Huazhong Tang 1 and Gerald Warnecke Introduction ANOTEON(2K + 1)-POINT CONSERVATIVE MONOTONE SCHEMES

Transcription

1 ESAIM: MAN Vol. 38, N o, 4, pp DOI:.5/man:46 ESAIM: Mahemaical Modelling and Nmerical Analysis ANOTEON(K + )-POINT CONSERVATIVE MONOTONE SCHEMES Hazhong Tang and Gerald Warnecke Absrac. Firs order accrae monoone conservaive schemes have good convergence and sabiliy properies, and hs play a very imporan role in designing modern high resolion shock-capring schemes. Do he monoone difference approimaions always give a good nmerical solion in sense of monooniciy preservaion or sppression of oscillaions? This noe will invesigae his problem from a nmerical poin of view and show ha a (K + )-poin monoone scheme may give an oscillaory solion even hogh he approimae solion is oal variaion diminishing, and saisfies maimm principle as well as discree enropy ineqaliy. Mahemaics Sbjec Classificaion. 35L65, 65M6, 65M. Received: December 5, 3.. Inrodcion Consider one-dimensional scalar hyperbolic conservaion laws: + f() =, (.) ogeher wih iniial daa (, ) = (), R. An eplici (K + )-poin finie-difference scheme approimaing (.) can be wrien as n+ j = G( n j K,, n j,, n j+k), (.) where K is any posiive ineger, K. We say ha he scheme (.) is monoone, if he fncion G is monoone wih respec o all is argmens, i.e. v i G(v K,,v,,v K ), K i K. (.3) We call (.) a conservaive scheme, if i can be cased in he form: ) n+ j = n j (h λ n j+ h n j, λ =, (.4) Keywords and phrases. Hyperbolic conservaion laws, finie difference scheme, monoone scheme, convergence, oscillaion. LMAM, School of Mahemaical Sciences, Peking Universiy, Beijing 87, PR China. hzang@mah.pk.ed.cn Insiü für Analysis nd Nmerik, Oo von Gericke Universiä Magdebrg, 396 Magdebrg, Germany. Gerald.Warnecke@mahemaik.ni-magdebrg.de c EDP Sciences, SMAI 4

2 346 H. TANG AND G. WARNECKE wih h j+ = h(n j K+,,n j+k ). (.5) Here and are sep sizes in ime and space, respecively, n j = (j, n ), and h j+ is a nmerical fl fncion. We assme ha λ is a consan and reqire he nmerical fl h j+ o be consisen wih he fl f() in he following sense: h(,,)=f(). There eis some sdies on he heoreical analysis of he monoone schemes. Haren, Hyman, and La [4] proved ha if he monoone difference approimaions converge as,, hey converge o he niqe enropy weak solion of hyperbolic conservaion laws (.). B monoone schemes are a mos firs-order accrae. Kznesov [6] proved ha monoone schemes for conservaion laws converge o he enropy solion in several space dimensions and provided siable error esimaes. Laer, Crandall and Majda [] proved a similar resl wiho he error esimaes. Sanders [9] proved convergence wih error esimaes for cerain hree-poin monoone schemes wih variable spaial differencing. The sharpness of he Kznesov s error bond was firs esablished by Tang and Teng [] who proved ha he bes L convergence rae for monoone schemes o (.) is one half if i incldes he linear fl case. This resl was hen eended o nonlinear fles by Sabac [8]. However, he half-order rae of convergence can be improved o order one for piecewise smooh solions wih conve fl []. De o he good propery of monoone schemes, hey have played a very imporan role in designing modern high resolion shock-capring schemes. However, o or knowledge, mos of sdies on nmerical approimaion and consrcions of he high resolion shock-capring schemes are condced by sing hree-poin monoone schemes. Do he monoone difference approimaions always give a good nmerical solion in sense of monooniciy preservaion or sppression of oscillaion? The prpose of his noe is o give an answer o he above problem from a nmerical poin of view. The resls will show ha a (K + )-poin monoone scheme may give an oscillaory solion even hogh he solion is oal variaion diminishing (TVD), and saisfies a maimm principle as well as a discree enropy ineqaliy. In his noe, we will invesigae he behavior of a special (K + )-poin scheme: n+ j = n j λ ( f( n K j+k ) f( n j K )) + αλ K (n j+k n j + n j K ), (.6) where α =ma { f () }. The scheme (.6) wih K = is considered as a generalized La-Friedrichs scheme. The scheme (.6) can be rewrien in he conservaive form (.4) wih he nmerical fl fncion: K f( j+ν ) α ( K ) (K ν ) j+ν, K >, h j+ = K ν= K+ K ν= K+ ( f(j+ )+f( j ) ) α j, K =, where j+ν = j+ν+ j+ν. Obviosly, (.6) is consisen wih he parial differenial eqaion (.). Moreover, i also saisfies he following properies: Lemma. (L sabiliy). If he iniial daa j, j Z, are bonded, i.e. hen he solion n j, j Z, o he scheme (.6) are also bonded: C j C, j Z, (.7) C n j C, j Z, (.8)

