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A fully discrete scheme for diffusive-dispersive conservation laws

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A fully discrete scheme for diffusive-dispersive conservation laws
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  Digital Object Identifier (DOI) 10.1007/s002110100267Numer. Math. 89: 493–509 (2001) NumerischeMathematik A fully discrete schemefor diffusive-dispersive conservation laws C. Chalons 1 , 2 , P.G. LeFloch 1 1 Centre de Math´ematiques Appliqu´ees & Centre National de la Recherche Scientifique,U.M.R. 7641, Ecole Polytechnique, 91128 Palaiseau Cedex, France;e-mail: lefloch@cmap.polytechnique.fr. 2 O.N.E.R.A., B.P. 72, 29 avenue de la Division Leclerc, 92322 Chˆatillon Cedex, France;e-mail: chalons@onera.fr.Received November 15, 1999 / Revised version received May 27, 2000 / Published online March 20, 2001 – c  Springer-Verlag 2001 Summary.  We introduce a fully discrete (in both space and time) schemefor the numerical approximation of diffusive-dispersive hyperbolic conser-vation laws in one-space dimension. This scheme extends an approach byLeFloch and Rohde [4]: it satisfies a cell entropy inequality and, as a conse-quence,thespaceintegraloftheentropyisadecreasingfunctionoftime.Thisis an important stability property, shared by the continuous model as well.Following Hayes and LeFloch [2], we show that the limiting solutions gen-erated by the scheme need not coincide with the classical Oleinik-Kruzkoventropy solutions, but contain nonclassical undercompressive shock waves.Investigatingthepropertiesofthescheme,westressvarioussimilaritiesanddifferences between the continuous model and the discrete scheme (dynam-ics of nonclassical shocks, nucleation, etc).  Mathematics Subject Classification (2000):  65M06, 35L65 1. Introduction Inthispaper,weareinterestedinthenumericalapproximationofthe limiting solutions generated by the diffusive-dispersive conservation law [3]: (1 . 1) ∂  t u + ∂  x f  ( u ) =  ǫβU  ′ ( u ) xx + ǫ 2 γU  ′ ( u ) xxx , u  =  u β,γ ǫ  ( x,t ) , x  ∈ R ,t >  0 , when  ǫ >  0  tends to zero. Here the flux  f   :  R  →  R  is a smooth givenfunction and  β,γ >  0  are fixed parameters. In the right-hand side of (1.1), Correspondence to : P.G. LeFloch  494 C. Chalons, P.G. LeFloch U   :  R  →  R  is a given, strictly convex function. The following is knownconcerning the limiting solutions  lim ǫ → 0 u β,γ ǫ  . (We refer to the review [3]and the references cited therein.) First of all, they generally depend on theparameters  β   and  γ  . Based on simple scaling arguments, one can see thatthey only depend on the ratio  δ   =  β/γ  . So we define (1 . 2)  u δ := lim ǫ → 0 u β,γ ǫ  , provided the limit exists in some strong topology. Based on (1.1) and ∂  t U  ( u ) + ∂  x F  ( u )=  ǫβ  2  U  ′ ( u ) 2  x  − ǫβU  ′ ( u ) 2 x  + ǫ 2 γ   U  ′ ( u ) U  ′ ( u ) xx  − U  ′ ( u ) 2 x / 2  x , it is easy to check that the function  u δ satisfies the hyperbolic conservationlaw (1 . 3)  ∂  t u + ∂  x f  ( u ) = 0 and the entropy inequality (1 . 4)  ∂  t U  ( u ) + ∂  x F  ( u )  ≤  0 , where  U   is regarded as an “entropy” of (1.3), and  F   :  R  →  R  is thecorresponding entropy flux defined by F  ′ ( u ) =  U  ′ ( u ) f  ′ ( u ) . Note also thatfrom (1.4) it follows that (1 . 