, where we wish to approximate discotios soltios, ad A is oly Lipschitz cotios, this calcls oly serves as a motivatio for the ew scheme. However, as we will show, i the preset case the scheme is stified ad spported by a covergece aalysis, sice the Lax-Wedroff-type argmet ivoked to show that the scheme coverges to a etropy soltio will appeal to 3.1 oly with g replaced by a smooth test fctio.

6 6 ACOSTA, BÜRGER, AND MEJÍA 3.2. Covergece aalysis of the mollified scheme. Lemma 3.1. The scheme 3.2 is coservative, ad its merical flx is cosistet with 1.1. Proof. Accordig to Lemma 2.1 the mollificatio operator is ideed coservative ad we ca write where we defie ψ := [J η A ] = A + ψ ψ 1, η ρ k A +k A k+1, where ρk = k=1 so the marchig formla 3.2 ca be rewritte as η w i, = λ + F EO 1, + 2µCη + ψ The cosistecy follows by recallig 3.1, ad by otig that F EO, = f ad ψ = if k = for k = η,..., + η. For later prposes, we remark that de to Lemma 2.1, we ca write ψ = η 1 k= η+1 where the qatities Q,..., Q η 1 are defied i 2.9. i=k Q k A +k+1 A +k, 3.5 Lemma 3.2. The scheme 3.2 is mootoe der the CFL coditio 1.5, ad we have the followig iform L bod: for all = 1,..., N. 3.6 Proof. We deote by ad v the data { i } i Z ad { i } i Z, respectively, ad assme that vi = i for i Z with the exceptio of i = k, for which we assme that k v k. We write the scheme 3.2 as +1 = S, where S deotes the right-had side of 3.2. First, we cosider the case η k 2, i which we simply get S S v [ = 2µC η Jη A Av ] η = 2µC η w i A +i A v +i, sice A is o-decreasig. I the case k = 1 we get i= η S S v = λ F EO 1, F EO v 1, + 2µCη [ Jη A Av ], where we se that F EO is o-decreasig i its first argmet. The cases k = + 1 ad + 2 k + η ca be hadled by similar argmets. The case k = reqires special attetio. We have S S v = v λ F EO, +1 F EO v, +1 + λ F EO 1, F EO 1, v [ + 2µC η Jη A Av ] 2µC η A A v. Cosiderig that [ Jη A Av ] η = w i A +i A v +i = w A A v i= η ad sig that by the defiitio of F EO we have we obtai F EO, +1 F EO v, +1 = f + f + v, F EO 1, F EO 1, v = f f v, S S v = v λ f + f + v + λ f f v

7 MOLLIFIED SCHEMES FOR STRONGLY DEGENERATE PARABOLIC EQUATIONS 7 η ε η e-3 Table 2. Vales of the coefficiet ε η i the CFL coditio 1.5 for p = 3 ad differet vales of η ε η η Figre 1. Vales of the coefficiet ε η i the CFL coditio 1.5 for p = 3 ad differet vales of η. + 2µC η w 1 A A v v k { = 1 λ max { f s, } + λ mi { f s, } } 2µC η 1 w as ds k v k k 1 λ f s 2µCη 1 w as ds. 3.7 Uder the coditio 1.5 the itegrad i 3.7 is o-egative, so S S v, ad the scheme 3.2 is mootoe. The secod assertio, 3.6, follows from the mootoicity by a stadard argmet if we take ito accot that if w = ± for all Z ad w +1 = S w, the w +1 = ± for all Z; therefore, if are arbitrary data with ad +1 = S, the +1. Remark 3.1. Sice < w 1 ad the costat C η defied i 3.1 satisfies C η 1 if η is sfficietly large, the CFL coditio 1.5 trs ot to be less restrictive tha 1.4. The actal vale of ε η = C η 1 w for the sal choice p = 3 ad a rage of η-vales is show i Table 2 ad illstrated i Figre 1. Sice mootoe schemes are total variatio dimiishig TVD, we obtai the followig corollary. Corollary 3.1. Uder the assmptios of Lemma 3.5 the merical soltio { } has the TVD property: for all =, 1,..., N Z Z