3 ANOTEON(K + )-POINT CONSERVATIVE MONOTONE SCHEMES 347 nder he CFL condiion Especially, he scheme (.6) is monoone nder he condiion (.9). Proof. In fac, nder he CFL condiion (.9), we have λ ma { f () } = λα K. (.9) G = αλ j K, G j±k = λ ( α f ( j±k ) ). (.) K Ths, he scheme (.6) is monoone nder (.9). Using (.) gives and n+ j C = G( n j K,, n j+k) G(C,,C ) = G( n j K,, n j K, n j+k) G( n j K,, n j K,C ) + + G( n j K,C,C ) G(C,,C ), n+ j C = G( n j K,,n j+k ) G(C,,C ) = G( n j K,,n j K,n j+k ) G(n j K,,n j K,C ) + + G( n j K,C,C ) G(C,,C ). This complees he proof of he firs par of Lemma.. Lemma. (TV sabiliy). If he oal variaion of he iniial daa j, j Z, is bonded, i.e. TV( )= j Z j+ j C, (.) hen nder he CFL resricion { λ ma j Z a j+ } λα K, (.) where a j+ saisfies a j+ j = f( j ), he solion n j, j Z, o he scheme (.6) is also TV-bonded: Proof. We rewrie he scheme (.6) in an incremenal form as follows: K n+ j = n j + C n j+ν+ n j+ν ν= TV( n ) C. (.3) ν= K D n j+ν+ n j+ν, (.4) where C n j+ν+ = λ ( ) α a n K j+ν+, D n j+ν+ = λ ( ) α + a n K j+ν+, (.5)

4 348 H. TANG AND G. WARNECKE and a j+ν+ saisfies a j+ν+ j+ν = f( j+ν ). Sbracing (.4) a j from (.4) a j +gives ( ) n+ j = C n j+ D n j+ n j + C n j+k+ n j+k + D n j K+ n j K. (.6) Taking he absole vale of (.6) and sing he riangle ineqaliy and he CFL resricion (.), we ge ( ) n+ j C n j+ D n j+ n j + C n j+k+ n j+k + D n j K+ n j K. (.7) Smming (.7) from j = o +, we ge by shifing indices TV( n+ ) TV( n ). I will complee he proof of Lemma.. Remark.. () From he proof of he Lemma., we can also conclde ha he scheme (.6) is monooniciy preserving, ha is o say, if he iniial daa j are monoone (eiher nonincreasing or nondecreasing) as a fncion of j, hen he solion n j shold have he same propery for all n. () Following he eising resls, e.g. [], he solion of he scheme (.6) shold also saisfy discree enropy condiion and converge o he niqe enropy solion of (.).. Nmerical analysis In his secion we condc some nmerical eperimens by sing he scheme (.6) o solve scalar conservaion laws (.) wih he fl f() =c or f() =,wherec is a consan. In he following, nless saed oherwise, λ is generally aken o be 3, and he compaional domain [ 8, ] is divided by 4 grid cells. Eample. The firs case is o solve iniial vale problem of scalar conservaion laws (.) wih he iniial daa {., 8 5, (, ) = (.8).4, 5. Or prpose of solving his eample o check he monooniciy-preserving propery of he solions calclaed by he scheme (.6). Figres and show he comped solions (lef) a = and he recorded oal variaion ( solid line ), maimm ( pls ), and minimm ( circle ) of he solions (righ) for f() =.8 and,andk = 8 and 5, respecively. In his case, he Coran nmber eqals o 3.6. We can see ha he recorded oal variaion, maimm, and minimm of he solions are kep consan, and he comped solions are monoone even if hey have become piecewise sep fncions. Eample. The second case is o solve he iniial vale problem of scalar conservaion laws (.) wih he iniial daa, 8.5, ( +.5)/,.5 <, (, ) = (.9) (.5)/, <.5,,.5.