5)   R U  ( u ( x,t )) dx  ≤   R U  ( u ( x,s )) dx, t  ≥  s. Inthispaper,weproposeafullydiscrete(inspaceandtime)finitediffer-ence scheme for the numerical approximation of the solutions  u δ in (1.2).We rely on the approach developed recently by LeFloch and Rohde [4] andbased on Tadmor’s notion of entropy conservative flux (see [6]) for the hy-perbolic part of (1.1). We will introduce here a fully discrete version of thesemi-discreteschemederivedin[4].Thehigh-orderaccuracyintimeispro-videdbyastandardRunge-Kuttatechnique.InSect.2,wecanprovethatourfully discrete scheme satisfies a cell entropy inequality, which implies thatthe entropy is a decreasing function of time. See Theorem 2.3 and Corollary2.4.In Sect.3, we investigate the properties of our scheme, especially interms of stability and nucleation. The stability condition derived in Sect.2is numerically investigated. We also demonstrate that the scheme admitsa nucleation threshold. Above the threshold value, nonclassical solutionsviolating the standard entropy criterion are observed. The dependence of the threshold in  δ   is investigated. We recall that these undercompressive,nonclassical solutions play an important role in many models of continummechanics when diffusive and dispersive effects are in balance [3].  Diffusive-dispersive conservation laws 495 2. A fully discrete scheme We start from the semi-discrete method derived by LeFloch and Rohde[4]. Consider the following scheme in conservative form (  j  describing theintegers) (2 . 1)  ddtu  j ( t ) =  − 1 h  g  j +1 / 2 ( t ) − g  j − 1 / 2 ( t )  , t  ≥  0 with g  j +1 / 2 ( t ) :=  g ( u  j − 1 ( t ) ,u  j ( t ) ,u  j +1 ( t ) ,u  j +2 ( t )  ,theparameter h >  0 being the mesh size. The numerical flux has the form  g  :=  g 1 +  g 2 +  g 3 ,where g 1 isconsistentwiththehyperbolicflux f  ( u ) , g 2 isanapproximationof   βhU  ′ ( u ) x , and  g 2 of   γh 2 U  ′ ( u ) xx . More precisely, using the entropyvariable v  :=  U  ′ ( u ) , g ( v ) :=  f  ( u ) , G ( v ) :=  F  ( u ) , and therefore  v 0  :=  U  ′ ( u 0 ) , etc, we define g 1 ( u − 1 ,u 0 ,u 1 ,,u 2 ) (2.2 1 ) :=    10 g ( v 0  + s ( v 1  − v 0 )) ds −  112  v 2  − v 1  − v 0  + v − 1  g ′ ( v 0 ) , (2 . 2 2 )  g 2 ( v 0 ,v 1 ) :=  − β  2 ( v 1  − v 0 ) , and (2 . 2 3 )  g 3 ( v − 1 ,v 0 ,v 1 ,v 2 ) :=  − γ  6  v 2  − v 1  − v 0  + v − 1  . Note that we are using the  same  notation for the exact flux  g  =  g ( v )  ex-pressed in the entropy variable and for the numerical flux. Theorem 2.1.  Thescheme (2 . 1) - (2 . 2) isconservatifandconsistentwiththehyperbolicconservationlaw (1 . 3) .Italsosatisfiesthecellentropyinequality( t  ≥  0  and   j  describing the integers) (2 . 3)  ddtU   u  j ( t )   + 1 h  G  j +1 / 2 ( t ) − G  j − 1 / 2 ( t )   ≤  0 , where the numerical entropy flux has the form G  :=  G 1 + G 2 + G 3  , where G 1 ( v − 1 ,v 0 ,v 1 ,v 2 ) (2.4 1 ) := ( v 0  + v 1 )2  g ( v − 1 ,v 0 ,v 1 ,v 2 ) −  12  ψ ( v 0 ,v 1 ,v 2 ) + ψ ( v − 1 ,v 0 ,v 1 )  with ψ ( v 0 ,v 1 ,v 2 ) :=  v 1 g ( v 1 ) − G ( v 1 ) + 112( v 1  − v 0 ) g ′ ( v 0 )( v 1  − v 2 ) ,  496 C. Chalons, P.G. LeFloch (2 . 4 2 )  G 2 ( v 0 ,v 1 ) :=  − β  4  v 21  − v 20  and  (2 . 4 3 )  G 3 ( v − 1 ,v 0 ,v 1 ,v 2 ) =  − γ  6  v − 1 v 1  + v 0 v 2  − 2 v 0 v 1  .  Moreover, the equivalent equation of the scheme  (2 . 1) - (2 . 2)  , up to the(third-order)termsin h 2  ,coincidewiththecontinuousmodel (1 . 1)  provided  ǫ  is replaced with  h .Proof.  It is straightforward to calculate that (2 . 