8 8 ACOSTA, BÜRGER, AND MEJÍA Lemma 3.3. Assme that the CFL coditio 1.5 is satisfied. The there exists a costat C 1, which is idepedet of t ad x, sch that m C1 λ m for all, m {, 1,..., N}. 3.9 Z Proof. Let {, 1,..., N 1}. From 3.4 we obtai that +1 = 1 λ F EO, +1 F EO 1, µCη ψ ψ 1. Notig that F EO, +1 F EO 1, 1 +1 = D 2, D 1, 1, 3.1 where D1, = 1F EO v 1/2, 1 +1 ad D 2, = 2F EO, ṽ 1/2 +1 for some vales v 1/2, ṽ 1/2 betwee ad 1, we obtai F EO, +1 F EO 1, 1 +1 = D2, D 1, D 1, D2, 1 1. Moreover, we have that ψ ψ 1 = [Jη A ] A [ Jη A 1 ] A 1 = = η w k A +k A 1 +k A A 1 k= η η k= η w k a 1/2 v +k +k 1 +k a v 1/2 1 for some v +k 1/2 betwee +k ad +k 1. Coseqetly, we obtai +1 = λd1, λd 2, λ D1, D2, 1 2µCη a v 1/2 1 w η 1/2 + 2µC η w k a v +k +k 1 +k. k= η k De to the CFL coditio 1.5, all coefficiets i 3.11 are o-egative, so we obtai +1 λd 1, λd 2, λ D1, D2, 1 2µCη a v 1/2 1 w 1 + 2µC η η k= η k w k a 1/2 v +k +k 1. Smmig this over all Z we obtai +1 1, which meas that Z x Z +1 Z +k x We eed to show that the right-had side of 3.12 is O t. From 3.4 for = we obtai Z 1 = λ F EO, µCη ψ for all Z.

9 MOLLIFIED SCHEMES FOR STRONGLY DEGENERATE PARABOLIC EQUATIONS 9 Similarly to 3.1 we obtai F EO, +1 = d 2,+1/2 + + d 1, 1/2 1, where d 1, 1/2 = 1 F EO v 1/2, ad d2, 1/2 = 2 F EO 1, ṽ 1/2 for some v 1/2, ṽ 1/2 betwee ad 1. Moreover, i light of 3.5 for = we get so we have x Z ψ = η 1 k= η+1 Q k + A +k, 1 { t d1, 1/2 1 d2,+1/2 +1 } + t Z Z Now, i light of the assmptio 2.4, we may write x Z 1 2 t f TV + t η 1 k= η+1 Q k Z η 1 k= η+1 Q k 2 A x +k. 1 + A x +k C 1 t 3.13 with a costat C 1 that is idepedet of x ad t. Ths, combiig 3.12 ad 3.13 we obtai +1 C1 λ, which immediately implies 3.9. Z Lemma 3.4. Assme that the CFL coditio 1.5 is satisfied. The there exists a costat C 2, which is idepedet of t ad x, sch that η 1 Q k A +k+1 A +k C 2 x k= η+1 Proof. For each discretizatio x, t ad time t, the merical soltio vales a fiite rage of idices. Ths, from 3.4 we dedce that ψ 2C η x F EO, +1 F EO, ψ C η x i= 1 +1 i i 1 +1 i i C λ 1, λ i= i Z are costat otside F EO i, ψ i i+1 + 2C η x where C 1 is the costat of Lemma 3.3. We coclde that ψ C 2 x, where C 2 = C 1 + F EO /2C η, from which 3.14 follows if we take ito accot 3.5. Lemma 3.5. Assme that 1.5 is satisfied, ad that the mollificatio weights w i satisfy the restrictio η 1 ζ := Q 2 Q i = w 1 i=1 η i 2 2iw i >, 3.15 where we recall that the qatities Q,..., Q η 1 are defied i 2.9. The there exists a costat C 3, which is idepedet of t ad x, sch that A A l C3 l x for all, l Z i=2