5 ANOTEON(K + )-POINT CONSERVATIVE MONOTONE SCHEMES oal variaion, maimm, and minimm of he solion (a) K = oal variaion, maimm, and minimm of he solion (b) K = Figre. The comped solion (lef) and he recorded oal variaion ( solid line ), maimm ( pls ), and minimm ( circle ) of he solions (righ). f() =. These iniial vales for f() = consis of wo rarefacion waves which are conneced by a saionary shock a =. The corresponding iniial vale problem has been sed in [5] o check abiliy of he large ime sep Godnov scheme. In Figres 5 8 we give he comped solions a = and he recorded oal variaion ( solid line ), maimm ( pls ), and minimm ( circle ) of he solions for f() =. and,andk =, 3, 8 and 5, respecively. The resls show ha he comped solions are TVD, and L sable. B nmerical oscillaions have been generaed in he comped solions when a large ineger K> is sed. Following he above nmerical eperimens, we can conclde ha a (K + )-poin monoone scheme for K> may give an oscillaory solion even hogh he approimae solion is TVD, monooniciy-preserving,

6 35 H. TANG AND G. WARNECKE oal variaion, maimm, and minimm of he solion (a) K = oal variaion, maimm, and minimm of he solion (b) K =5 Figre. The comped solion (lef) and he recorded oal variaion ( solid line ), maimm ( pls ), and minimm ( circle ) of he solions (righ). f() =.8. and saisfies maimm principle as well as discree enropy ineqaliy. Moreover, he oscillaion canno be sppressed or eliminaed aomaically a a laer ime. Is a hree-poin monoone scheme non-oscillaory in he sense ha he nmber of erema of he solion h (, ) does no eceed ha of he solion h (, ), where >? To answer his qesion, we se he scheme (.6) o solve he Brgers eqaion wih iniial daa j = {, j,, j =. (.)

7 ANOTEON(K + )-POINT CONSERVATIVE MONOTONE SCHEMES 35 In his case, he Coran nmber,, andk are aken as.8,, and K =, respecively. In Table we lis he solions n j a several differen ime levels. The resl shows ha he nmber of erema of he solion j is larger han ha of he solion j, b i is redced a he laer ime. Tha means ha he hree-poin scheme (.6) may also generae a new eremm, b he oscillaion generaed by i can be eliminaed essenially by iself a laer ime. I is worh noing ha a similar eample has been applied by Tadmor in []. He se he La-Friedrichs (LF) scheme wih λ = / o solve he iniial vale problem of (.) wih f() = and j = δ j, {, j,, j =. (.) Table. The comped solion of he Brgers eqaion wih iniial daa given in (.). n TV( n ) n 4 n 3 n n n n n n 3 n His resl showed ha, in his case, he LF scheme amoned o a pre ranslaion, lacking he dissipaion o case any decay. However, if λ<, hen we can find ha he solion consrced by he LF scheme will no be a pre ranslaion; he new erema will also appear in i. Moreover, nmber of local erema in he solion of he LF scheme will no be diminished in his case. The main reason for generaion of local erema by he LF scheme is ha he LF scheme is in a saggered form, i.e., n+ j only depends on n j+ and n j, for all j Z. In oher words, he solions a = n+ wih odd (or even) inde j only depend on he solions a = n wih even (or odd) inde j. This phenomenon can also be observed when we se he LF scheme o solve he iniial vale problem of (.) wih f() =c and he iniial daa given in (.) or (.), where c is a consan. I means ha generaion of local erema by he LF scheme is no de o nonlineariy of he fl f() in (.). This seems o be differen from he scheme (.6). In Figre 3 we give a comparison of he solions of he iniial vale problem of (.) wih f() = and he iniial daa (.) calclaed by he LF scheme and he scheme (.6) wih K =, respecively. We also se he LF scheme o solve Eample wih p o = as before. The resls are shown in Figre 4. We can see ha he solion of he LF scheme is oscillaory. The main difference beween he hree-poin scheme (.6) and is (K + )-poin version is ha nder a siable CFL resricion, he hree-poin scheme saisfies he following local maimm principle: { } { } min n j+p n+ p=,± j ma n j+p, (.) p=,± which is sronger han he global one shown in Lemma.. I seems o be necessary o garanee ha he (K + )-poin scheme (.6) is nonoscillaory, a leas for his special iniial daa.

8 35 H. TANG AND G. WARNECKE Figre 3. The solions of he iniial vale problem of (.) wih f() = and he iniial daa (.). Lef: he LF scheme; righ: he scheme (.6) wih K = oal variaion, maimm, and minimm of he solion Figre 4. The solion (lef) comped by he LF scheme for f() =, and he recorded oal variaion ( solid line ), maimm ( pls ), and minimm ( circle ) of he solions (righ). De o nonlineariy, i is sill difficl now o give a sfficien condiion o garanee ha a generally conservaive difference scheme is nonoscillaory. Here, we wan o sae ha he above observed resls do no conradic he convergence of a (K + )-poin conservaive monoone scheme o he physically relevan limi solion. Many pracical compaions have also shown ha he solion o (.) calclaed by a hree-poin

9 ANOTEON(K + )-POINT CONSERVATIVE MONOTONE SCHEMES oal variaion, maimm, and minimm of he solion (a) K = oal variaion, maimm, and minimm of he solion (b) K = Figre 5. The comped solion (lef) and he recorded oal variaion ( solid line ), maimm ( pls ), and minimm ( circle ) of he solions (righ). f() =. scheme is essenially nonoscillaory. Ths, he hree-poin monoone scheme can be considered as essenially nonoscillaory scheme.