5)  ddtU   u  j ( t )   + 1 h  G  j +1 / 2 ( t ) − G  j − 1 / 2 ( t )   =  − D  j ( t ) , where D  j ( t ) =  β  4  v  j ( t ) − v  j − 1 ( t )  2 +  v  j +1 ( t ) − v  j ( t )  2   ≥  0 . The second claim in the theorem follows by performing a Taylor expansionin (2.1) and by using the definitions (2.2).  ✷ We now turn to defining our fully discrete scheme, having in mind togeneralize Theorem 2.1, under some CFL stability restriction on the timediscretization. To begin with, we analyze a first order time-discretization.Denote by  k >  0  the size of the regularly spaced time-mesh. Consider thefollowing scheme ( n  ≥  0  and  j  describing the integers) (2 . 6)  u n +1  j  =  u n j  − λ  g n j +1 / 2  − g n j − 1 / 2  , wheretheratio λ  =  k/h iskeptfixed.In(2.6),weusethenotation g n j +1 / 2  := g ( u n j − 1 ,u n j ,u n j +1 ,u n j +2  , and we define  G n j +1 / 2  similarly, etc. Theorem 2.2.  The scheme  (2 . 6)  satisfies the cell entropy inequality U  ( u n +1  j  ) − U  ( u n j  ) + λ  G n j +1 / 2  − G n j − 1 / 2  + βλ 4  v n j  − v n j − 1  2 +  v n j +1  − v n j  2  ≤  3 λ 2  U  ′′  ∞   g ′  2 ∞ D 1 ,n j  + β  2 D 2 ,n j  + γ  2 D 3 ,n j  , (2.7) where D 1 ,n j  := 136  | v n j − 1  − v n j − 2 | 2 + 73 | v n j  − v n j − 1 | 2 +73 | v n j +1  − v n j | 2 + | v n j +2  − v n j +1 | 2  ,D 2 ,n j  := 14  | v n j  − v n j − 1 | 2 + | v n j +1  − v n j | 2  ,  Diffusive-dispersive conservation laws 497 and  D 3 ,n j  := 118  | v n j − 1 − v n j − 2 | 2 + | v n j  − v n j − 1 | 2 + | v n j +1 − v n j | 2 + | v n j +2 − v n j +1 | 2  .  Hence, for   λ  sufficiently small, the entropy is a strictly decreasing func-tion of time, in the sense (2 . 8) ∞   j = −∞ U  ( u n +1  j  ) + K λ ∞   j = −∞  v n j  − v n j − 1  2 ≤ ∞   j = −∞ U  ( u n j  ) , with K   :=  β  2  − 3 λ  U  ′′  ∞  379   g ′  2 ∞  +  β  2 2 + 29  γ  2  . Furthermore, for smooth solutions, the equivalent equation of the scheme (2 . 6)  is (2 . 9)  ∂  t u + ∂  x f  ( u ) =  hβU  ′ ( u ) xx  + h 2 γU  ′ ( u ) xxx  + O ( h 3 + k ) . Observe that the equation (2.9) no longer correspond to the continousmodel (1.1). Namely the time-discretization generates a term of order O ( k ) which can not be absorbed in the cubic error  O ( h 3 )  (except of course if wewere to impose the very drastic restriction  k  =  O ( h 3 ) .) Therefore a moreaccurate time-discretization will be necessary. Proof.  By a Taylor expansion in  λ  we can deduce from (2.6) that U  ( u n +1  j  )  −  U  ( u n j  ) + λU  ′ ( u n j  )  g n j +1 / 2  − g n j − 1 / 2  ≤  λ 2 2   U  ′′  ∞  g n j +1 / 2  − g n j − 1 / 2  2 . (2.10)We determine the right hand side of (2.10) by treating successively theterms in 12  g n j +1 / 2  − g n j − 1 / 2  2 ≤  3  g n j +1 / 2  − g ( v n j  )  2 +  g ( v n j  ) − g n j − 1 / 2  2 + | g 2 ,n j +1 / 2 | 2 + | g 2 ,n j − 1 / 2 | 2 + | g 3 ,n j +1 / 2 | 2 + | g 3 ,n j − 1 / 2 | 2  . In view of (2.2) we find 12  g n j +1 / 2  − g ( v n j  )  2 ≤  Dg  2 ∞   172  | v  j  − v  j − 1 | 2 + | v  j +1  − v  j | 2 + 172  | v  j +2  − v  j +1 | 2  , and the same inequality remains true by replacing  g ( v n j  )  with  g ( v n j +1 ) . Onthe other hand, | g 2 ,n j +1 / 2 | 2 + | g 2 ,n j − 1 / 2 | 2 ≤  β  2 4  | v  j  − v  j − 1 | 2 + | v  j +1  − v  j | 2  ,

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Jan 14, 2019
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