10 1 ACOSTA, BÜRGER, AND MEJÍA Proof. We begi by settig z := A +1 A. Sice the iitial datm is assmed to be costat otside a boded iterval see 2.4, for a give pair of discretizatio parameters x, t there exists a iteger K >, which i geeral depeds o, sch that z = z = for > K. Additioally, from the triaglar ieqality ad Lemma 3.4 we obtai η 1 η 1 η 1 Q z Q i z i Q i z +i Q k z+k C 2 x, = K,..., K i=1 i=1 k= η+1 Actally, 3.17 is valid for Z, bt is trivially satisfied for > K sice z defiig the vectors d := z K,..., z K T R 2K+1, e := 1,..., 1 T R 2K+1 ad the 2η 1-diagoal 2K + 1 2K + 1-matrix Q Q 1 Q η 1. Q 1 Q M =. Q η 1 Qη 1, Q1 Q η 1 Q 1 Q = for these. Coseqetly, we ca rewrite 3.17 as the system of ieqalities Md C 2 xe, where holds i a compoet-wise sese. Clearly, de to its sig strctre, M is a L-matrix. Moreover, if 3.15 is satisfied, the M becomes a M-matrix, which meas that M 1 exists, M 1 i a compoet-wise sese, ad M 1 ζ 1. This implies that d C 2 xm 1 e i a compoet-wise sese, i particlar d C 2 ζ 1 x. Ths, 3.16 follows by takig C 3 := C 2 /ζ, ad otig that ζ does ot deped o x, t, or K. Remark 3.2. We have st proved that by imposig that the mollificatio weights satisfy the additioal coditio 3.15, oe ca establish the spatial reglarity property For p = 3, 3.15 is satisfied for η = 1,..., 5, see Table 1, which also shows the correspodig vales of ζ. I light of Lemma 3.5 there exists a costat C 4, which is idepedet of t ad x, sch that λ Z N 1 A +1 A 2 C = Lemma 3.6. Uder the assmptios of Lemma 3.5 there exists a costat C 5, which is idepedet of t ad x, sch that A A m C5 m t 1/2 for all, m {,..., N}. The proof of Lemma 3.6 is give by the proof of [17, Lemma 4.2], which i tr is based o a techiqe itrodced i [1]. The proof is based o a iterpolatio techiqe that exploits 3.16 ad does ot deped o the particlar scheme beig cosidered. As i [1], we deote by where = x, t the iterpolat of degree oe associated with the data poits { }. The fctio is cotios everywhere ad differetiable almost everywhere. From 3.6 i Lemma 3.2, Corollary 3.1 ad Lemma 3.3 we dedce that there is a costat C 6 sch that L Π T + TV ΠT C 6, 3.19 while Lemmas 3.5 ad 3.6 imply that there is a costat C 7 sch that A y, τ A x, t C7 x y + t τ 1/2 + x + t 1/2 for all x, t, y, τ Π T. 3.2

11 MOLLIFIED SCHEMES FOR STRONGLY DEGENERATE PARABOLIC EQUATIONS 11 Lemma 3.7. Let s recall the stadard otatio a b := mi{a, b} ad a b := max{a, b}, ad defie A ψ := ρ k +k Ac A k+1 Ac, k=1 where ρ k is defied as i 3.3. The the mollified scheme 3.2 satisfies the followig cell etropy ieqality: c R : +1 c c λ F EO c, +1 c F EO c, +1 c 2C η x ψ 3.21 k. Proof. Recallig the defiitio of ψ i 3.3, we obtai by replacig every ocrrece of i the defiitio of S by c, where c := { c} Z, the idetity S c = c λ F EO c, +1 c 2C η ρ k A x +k c A k+1 c. The same idetity holds if every is replaced by ad we defie c := { c} Z. Sbtractig the secod versio from the first, we obtai S c S c = c λ F EO c, +1 c F EO c, +1 c λ 2C η ρ k A +k c A +k c A k+1 c A k+1 c. x k=1 Sice A is o-decreasig, we ca rewrite this as S c S c = c λ F EO c, +1 c F EO c, +1 c + λ 2C η x ψ. O the other had, the mootoicity of the scheme implies that k= S c S c S c S c = +1 c Combiig 3.22 ad 3.23 we obtai the desired etropy ieqality Theorem 3.2. Assme that t ad x satisfy the CFL coditio 1.5, that the weights w η,..., w η of the discrete mollificatio operator satisfy the restrictio 3.15, ad that the iitial datm satisfies 2.4. The the iterpolated approximate soltio obtaied from the mollified scheme 3.2 coverges i the strog topology of L 1 Π T to a etropy soltio of 1.1, 1.2. Proof. De to the embeddig of L Π T BV Π T i L 1 Q T [22]; see [1], we dedce from 3.19 that there exists a seqece { i } i N with i for i ad a fctio L Π T BV Π T sch that a.e. o Π T. Moreover, i light of 3.2 the Arzelà-Ascoli theorem implies that A A iformly o Π T, ad we have that A C 1,1/2 Π T. It remais to prove that satisfies the etropy ieqality 2.1. This ca be doe by a stadard Lax- Wedroff-type argmet; amely, we choose a o-egative test fctio φ C Π T with compact spport o R [, T, mltiply the discrete etropy ieqality 3.21 by xφ, where φ = φx, t, sm the reslt over all ad, apply smmatio by parts, ad let. Details cf. e.g. [1, 15] will be omitted here, bt we metio that the smmatio by parts for the discretizatio of the diffsive terms i 3.21 ca be doe as follows: x Z N 1 = λφ 2Cη x ψ N 1 = x t = 2C η x 2 Z φ η w i A +i Ac A Ac i= η

12 12 ACOSTA, BÜRGER, AND MEJÍA Mollified Referece x Figre 2. Example 1: merical soltio at T =.5 calclated by the basic scheme 2.3 ad the mollified scheme 3.2 with η = 5 with x = 1/128. The solid lie is the referece soltio correspodig to x = 1/496. = x t = x t N 1 = Z N 1 = Z A Ac 2C η x 2 η w i φ i φ i= η A Ac φxx x, t + O x 2, where the last eqality follows from applyig 3.1 to the smooth test fctio φ. 4. Nmerical Examples I this sectio we preset merical soltios of some test problems to evalate the performace of the mollified scheme 3.2. For compariso prposes we will se the basic scheme 2.3 as referece. I both cases, the time step t is selected by cosiderig eqality i the respective CFL coditios 1.4 ad 1.5 with the right-had sides set to.98. Ths, for the mollified scheme we employ x 2 t =.98 x f. + 2ε η a I all examples a referece soltio was compted sig the basic scheme o a very fie grid. This referece soltio was the sed for approximatig the error of the schemes o coarser grids. More precisely, the relative L 1 -error was approximated by the qatity M 1 M U x U x i=1 where M is the total mber of grid poits, is the compted soltio at x i the simlated time t ad U x is vale of the referece soltio at the same time t i the grid poit x. The grids were bilt i sch a way that o spatial iterpolatio is eeded to evalate U x. However, the last time step was fixed for both methods so that the desired fial time is attaied exactly. i=1

13 MOLLIFIED SCHEMES FOR STRONGLY DEGENERATE PARABOLIC EQUATIONS 13 scheme 2.3 Mollified Scheme 3.2 with η = 3 1/ x L 1 -error cov. rate CPU time [s] L 1 -error cov. rate CPU time [s] e e e e e e e e e e Mollified Scheme 3.2 with η = 5 Mollified Scheme 3.2 with η = 8 1/ x L 1 -error cov. rate CPU time [s] L 1 -error cov. rate CPU time [s] e e e e e e e e e e Table 3. Example 1: approximate relative L 1 errors ad CPU times for the basic scheme 2.3 ad the mollified scheme 3.2 with η = 3, η = 5 ad η = 8, for the simlated time T =.5. a b.3 η = 3 η = 5 η = 8 Referece.15 η = 3 η = 5 η = 8 Referece Critical x x Figre 3. Example 2: a merical soltio at T = 1 calclated by the mollified scheme 3.2 with η = 3, η = 5 ad η = 8 for x = 1/16, b elarged view of a portio of a, showig also the soltio obtaied by the basic scheme 2.3 with x = 1/16. I both plots, the solid lie is the referece soltio calclated by the basic scheme 2.3 with x = 1/248. Example 1 Bckley-Leverett Problem. Cosider the satratio eqatio give by 1.1 with f = 2 2, a = 4ε 1, + 1 2

14 14 ACOSTA, BÜRGER, AND MEJÍA a b.3.25 Δ x = 1/16 Δ x = 1/32 Δ x = 1/64 Referece.15 Δ x = 1/16 Δ x = 1/32 Δ x = 1/64 Referece Critical x x Figre 4. Example 2: a merical soltio at T = 1 calclated by the mollified scheme 3.2 with η = 5 ad several vales of x, b elarged view of a portio of a. I both plots, the solid lie is the referece soltio calclated by the basic scheme 2.3 with x = 1/248. scheme 2.3 Mollified Scheme 3.2 with η = 3 1/ x L 1 -error cov. rate CPU time [s] L 1 -error cov. rate CPU time [s] e e e e e e e e e e e e Mollified Scheme 3.2 with η = 5 Mollified Scheme 3.2 with η = 8 1/ x L 1 -error cov. rate CPU time [s] L 1 -error cov. rate CPU time [s] e e e e e e e e e e e e Table 4. Example 2: approximate relative L 1 errors ad CPU times for the referece scheme 2.3 ad the mollified scheme 3.2 with η = 3, η = 5 ad η = 8, for the simlated time T = 1. o the domai [, 1] with iitial data x = 1 for x <.1 ad x = otherwise, ad ε =.1. This example was solved by a operator splittig method i [23]. Here, the bodary coditios sed for the comptatio of the merical soltio were, t = 1 ad 1, t =. For comptig the discrete mollificatio of A a reflectio of the data i the iterior was

15 MOLLIFIED SCHEMES FOR STRONGLY DEGENERATE PARABOLIC EQUATIONS Example 1 η = 3 η = 5 η = Example 2 η = 3 η = 5 η = L 1 Error L 1 Error CPU-time [s] CPU-time [s] Figre 5. Examples 1 left ad 2 right: approximate L 1 error of the basic scheme 2.3 ad the of mollified scheme 3.2 with several vales of η verss the CPU time i secods, for a simlated time T =.5 left ad T = 1 right with x = 2 for = 6,..., 11 left ad x = 2 for = 4,..., 1 right. implemeted. For details o the treatmet of bodary coditios with discrete mollificatio we refer to [3]. The reslts obtaied by the basic scheme 2.3 ad the mollified versio 3.2 are smmarized i Table 3 ad illstrated i Figres 2 5. The error is compted sig as referece soltio the reslt of 2.3 o a very fie grid with x = 1/496. For the mollified scheme we sed η = 3, η = 5 ad η = 8. Example 2. Next, we cosider 1.1 with f = 1, a = { for.1, 1 for >.1, x = { 1 for x [, 1], otherwise. Uder these assmptios, 1.1 trs ito a algebraically simplified versio of a diffsively corrected kiematic-wave traffic model [24]. We solve the problem p to T = 1. Agai we choose η = 3, η = 5 ad η = 8. I this case, the bodary coditios for ad the discrete mollificatio of A were of type Dirichlet, 3, t = 5, t =. However, x = 3 ad x = 5 are far away from the actal spport of the merical soltio, so the merical soltio coicides with that of the iitial vale problem 1.1, 1.2. See Table 4 for approximate errors, CPU times ad covergece rates, ad Figres 3 ad 4 for merical soltios where we illstrate the effects of differet vales of η at a give fixed spatial discretizatio ad of redcig x for a fixed vale of η, respectively. Figre 5 displays the approximate L 1 error verss the CPU time for the referece scheme ad the mollified versio with several vales of η.

16 16 ACOSTA, BÜRGER, AND MEJÍA a T = 8 s b T = 16 s Referece Mollified Critical Referece Mollified Critical x [m].8 x [m] c T = 24 s d T = 4 s Referece Mollified Critical Referece Mollified Critical x [m].8 x [m] Figre 6. Example 3: merical soltio for differet simlated times calclated by the basic scheme 2.3 ad the mollified scheme 3.2 with η = 5 ad x = L/64. I all plots, the solid lie is the referece soltio calclated by the basic scheme 2.3 with x = L/248. Example 3 Sedimetatio. Oe of the mai applicatios of strogly degeerate parabolic eqatios is a model of sedimetatio-cosolidatio processes of solid-liqid sspesios, see e.g. [25], where the fctios f ad A model the effects of hidered settlig ad sedimet compressibility, respectively, of a sspesio of local solids volme fractio. Or example is the same as [26, Sect ] ad [27, Example 1] where this problem was treated by a adaptive mltiresoltio techiqe, so reslts may be compared. We

17 MOLLIFIED SCHEMES FOR STRONGLY DEGENERATE PARABOLIC EQUATIONS T = 4 η = 3 η = 5 η = T = 24 η = 3 η = 5 η = T = 4 η = 3 η = 5 η = L 1 Error 1-2 L 1 Error 1-2 L 1 Error CPU-time [s] CPU-time [s] CPU-time [s] Figre 7. Example 3: approximate L 1 error of the basic scheme 2.3 ad the of mollified scheme 3.2 with several vales of η verss the CPU time i secods, for the simlated time T = 4 s left, T = 24 s middle ad T = 4 s right with x = 2 L for = 6,..., 1. cosider 1.1 with the choice { v 1 C for < < max, f = otherwise, ad a diffsio fctio A defied by 1.3 with a = fσ e ϱ g, where σ e = with parameters v < ad C > 1, { for < c, σ [/ c β 1] for > c, σ >, β > 1, where ϱ ad g are costats ad c the so-called critical cocetratio, β ad σ are parameters. If β is a iteger, the A ca be evalated i closed form as follows: { for < c, A = where A = v σ β k β + 1 l A A c for > c, ϱ g β 1 C+k β k. c C + l We cosider x [, L] ad time t [, T ]. The iitial ad bodary coditios are of the form x, = x for x [, L], f A x =, for x = ad x = L. k=1 l=1

18 18 ACOSTA, BÜRGER, AND MEJÍA a T = 4 s scheme 2.3 Mollified Scheme 3.2 with η = 3 1/ x L 1 -error cov. rate CPU time [s] L 1 -error cov. rate CPU time [s] e e e e e e e e Mollified Scheme 3.2 with η = 5 Mollified Scheme 3.2 with η = 8 1/ x L 1 -error cov. rate CPU time [s] L 1 -error cov. rate CPU time [s] e e e e e e e e b T = 24 s scheme 2.3 Mollified Scheme 3.2 with η = 3 1/ x L 1 -error cov. rate CPU time [s] L 1 -error cov. rate CPU time [s] e e e e e e e e Mollified Scheme 3.2 with η = 5 Mollified Scheme 3.2 with η = 8 1/ x L 1 -error cov. rate CPU time [s] L 1 -error cov. rate CPU time [s] e e e e e e e e c T = 4 s scheme 2.3 Mollified Scheme 3.2 with η = 3 1/ x L 1 -error cov. rate CPU time [s] L 1 -error cov. rate CPU time [s] e e e e e e e e Mollified Scheme 3.2 with η = 5 Mollified Scheme 3.2 with η = 8 1/ x L 1 -error cov. rate CPU time [s] L 1 -error cov. rate CPU time [s] e e e e e e e e Table 5. Example 3: approximate relative L 1 errors ad CPU times for the referece scheme 2.3 ad the mollified scheme 3.2 with η = 3, η = 5 ad η = 8, for the simlated times a T = 4 s, b T = 24 s ad c T = 4 s. The vales for the parameters are: v = m/s, c =.7, max =.5, C = 21.5, σ = 1.2 Pa, β = 5, ϱ = 166 kg/m 3, L =.16 m, g = 9.81 m/s 2 ad the iitial coditio x =.5 for x [, L]. The implemetatio of the bodary coditios for is the same i [26, Sect ] ad [27, Example 1].

19 MOLLIFIED SCHEMES FOR STRONGLY DEGENERATE PARABOLIC EQUATIONS 19 For the discrete mollificatio of A, we obtaied a liear extrapolatio at the borders. For istace at x =, we se the kowledge of A ad A x = f for gettig a iterpolatig straight lie with slope f. See Table 5 ad Figres 6 ad 7 for reslts. 5. Coclsios The covergece aalysis shows that stadard compactess ad etropicity argmets for fiite differece schemes for o-liear first-order coservatio laws ad strogly degeerate parabolic eqatios ca be applied to establish covergece of 3.2 to the etropy soltio of 1.1, 1.2 der a CFL coditio that allows a larger time step tha the basic mollified scheme. Althogh the covergece statemet holds oly for η 5 whe p = 3, ecoragig merical reslts were also obtaied i the case η > 5, as illstrated above with η = 8. A possibly sharper bod i Lemma 3.5 cold lead to a less restrictive covergece coditio tha We have kept here the argmets fairly simple, ad limited orselves to oe set of mollificatio weights, amely those obtaied i Table 1 for p = 3; this vale of p is also employed i [3, 1]. Variatios of p ad of the correspodig weights cold eqally tr ot more favorable coditios of satisfactio of Overall, the reslts of the merical soltios look ecoragig ad i the cases of Examples 1 ad 3 illstrate that the mollified scheme works well also for problems with bodary coditios, which we have ot iclded i or covergece aalysis see [15]. The comptatios have bee performed with the maximal time step allowed by the correspodig CFL coditios for the basic scheme ad its mollified versios. Oe shold keep i mid that the mollified schemes permit a larger time step, bt the evalatio of the discretizatio of A xx for the mollified scheme is algebraically slightly more ivolved that that for the basic scheme. I fact, i Examples 1 ad 2 a sigificat gai i CPU time is achieved oly for η = 5 ad η = 8. I all examples, the approximate errors for the basic ad mollified schemes at a give discretizatio are similar. Probably the resltig speed-p is ot as good as for implicit versios of the basic scheme, bt the mollificatio-based optio preseted herei is easy to implemet ad avoids, for example, the ecessity to solve systems of oliear eqatios that appear with implicit schemes. Frthermore, we metio that i ay case a merical soltio calclated by the mollified scheme o a portio of Π T reqires less storage space tha the soltio obtaied from the basic scheme. This makes it potetially iterestig to se the mollified scheme 3.2 as the basic forward solver for parameter idetificatio problems see [2], i which the coefficiets of the so-called adoit backward-i-time scheme deped o the properly stored soltio of the direct forward-i-time problem. Ackowledgemets CDA ad RB ackowledge spport by Fodap i Applied Mathematics, proect 151, BASAL proect CMM, Uiversidad de Chile ad Cetro de Ivestigació e Igeiería Matemática CI 2 MA, Uiversidad de Cocepció, ad proect AMIRA P996/INNOVA 8CM1-17 Istrmetació y Cotrol de Espesadores. I additio, RB is spported by Fodecyt proect ad CDA by Uiversidad Nacioal de Colombia, DIMA Proect Refereces [1] S. Eve ad K.H. Karlse, Mootoe differece approximatios of BV soltios to degeerate covectio-diffsio eqatios, SIAM J Nmer Aal 37 2, [2] B. Egqist ad S. Osher, Oe-sided differece approximatios for oliear coservatio laws, Math Comp , [3] C.D. Acosta ad C.E. Meía, Stabilizatio of explicit methods for covectio-diffsio problems by discrete mollificatio, Compt Math Applic 55 28, [4] C.D. Acosta ad C.E. Meía, Approximate soltio of hyperbolic coservatio laws by discrete mollificatio, Appl Nmer Math 59 29, [5] D.A. Mrio, The Mollificatio Method ad the Nmerical Soltio of Ill-Posed Problems, Joh Wiley, New York, [6] D.A. Mrio, Mollificatio ad space marchig, i: K. Woodbry Ed., Iverse Egieerig Hadbook, CRC Press, Boca Rato, 22.

20 2 ACOSTA, BÜRGER, AND MEJÍA [7] C.E. Meía ad D.A. Mrio, Nmerical idetificatio of diffsivity coefficiet ad iitial coditio by discrete mollificatio, Compt Math Applic , [8] C.E. Meía ad D.A. Mrio, Nmerical soltio of geeralized IHCP by discrete mollificatio, Compt Math Applic , [9] D.A. Mrio, C.E. Meía ad S. Zha, Some applicatios of the mollificatio method, i: M. Lassode Ed., Approximatio, Optimizatio ad Mathematical Ecoomics, Physica-Verlag, 21, pp [1] C.D. Acosta ad C.E. Meía, A mollificatio-based operator splittig method for covectio-diffsio eqatios, Compt Math Applic, to appear. [11] K.H. Karlse ad N.H. Risebro, A operator splittig method for oliear covectio-diffsio eqatios, Nmer Math , [12] M.G. Cradall ad A. Mada, Mootoe differece approximatios for scalar coservatio laws, Math Comp , [13] A. Harte, J.M. Hyma ad P.D. Lax, O fiite differece approximatios ad etropy coditios for shocks, Comm Pre Appl Math , [14] S. Eve ad K.H. Karlse, Degeerate covectio-diffsio eqatios ad implicit mootoe differece schemes. I: H. Freistühler ad G. Warecke eds., Hyperbolic Problems: Theory, Nmerics, Applicatios, Birkhäser-Verlag, Basel, Switzerlad, 1999, pp [15] R. Bürger, A. Coroel ad M. Sepúlveda, A semi-implicit mootoe differece scheme for a iitial-bodary vale problem of a strogly degeerate parabolic eqatio modellig sedimetatio-cosolidatio processes, Math Comp 75 26, [16] K.H. Karlse ad N.H. Risebro, Covergece of fiite differece schemes for viscos ad iviscid coservatio laws with rogh coefficiets, M2AN Math Model Nmer Aal 35 21, [17] K.H. Karlse, N.H. Risebro ad J.D. Towers, Upwid differece approximatios for degeerate parabolic covectiodiffsio eqatios with a discotios coefficiet, IMA J Nmer Aal 22 22, [18] K.H. Karlse, N.H. Risebro ad J.D. Towers, L 1 stability for etropy soltios of oliear degeerate parabolic covectiodiffsio eqatios with discotios coefficiets, Skr K Nor Vid Selsk 3 23, 49 pp. [19] R. Bürger, K.H. Karlse ad J.D. Towers, A mathematical model of cotios sedimetatio of flocclated sspesios i clarifier-thickeer its, SIAM J Appl Math 65 25, [2] A. Coroel, F. James ad M. Sepúlveda, Nmerical idetificatio of parameters for a model of sedimetatio processes, Iverse Problems 19 23, [21] K.H. Karlse ad N.H. Risebro, O the iqeess ad stability of etropy soltios of oliear degeerate parabolic eqatios with rogh coefficiets, Discr Cot Dy Syst 9 23, [22] L.C. Evas ad R.C. Gariepy, Measre Theory ad Fie Properties of Fctios. CRC Press, Boca Rato, FL, [23] H. Holde, K.H. Karlse ad K.-A. Lie, Operator splittig methods for degeerate covectio-diffsio eqatios II: merical examples with emphasis o reservoir simlatio ad sedimetatio, Compt Geosci 4 2, [24] R. Bürger ad K.H. Karlse, O a diffsively corrected kiematic-wave traffic model with chagig road srface coditios, Math Models Methods Appl Sci 13 23, [25] S. Berres, R. Bürger, K.H. Karlse ad E.M. Tory, Strogly degeerate parabolic-hyperbolic systems modelig polydisperse sedimetatio with compressio, SIAM J Appl Math 64 23, [26] R. Bürger ad K.H. Karlse, O some pwid schemes for the pheomeological sedimetatio-cosolidatio model, J Eg Math 41 21, [27] R. Bürger, A. Kozakevicis ad M. Sepúlveda, Mltiresoltio schemes for strogly degeerate parabolic eqatios i oe space dimesio, Nmer Meth Partial Diff Eqs 23 27,