10 354 H. TANG AND G. WARNECKE oal variaion, maimm, and minimm of he solion (a) K = oal variaion, maimm, and minimm of he solion (b) K = Figre 6. Same as Figre 5. The eising high-resolion TVD schemes, e.g. Haren s 5-poin TVD scheme [], can also give a nmerical solion o many pracical problem ha is essenially nonoscillaory, becase hey may be acally considered as a hree-poin like scheme, for eample, we can wrie hem in he following incremenal form: n+ j = n j + C n j+ n j D n j n j. (.3)

11 ANOTEON(K + )-POINT CONSERVATIVE MONOTONE SCHEMES oal variaion, maimm, and minimm of he solion (a) K = oal variaion, maimm, and minimm of he solion (b) K = Figre 7. The comped solion (lef) and he recorded oal variaion ( solid line ), maimm ( pls ), and minimm ( circle ) of he solions (righ). f() =.. Under a siable CFL resricion, he incremenal coefficiens saisfy: C n j+, D n j, C n j+ + D n j, (.4) and hence he corresponding schemes saisfy he local maimm principle (.).

12 356 H. TANG AND G. WARNECKE oal variaion, maimm, and minimm of he solion (a) K = oal variaion, maimm, and minimm of he solion (b) K = Figre 8. Same as Figre 7. In he lierare, here have eised some niformly high order nonoscillaory schemes for hyperbolic conservaion laws, for eample, Haren and Osher s UNO scheme [3], and Li and Tadmor s hird order non-oscillaory cenral scheme [7]. However, consrcion of heir nonoscillaory schemes is based on he eac solion of he Riemann problem. In fac, heir schemes proceed in hree seps: firs, reconsrcing he solion o of is approimae cell-averages o he appropriae accracy; second, solving eacly local Riemann problem; and finally,

13 ANOTEON(K + )-POINT CONSERVATIVE MONOTONE SCHEMES 357 aking cell averages of he solion given in he second sep. The second and hird seps are also nonoscillaory. Ths, if he reconsrcion is also nonoscillaory, hen he approimae solion a a new ime level is also nonoscillaory. Acknowledgemens. This research was parially sppored by he Special Fnds for Major Sae Basic Research Projecs of China, he Naional Naral Science Fondaion of China, and he Aleander von Hmbold fondaion. The ahors also wish o hank Professor Randy LeVeqe, Tao Tang, and Jingha Wang for many ineresing discssions. References [] M.G. Crandall and A. Majda, Monoone difference approimaions for scalar conservaion laws. Mah. Comp. 34 (98). [] A. Haren, High resolion schemes for hyperbolic conservaion laws. J. Comp. Phys. 49 (983) [3] A. Haren and S. Osher, Uniformly high order accrae non-oscillaory schemes I. SIAM J. Nmer. Anal. 4 (987) [4] A. Haren, J.M. Hyman and P.D. La, On finie difference approimaions and enropy condiions for shocks. Comm. Pre Appl. Mah. 9 (976) [5] C. Helzel and G. Warnecke, Uncondiionally sable eplici schemes for he approimaion of conservaion laws, in Ergodic Theory, Analysis, and Efficien Simlaion of Dynamical Sysems, B. Fiedler Ed., Springer (). Also available a hp:// danse/bookpapers/ [6] N.N. Kznesov, Accracy of some approimae mehods for comping he weaks solions of a firs-order qasi-linear eqaion. USSR. Comp. Mah. Phys. 6 (976) 5 9. [7] X.D. Li and E. Tadmor, Third order nonoscillaory cenral scheme for hyperbolic conservaion laws. Nmer. Mah. 79 (998) [8] F. Sabac, The opimal convergence rae of monoone finie difference mehods for hyperbolic conservaion laws. SIAM J. Nmer. Anal. 34 (997) [9] R. Sanders, On he convergence of monoone finie difference schemes wih variable spaial differencing. Mah. Comp. 4 (983) 9 6. [] E. Tadmor, The large-ime behavior of he scalar, geninely nonlinear La-Friedrichs schemes. Mah. Comp. 43 (984) [] T. Tang and Z.-H. Teng, The sharpness of Kznesov s O( )L -error esimae for monoone difference schemes. Mah. Comp. 64 (995) [] T. Tang and Z.-H. Teng, Viscosiy mehods for piecewise smooh solions o scalar conservaion laws. Mah. Comp. 66 (997) To access his jornal online: