Riemann problems with non--local point constraints and capacity drop
Boris Andreianov, Carlotta Donadello, Ulrich Razafison, Massimiliano D. Rosini
RRiemann problems with non–local point constraintsand capacity drop
Boris Andreianov , Carlotta Donadello , Ulrich Razafison , andMassimiliano D. Rosini Institut f¨ur Mathematik, Technische Universit¨at Berlin,Str. des 17. Juni 136, 10623 Berlin, Germany Laboratoire de Math´ematiques CNRS UMR 6623,Universit´e de Franche-Comt´e,16 route de Gray, 25030 Besan¸con Cedex, France ICM, Uniwersytet Warszawski,ul. Prosta 69, 00838 Warsaw, Poland
October 30, 2018
Abstract
In the present note we discuss in details the Riemann problem for a one–dimensional hyperbolic conservation law subject to a point constraint. Weinvestigate how the regularity of the constraint operator impacts the well–posedness of the problem, namely in the case, relevant for numerical appli-cations, of a discretized exit capacity. We devote particular attention to thecase in which the constraint is given by a non–local operator depending onthe solution itself. We provide several explicit examples.We also give the detailed proof of some results announced in the paper[Andreainov, Donadello, Rosini,
Crowd dynamics and conservation laws withnon–local point constraints and capacity drop ], which is devoted to existenceand stability for a more general class of Cauchy problems subject to Lipschitzcontinuous non–local point constraints.
MSC 2010:
Keywords:
Riemann problem, non–local constrained hyperbolic PDE’s, loss ofself–similarity, loss of uniqueness, crowd dynamics, capacity drop.
Traffic modeling is an exciting and fast–developing field of research with plentifulapplications to real life. While this subject was initially limited to the descriptionand the management of vehicular traffic, we see a growing interest nowadays ondifferent applications as crowd dynamics and bio–mathematics. This note is related1 a r X i v : . [ m a t h . A P ] M a r o an extensive on–going research project concerning the theoretical and the nu-merical study of macroscopic models for which the definition of solution involvesan artificial limitation of the flux in a finite number of points. From the modelingpoint of view, this may correspond to a narrow exit in crowd modeling, a toll gatein vehicular traffic, a cell membrane in bio-medical modeling.In the pioneering paper [7], R. Colombo and P. Goatin introduced point con-straints in the classical one–dimensional LWR road traffic model [17, 18], with thegoal to model the presence of obstacles on the road as toll gates and road lights.This model reads as ∂ t ρ + ∂ x f ( ρ ) = 0 ( t, x ) ∈ R + × R (1a) f ( ρ ( t, ± )) ≤ q ( t ) t ∈ R + (1b) ρ (0 , x ) = ρ ( x ) x ∈ R , (1c)where ρ = ρ ( t, x ) ∈ [0 , R ] is the unknown (mean) density at time t ∈ R + of vehiclesmoving along the road parameterized by the coordinate x ∈ R . Then, R ∈ R + is the maximal road density, f : [0 , R ] → R is the nonlinearity relating the fluxin the direction of increasing x to the density, q : R + → R + is a given functionprescribing the maximal flow allowed through the point x = 0, and ρ : R → [0 , R ]is the initial (mean) density. Finally, ρ ( t, − ) denotes the left measure theoretictrace along the constraint implicitly defined bylim ε ↓ ε (cid:90) + ∞ (cid:90) − ε | ρ ( t, x ) − ρ ( t, − ) | φ ( t, x ) d x d t = 0for all φ ∈ C ∞ c ( R ; R ). The right measure theoretic trace, ρ ( t, q is not given beforehand, but depends onthe solution ρ itself in a neighborhood of x = 0. In such situation we say thatthe point constraint is non–local. In this way we obtain crowd and cell membranedynamics models described by coupled PDE–ODE systems for which the existenceand well–posedness of solutions are not trivial matter. Nevertheless, this case hasa practical interest. In crowd dynamics, as an example, the experimental obser-vations by E. Cepolina in [6] prove that the irrational behavior of pedestrians atbottlenecks ends up by reducing the maximal possible outflow. This phenomenon,called capacity drop , is also related to other effects observed in crowd dynamics,such Faster Is Slower and the Braess’ paradox.In full generality, we may consider the constraint function q as follows q ( t ) = Q [ ρ ]( t ) , where Q : C (cid:0) [0 , T ]; L ( R ; [0 , R ]) (cid:1) → L ([0 , T ]; [0 , R ]). The minimal regularityproperties to impose on Q in order to achieve well–posedness of solutions are notknown at the moment, and they are the object of one of our current researchprojects. 2 .2 An example of non–local point constraint In the paper [3], B. Andreianov, C. Donadello and M. D. Rosini proposed a modelwhich generalizes the one in [7] and consists of a Cauchy problem for a one–dimensional hyperbolic conservation law as 1 subject to a non–local point constraintof the form q ( t ) = p (cid:32)(cid:90) R − w ( x ) ρ ( t, x ) d x (cid:33) t ∈ R + . (2)Here p : R + → R + prescribes the maximal flow allowed through an exit placedin x = 0 as a function of the weighted average density of pedestrians, ρ , in a leftneighborhood of the exit and w : R − → R + is the weight function used to averagethe density. The authors of [3] proved well-posedness of the Cauchy problem in L ∞ ( R ; [0 , R ]) for the model under the following assumptions on the regularity of f , w and p , see Figure 1, (F) f ∈ Lip ([0 , R ]; [0 , + ∞ [), f (0) = 0 = f ( R ) and there exists ¯ ρ ∈ ]0 , R [ suchthat f (cid:48) ( ρ ) (¯ ρ − ρ ) > ρ ∈ [0 , R ] \ { ¯ ρ } . (W) w ∈ L ∞ ( R − ; R + ) is an increasing map, (cid:107) w (cid:107) L ( R − ; R + ) = 1 and there existsi w > w ( x ) = 0 for any x ≤ − i w . (P1) p belongs to Lip ([0 , R ] ; ]0 , f (¯ ρ )]) and it is a non-increasing map.We recall that, in the previous literature, the only macroscopic model able to re-produce the capacity drop at bottlenecks is the CR model introduced by R. Colomboand M. D. Rosini in [9]. The Riemann solver for the model described in [9] is fairlyintricate; in this note, our main goal is to describe in an exhaustive way the Rie-mann solver (or, rather, solvers) for the model of [3]. Notice that, in contrast to [9],a specific nonclassical Riemann solver has to be used only at the exit point x = 0while the simple classical Riemann solver is used elsewhere.The notion of solution we adopt is a natural extension of the one introduced in[7], for a constrained Cauchy problem of the form 1. Definition 1.1.
Assume conditions (F) , (W) and that p is a non–increasing,possibly multivalued, map with values in ]0 , f (¯ ρ )] . A map ρ ∈ L ∞ ( R + × R ; [0 , R ]) ∩ C ( R + ; L ( R ; [0 , R ])) is an entropy weak solution to 1, 2 if the following conditionshold:1. There exists q ∈ L ∞ ( R + ; [0 , f (¯ ρ )]) such that for every test function φ ∈ C ∞ c ( R ; R + ) and for every k ∈ [0 , R ] (cid:90) R + (cid:90) R [ | ρ − k | ∂ t φ + sign( ρ − k ) ( f ( ρ ) − f ( k )) ∂ x φ ] d x d t (3a)+ 2 (cid:90) R + (cid:20) − q ( t ) f (¯ ρ ) (cid:21) f ( k ) φ ( t,
0) d t (3b)+ (cid:90) R | ρ ( x ) − k | φ (0 , x ) d x ≥ , (3c) and f ( ρ ( t, ± )) ≤ q ( t ) for a.e. t ∈ R + . (3d)3 . In addition q is linked to ρ by the relation 2. If q is given a priori , then 3 is the definition of entropy weak solution to prob-lem 1. We refer to Proposition 2.6 in [4] for a series of equivalent formulations ofconditions 3.The next proposition lists the basic properties of a entropy weak solution of 1,2, for the case of a single valued p , the proof is given in [3]. Proposition 1.
Let [ t (cid:55)→ ρ ( t )] be an entropy weak solution of 1, 2 in the sense ofDefinition 1.1. Then(1) It is also a weak solution of the Cauchy problem 1a, 1c.(2) Any discontinuity satisfies the Rankine–Hugoniot jump condition, see [12].(3) Any discontinuity away from the constraint is classical, i.e. satisfies the Laxentropy inequalities, see [12].(4) Nonclassical discontinuities, see [16], may occur only at the constraint location x = 0 , and in this case the flow at x = 0 is the maximal flow allowed by theconstraint. Namely, if the solution contains a nonclassical discontinuity for alltimes t ∈ I , I open in R + , then for a.e. t in If ( ρ ( t, − )) = f ( ρ ( t, p (cid:32)(cid:90) R − w ( x ) ρ ( t, x ) d x (cid:33) . (4)If the constraint function p is multivalued the equalities in 2 and 4 should beinterpreted as inclusions, and the result of the proposition remains true.The existence result in [3] is achieved by a procedure which couples the operatorsplitting method [13], with the wave–front tracking algorithm, [11], see also [1] fora similar technique. This approach allows us to approximate our problem by aproblem with “frozen” constraint, as 1, at each time step.The regularity of p plays a central role in the well-posedness result. Whileexistence still holds for the Cauchy problem when p is merely continuous, it isdifficult to justify uniqueness in this case. Further, in [3], the authors give somebasic examples illustrating that solutions of a Riemann problem for the case of anon-decreasing piecewise constant p , see (P2) below, may fail to be unique, L –continuous and consistent.The case in which p is piecewise constant is extremely important both for thetheoretical study of the problem and its numerical applications. First, it is relatedto the construction of the Riemann solver, which is the basic building block for thewave–front tracking algorithm, a precious tool in the study of existence and stabilityof the solutions for the general Cauchy problem. Moreover, the piecewise constantcase is essentially the only case in which solutions can be computed explicitly, whichis an undeniable atout when looking for examples and applications. To this aim,it is relevant to provide a detailed study of the different pathological behaviors onemay encounter. Remarkably, we show that these behaviors can be easily forecastand avoided when looking for explicit examples of solutions.In the following section we develop a detailed proof of the fact that, when p ispiecewise constant, the Riemann solver for 1 with a constraint of the form 2, is notunique and does not satisfy the minimal requirements needed to develop the classical4ave–front tracking approach. Additionally, we compare the two extreme Riemannsolvers: the one that minimizes the capacity drop, and the one that maximizes it.In particular, for any time T >
In this section we study constrained Riemann problems of the form ∂ t ρ + ∂ x f ( ρ ) = 0 ( t, x ) ∈ R + × R (5a) f ( ρ ( t, ± )) ≤ p (cid:32)(cid:90) R − w ( x ) ρ ( t, x ) d x (cid:33) t ∈ R + (5b) ρ (0 , x ) = (cid:26) ρ L if x < ρ R if x ≥ x ∈ R , (5c)with ρ L , ρ R ∈ [0 , R ]. The flux f and the weight function w satisfy (F) and (W) ,moreover, we adopt the following assumption on p (instead of (P1) ) to allow anexplicit construction of solutions to 5 (P2) p : [0 , R ] → ]0 , f (¯ ρ )] is piecewise constant non–increasing map with a finitenumber of jumps.Figure 1: Examples of functions satisfying conditions (F) , (W) , and (P2) .Unfortunately, the regularity of p required by (P2) is not enough to apply thewell–posedness results in [3]. In particular, Example 2 in [3] illustrates the lossof uniqueness and stability of entropy weak solutions. In this section we presenta systematic study of the possible pathological behaviors. We denote by R theclassical Riemann solver. This means that the map [( t, x ) (cid:55)→ R [ ρ L , ρ R ]( x/t )] isthe unique entropy weak solution for the unconstrained problem 5a, 5c, see forexample [5] for its construction. Whenever the classical weak solution given by R does not satisfy the constraint 5b, we replace it by a nonclassical weak solution,see [16] as a general reference, ρ ( t, x ) = (cid:26) R [ ρ L , ˆ ρ ( q )]( x/t ) if x < R [ˇ ρ ( q ) , ρ R ]( x/t ) if x ≥ , (6)where the maps ˇ ρ, ˆ ρ : [0 , f (¯ ρ )] → [0 , R ] are implicitly defined by f (ˇ ρ ( q )) = q = f (ˆ ρ ( q )) and ˇ ρ ( q ) ≤ ¯ ρ ≤ ˆ ρ ( q ) . R + × R − and R + × R + . However, the jump at x = 0 is a nonclassical shock, in the sense thatit does not satisfy the Lax entropy inequalities.First, we should notice that as soon as the constraint function t → q ( t ) is notconstant, the solutions of the Riemann problem may not be self–similar. Example 2.1.
We use a Cauchy problem of the form 1 to model vehicular traffic inpresence of a traffic light. Assume f ( ρ ) = ρ (1 − ρ ) and q ( t ) = 0 . (cid:80) k ∈ N χ [2 k, k +1[ ( t ) .When the traffic light is green, i.e. for t ∈ [2 k, k + 1[ , the flow at x = 0 isfree from any constraint. Conversely, when the traffic light is red, i.e. for t ∈ [2 k + 1 , k + 1)[ , the admissible flow at x = 0 become zero. This means that theRiemann problem with initial condition at t = 0 given by ρ L = ρ R (cid:54) = 0 will notbe self–similar, because the constant solution will not satisfy the constraint startingfrom t = 1 .The above example also shows that as soon as we use a nonclassical Riemannsolver we lose the a priori BV bounds on the solution. In the proof of Proposition 2 we show that any entropy weak solution of 5is self–similar for sufficiently small times. Therefore, it makes sense to introducenonclassical local
Riemann solvers, see Definition 2.3. Then, the availability ofa local Riemann solver allows us to construct a global solution to the Riemannproblem 5 by a wave–front tracking algorithm in which the jumps in the map[ t (cid:55)→ q ( t )] are interpreted as non–local interactions.Aiming for a general construction of the solutions to 5, we allow p to be amulti–valued piecewise constant function, namely, see Fig. 1, right: • there exist ξ , . . . , ξ n ∈ ]0 , R [ and p , . . . , p n ∈ ]0 , f (¯ ρ )], with ξ i < ξ i +1 and p i > p i +1 , such that p (0) = p , p ( R ) = p n , p χ ] ξ i , ξ i +1 [ = p i for i = 0 , . . . , n − p ( ξ i ) = [ p i , p i − ] for i = 1 , . . . , n , being ξ = 0 and ξ n +1 = R .In the following we will use the notations ˇ ρ i = ˇ ρ ( p i ) and ˆ ρ i = ˆ ρ ( p i ).As it will become clear in Proposition 2, the possible loss of uniqueness andstability can be easily forecast once the piecewise constant constraint p and the flux f are given. In particular, for some respective configurations of p and f the solutionof the Riemann problem exists and is unique, locally in time, for any initial data( ρ L , ρ R ) in [0 , R ] . 6 efinition 2.2. Introduce the subset of [0 , R ] C = (cid:110) ( ρ L , ρ R ) ∈ [0 , R ] : ( ρ L , ρ R ) satisfies condition (C) (cid:111) , where we say that ( ρ L , ρ R ) satisfies condition (C) if it satisfies one of the followingconditions: (C1) ρ L < ρ R , f ( ρ R ) < f ( ρ L ) and f ( ρ R ) ≤ p ( ρ L +) ; (C2) ρ L < ρ R , f ( ρ L ) ≤ f ( ρ R ) and f ( ρ L ) ≤ p ( ρ L +) ; (C3) ρ R ≤ ρ L ≤ ¯ ρ and f ( ρ L ) ≤ p ( ρ L +) ; (C4) ρ R ≤ ¯ ρ < ρ L and f (¯ ρ ) = p ( ρ L +) ; (C5) ¯ ρ < ρ R ≤ ρ L , f ( ρ R ) ≤ p ( ρ L − ) and f ( ρ L ) < p ( ρ L +) .Analogously, introduce the subset of [0 , R ] N = (cid:110) ( ρ L , ρ R ) ∈ [0 , R ] : ( ρ L , ρ R ) satisfies condition (N) (cid:111) , where we say that ( ρ L , ρ R ) satisfies condition (N) if it satisfies one of the followingconditions: (N1) ρ L < ρ R and f ( ρ L ) > f ( ρ R ) > p ( ρ L +) ; (N2) ρ L < ρ R , f ( ρ L ) ≤ f ( ρ R ) and f ( ρ L ) > p ( ρ L − ) ; (N3) ρ R ≤ ρ L ≤ ¯ ρ and f ( ρ L ) > p ( ρ L − ) ; (N4a) ρ R ≤ ¯ ρ < ρ L , f (¯ ρ ) (cid:54) = p ( ρ L − ) and f ( ρ L ) < p ( ρ L +) ; (N4b) ρ R ≤ ¯ ρ < ρ L , f (¯ ρ ) (cid:54) = p ( ρ L − ) and f ( ρ L ) > p ( ρ L − ) ; (N5a) ¯ ρ < ρ R ≤ ρ L , f ( ρ R ) > p ( ρ L − ) and f ( ρ L ) < p ( ρ L +) ; (N5b) ¯ ρ < ρ R ≤ ρ L and f ( ρ L ) > p ( ρ L − ) . Here, C stands for classical and N , for nonclassical , in relation with the natureof the shock appearing in the solution of 5 at x = 0. Observe that if the constraintfunction p is constant in a neighborhood of the state ρ L , then p ( ρ L − ) = p ( ρ L +)and this simplifies the above conditions. Also a right or left continuity assumptionon p would simplify the above definitions.The next proposition says that uniqueness holds at least for small times if andonly if the initial data are in C ∪ N . It is fundamental to remark that since non–uniqueness is possible only when p ( ρ L − ) (cid:54) = p ( ρ L +) and p ( ρ L − ) ≥ f ( ρ L ) ≥ p ( ρ L +),non–uniqueness concerns at most a finite number of left states and therefore theregion [0 , R ] \ ( C ∪ N ) is the union of a finite number of line segments.
Proposition 2.
Consider the constrained Riemann problem 5. • If ( ρ L , ρ R ) ∈ C , then the map [( t, x ) (cid:55)→ R [ ρ L , ρ R ]( x/t )] is the unique entropy weaksolution at least for t > sufficiently small. • If ( ρ L , ρ R ) ∈ N , then there exists a unique ¯ p ∈ [ p ( ρ L +) , p ( ρ L − )] such that themap (cid:20) t (cid:55)→ (cid:26) R [ ρ L , ˆ ρ (¯ p )]( x/t ) if x < R [ˇ ρ (¯ p ) , ρ R ]( x/t ) if x ≥ (cid:21) is the unique entropy weak solution at least for t > sufficiently small. • If ( ρ L , ρ R ) ∈ [0 , R ] \ ( C ∪ N ) , then the corresponding constrained Riemann prob-lem 5 admits more than one entropy weak solution. The proof is deferred to Section 4.1 7 emark 1.
Once the function p is fixed, the time interval [0 , τ ] on which thesolution to the Riemann problem 5 is self-similar can be estimated from the initialdatum ( ρ L , ρ R ) provided it belongs to C ∪ N .On the contrary, when ( ρ L , ρ R ) ∈ [0 , R ] \ ( C ∪ N ) , we are not always able toforecast when the next “interaction with the constraint” will take place. In somesituations, a whole one–parameter family of solutions exists, we refer to Example 2in [3] for a detailed discussion of this case. As the local solutions of the Riemann problem 5 are not unique in general, we arenaturally led to question the existence of suitable selection criteria. All the solutionswe introduce are solutions in the Kruˇzkov sense in the open half–planes R + × R + and R + × R − , so they satisfy the basic requirement of entropy dissipation. However,coming back to the real situations which our model aims to describe, we argue thatthe most interesting behaviors to track correspond to the extreme cases in whichthe flux at the exit is either the highest or the lowest possible from a given initialcondition.If, as an example, our model describes the evacuation of a narrow corridor, itis clear that the optimal solution corresponds to the highest admissible values ofthe flux at the exit. By opposition to the next case, we describe this situationas quiet behaviour. In analogy to the discussion in [14] we interpret all otherpossible solutions as consequences of an irrational behavior, which in literature isoften described as panic . In particular, we can use the solution corresponding tothe lowest admissible values of the flux at the exit to find an upper bound for theevacuation time.From now on we restrict ourselves to the case in which p ( ξ i ) can only take thevalues p i and p i +1 and not the intermediate values. Definition 2.3.
Two Riemann solvers R q and R p for 5 are defined as follows for t > sufficiently small and x ∈ R : (C) If ( ρ L , ρ R ) ∈ C then R q [ ρ L , ρ R ]( t, x ) = R p [ ρ L , ρ R ]( t, x ) = R [ ρ L , ρ R ]( x/t ) . (N) If ( ρ L , ρ R ) ∈ N then R q [ ρ L , ρ R ]( t, x ) = R p [ ρ L , ρ R ]( t, x ) = (cid:26) R [ ρ L , ˆ ρ (¯ p )]( x/t ) if x < R [ˇ ρ (¯ p ) , ρ R ]( x/t ) if x ≥ , where ¯ p = p ( ρ L − ) if ( ρ L , ρ R ) satisfies (N4a) or (N5a) , otherwise ¯ p = p ( ρ L +) . (CN2), (CN3), (CNN5) If ( ρ L , ρ R ) satisfies one of these sets of conditions then R q [ ρ L , ρ R ]( t, x ) = R [ ρ L , ρ R ]( x/t ) , R p [ ρ L , ρ R ]( t, x ) = (cid:26) R [ ρ L , ˆ ρ ( p ( ρ L +))]( x/t ) if x < R [ˇ ρ ( p ( ρ L +)) , ρ R ]( x/t ) if x ≥ . (NNN4), (NNN5) If ( ρ L , ρ R ) satisfies one of these sets of conditions then R q [ ρ L , ρ R ]( t, x ) takes the form 11 with ¯ p = p ( ρ L − ) and R p [ ρ L , ρ R ]( t, x ) takes the form 11 with ¯ p = p ( ρ L +) .
8n the next proposition we collect the main properties of the Riemann solvers R q and R p . In particular (R6) means that the Riemann solver R q is the one whichallows for the fastest evacuation, while R p is associated to the slowest one. Proposition 3.
Let ( ρ L , ρ R ) ∈ [0 , R ] . Then, for (cid:63) = q, p : (R1) [( t, x ) (cid:55)→ R (cid:63) [ ρ L , ρ R ]( t, x )] is a weak solution to 5a, 5c. (R2) R (cid:63) [ ρ L , ρ R ] satisfies the constraint 5b in the sense that f ( R (cid:63) [ ρ L , ρ R ]( t, ± )) ≤ p (cid:32)(cid:90) R − w ( x ) R (cid:63) [ ρ L , ρ R ] ( t, x ) d x (cid:33) . (R3) R (cid:63) [ ρ L , ρ R ]( t ) ∈ BV ( R ; [0 , R ]) . (R4) The map R (cid:63) : [0 , R ] → L ( R + × R ; [0 , R ]) is continuous in C ∪ N but notin all [0 , R ] . (R5) R (cid:63) is consistent, see [7], [8] and the comment below. (R6) R q [ ρ L , ρ R ] maximizes the flux at the exit, in the sense that if E is the set ofall entropy weak solutions of the Riemann problem 5, we have max ρ ∈E { f ( ρ ( t, ± )) } = f ( R q [ ρ L , ρ R ](0 ± )) . Analogously, R p [ ρ L , ρ R ] minimizes the flux at the exit, in the sense that min ρ ∈E { f ( ρ ( t, ± )) } = f ( R p [ ρ L , ρ R ](0 ± )) . We recall that a Riemann solver is said to be consistent when the juxtapositionof the solutions of two Riemann problems with respective initial conditions ( ρ L , ρ M )and ( ρ M , ρ R ) is the solution of the Riemann problem with datum ( ρ L , ρ R ). More-over the vice versa also holds true, in the sense that whenever the state ρ M is anintermediate state in the solution of the Riemann problem with initial condition( ρ L , ρ R ), then the solution consist of exactly the same states and waves which wewould obtain by solving side by side the two Riemann problems with data ( ρ L , ρ M )and ( ρ M , ρ R ).The proof of Proposition 3 is deferred to Section 4.2. Remark 2.
It is important to observe that even if p ( ξ i ) can only take the two values p i and p i +1 , this is not enough to rule out the existence of infinitely many differentsolutions as the ones described in Example 2 of [3], in the case p i > f ( ξ i ) = p i +1 .However, each of the extremes Riemann solvers R (cid:63) , (cid:63) = p, q , selects one of thembecause it sticks to the constant level of constraint prescribed by Definition 2.3, thelevel p i for R q and the level p i +1 for R p , until a non–local interaction takes place. R (cid:63) , (cid:63) = q, p Although the Riemann solvers R (cid:63) are not L –continuous, an existence result forthe Cauchy problem 1 can be obtained from a wave–front tracking algorithm basedon R (cid:63) , see for instance [10], [19]. Such approach using R (cid:63) does not require the9perator splitting method. However, the non–local nature of the approximatingproblems prevents us from a direct application of the Riemann solvers R (cid:63) . In fact,even in a arbitrary small neighborhood of x = 0, to prolong the approximatingsolution ρ n beyond a time t = ¯ t > ρ n (¯ t, − ), ρ n (¯ t, (cid:82) − i w w ( x ) ρ n (¯ t, x ) d x is needed. Roughly speaking,because of the non–local character of the constraint one cannot merely juxtaposethe solution to the Riemann problem associated to the values of the traces at x = 0with the solution to the Riemann problems away from the constraint. Finally, alsojumps in [ t (cid:55)→ p ( ξ ( t ))] have to be considered as (non–local) interactions. Therefore,the approach using R (cid:63) is considerably heavier and more technical than the onepresented in [3]. R p and R q In this section we aim to compare the solutions obtained by the two Riemann solversintroduced above, starting from the same initial condition ( ρ L , ρ R ). It is clear fromDefinition 2.3 that as soon as ( ρ L , ρ R ) belongs to C ∪ N the solutions obtained bythe two Riemann solvers coincide.As a preliminary remark we stress that adapting the proof of Theorem 3.1 of [3]to the case in which p is discontinuous, one can get a rough upper bound, exponentialin time, for the L distance between two solutions ρ , ρ obtained from the sameinitial condition (not necessarily of Riemann type). Indeed, instead of the bound | p ( ρ ) − p ( ρ ) | ≤ Lip( p ) | ρ − ρ | , (7)valid when p is Lipschitz continuous, in the discontinuous case one can use thebound | p ( ρ ) − p ( ρ ) | ≤ h + N | ρ − ρ | , (8)where h is the maximal size of jump in p ( · ) and N is a constant (observe thatif p ( · ) is a discretization of some Lipschitz function, see Section 4.1 of [3], then N can be taken independent of h ). From the fundamental stability estimate ofProposition 2.10 of [4], using 8 and the Gronwall inequality one easily gets thebound (cid:107) ρ ( t ) − ρ ( t ) (cid:107) L ( R ; R ) ≤ h N [exp(2 N t ) − , (9)whenever ρ (0 , · ) = ρ (0 , · ). This rough estimate is enough to show that as h goes tozero, the discrepancy between different solutions vanishes and this argument appliesto any initial datum, not necessarily of the Riemann type.However, the exponential growth with respect to t of the upper bound 9 is clearlynot optimal when we aim to compare the solutions of a Riemann problem, sinceit does not take into account the specific self-similar structure of solutions valid atleast on a small interval of time [0 , T ].Let us demonstrate that in the case where different Riemann solvers co-exist,the L distance of the associated solution grows at most linearly both in h and in t ∈ [0 , T ] (see also the numerical experiment on Figure 6).In order to keep our presentation as light as possible, we focus on only one of thepossible cases in which R p and R q differ. All other cases can be handed in a similarway. Assume that p ( ξ ) = p χ [0 , ¯ ξ ] ( ξ ) + p χ ]¯ ξ,R ] ( ξ ), where ¯ f > p > f ( ¯ ξ ) > p > f , the constraint function p | ]ˆ ρ , ˆ ρ [ and the values of the density ρ considered in Section 2.3.and ρ L = ¯ ξ , ρ R = ¯ ρ , see Figure 3. We only consider solutions in a small interval oftime [0 , T ] in which they are self–similar. Then we get R p [ ρ L , ρ R ]( t, x ) = (cid:26) R [ ρ L , ˆ ρ ]( x/t ) if x < R [ˇ ρ , ¯ ρ ]( x/t ) if x ≥ , R q [ ρ L , ρ R ]( t, x ) = (cid:26) R [ ρ L , ˆ ρ ]( x/t ) if x < R [ˇ ρ , ¯ ρ ]( x/t ) if x ≥ , where the values ˆ ρ i and ˆ ρ i , for i = 1 ,
2, are implicitly defined by the relations f (ˆ ρ i ) = f (ˇ ρ i ) = p i and ˇ ρ i ≤ ¯ ρ ≤ ˆ ρ i . More explicitly, we can say that the solutioncorresponding to R p consists of a shock of negative speed λ p between ρ L and ˆ ρ , astationary nonclassical shock between ˆ ρ and ˇ ρ and a shock of positive speed µ p between ˇ ρ and ¯ ρ . The solution corresponding to R q consists of a rarefaction wavebetween ρ L and ˆ ρ , a stationary nonclassical shock between ˆ ρ and ˇ ρ and a shockof positive speed µ q between ˇ ρ and ¯ ρ , see Figure 4.As the characteristics of this problem propagate with finite speed, we expectthe solutions associated to the two solvers coincide outside a bounded interval.The geometry of the problem, see Figure 3, implies that µ p > µ q and that λ p issmaller than all the propagation speeds in the rarefaction wave between ρ L and ˆ ρ .Therefore, at time t ∈ [0 , T ] fixed, the two solutions coincide outside the interval[ λ p t, µ p t ].The value of the distance (cid:107)R p [ ρ L , ρ R ]( t ) − R q [ ρ L , ρ R ]( t ) (cid:107) L ( R ; R ) corresponds,loosely speaking, to the value of the area between the profiles of solutions, seeFigure 4.Following the same technique as in [5], Chapter 7, we can estimate the distancebetween the solutions profiles at a fixed time t ∈ [0 , T ]. For the reader convenience,we recall that the propagation speed of the shock discontinuity between states the ρ a and ρ b is given by the Rankine-Hugoniot condition σ ( ρ a , ρ b ) = f ( ρ a ) − f ( ρ b ) ρ a − ρ b , (10)and that the propagation speed of the characteristics in a rarefaction wave joiningthe states ρ a and ρ b varies between the values f (cid:48) ( ρ a ) and f (cid:48) ( ρ b ). Also, by definition11igure 4: The solutions corresponding to R p , left, and to R q , center, and thecomparison between their profiles at fixed time, right, as described in Section 2.3. f (ˆ ρ i ) = f (ˇ ρ i ) = p i , for i = 1 ,
2. A direct calculation gives us (cid:107)R p [ ρ L , ρ R ]( t, · ) − R q [ ρ L , ρ R ]( t, · ) (cid:107) L ( R ; R ) ≤(cid:107)R [ ρ L , ˆ ρ ]( · /t ) − R [ ρ L , ˆ ρ ]( · /t ) (cid:107) L ( R ; R ) + (cid:107)R [ˇ ρ , ρ R ]( · /t ) − R [ˇ ρ , ρ R ]( · /t ) (cid:107) L ( R ; R ) ≤ [ λ p − f (cid:48) ( ρ L )] (ˆ ρ − ρ L ) t + f (cid:48) ( ρ L ) (ˆ ρ − ˆ ρ ) t + (ˇ ρ − ˇ ρ ) µ q t + (¯ ρ − ˇ ρ ) ( µ q − µ p ) t ≤ t [ p − p + o (ˆ ρ − ˆ ρ )] . This means that whenever the piecewise constant function p we consider is thediscretization of a smooth function we can bound a priori the size of the error dueto the lack of uniqueness and we can make it smaller and smaller as h tends to 0. We present here some numerical experiments in order to illustrate the results ofthe above section. The scheme used for the simulations combines the ideas of [4]with an explicitly updated constraint computed from weighted space averages ofthe discrete solution at previous time step. We will justify in the future work [2]convergence of this scheme to an entropy solution of the nonlocally constraintedproblem in the sense of Definition 1.1, where the constraint function p ( · ) must betaken multi-valued. While it is delicate or even impossible to identify a uniqueRiemann solver to which the scheme would converge, we can use the simulations onFigure 5 to illustrate the fact that non-uniqueness for the Riemann problem resultsas unstable behavior in a vicinity of some specific data.For the examples, we consider the flux f ( ρ ) = ρ (1 − ρ ). The domain of computationis x ∈ [ − , p ( ξ ) = p χ [0 , . ( ξ ) + p χ ]0 . , ( ξ ), where p = 0 . p = 0 .
05, the weight function is w ( x ) = 2( x + 1) χ ] − , ( x ). The finaltime of computation is T = 1. In Figure 5 is shown the computed solutions ρ and ρ corresponding respectively to the following initial states ρ ( x ) = (cid:26) . x < . x > ρ ( x ) = (cid:26) . x < . x > . Note that we have considered ρ L and ρ L such that ρ L − ρ L (cid:39) ∆ x/
8, where thespace step ∆ x = 0 . t = ∆ x/ solutions x rho rho Figure 5: The computed densities ρ and ρ Nonetheless, as shown in Figure 6, in practice the instability is limited to a behaviorof kind (cid:107) ρ − ρ (cid:107) L ([ − , R ) ≤ (cid:13)(cid:13) ρ − ρ (cid:13)(cid:13) L ([ − , R ) + C h α , where h is the maximal size of jump in p ( · ), α is close to 1 and C > L -discrete norms of thedifference ρ − ρ when we take, p = 0 .
05, 0 . .
1, 0 .
125 and 0 .
15 in the definitionof the constraint function. Using logarithmic scales, we deduce that the distancebetween the two solutions is approximatively proportional to | p − p | . . First, we introduce the notation ξ ( t ) = (cid:90) R − w ( x ) ρ ( t, x ) d x . Therefore ξ (0) = ρ L and the map [ t (cid:55)→ ξ ( t )] is continuous.We stress that any nonclassical entropy weak solution in the sense of Defini-tion 1.1 is also a classical entropy weak solution in the Kruˇzkov sense in the half–planes R + × R − and R + × R + . Therefore, at least for t > (P2) and the continuity of the map [ t (cid:55)→ ξ ( t )], anynonclassical entropy weak solution of 5 must have the form, see Fig. 2, ρ ( t, x ) = (cid:26) R [ ρ L , ˆ ρ (¯ p )]( x/t ) if x < R [ˇ ρ (¯ p ) , ρ R ]( x/t ) if x ≥ . (11a)13 L - d i sc r e t e no r m s o f r ho -r ho i n l og sc a l e ln|p -p | Figure 6: The norm (cid:107) ρ − ρ (cid:107) L with respect to | p − p | in log/log scale.Observe that 11a is uniquely identified once we know ¯ p which, by 4, satisfies¯ p = f (ˇ ρ (¯ p )) = f (ˆ ρ (¯ p )) . (11b)We recall that 11b means in particular that the Rankine–Hugoniot jump conditionis satisfied at x = 0 even when the solution to the Riemann problem is nonclassical.As a consequence of 11b, of assumption (P2) and of the continuity of [ t (cid:55)→ ξ ( t )],we have that ¯ p ∈ [ p ( ρ L +) , p ( ρ L − )] . (11c)This implies that if p ( ρ L +) = p ( ρ L − ), then p ( ξ ) is constant in a neighborhood of ρ L and, since the solution is in C (cid:0) R + ; L ( R ; [0 , R ]) (cid:1) , uniqueness is ensured bythe results in [7]. However, the continuity of p at ρ L is not a necessary conditionfor uniqueness as we show in the following section. In this section we prove that: If ( ρ L , ρ R ) ∈ C the corresponding classical solution satisfies 5 for all t > If ( ρ L , ρ R ) ∈ N the corresponding classical solution does not satisfy 5b, and thereexists a unique nonclassical solution that satisfies 5.We list here two basic properties which will be of great help in the followingcase by case analysis.By assumption (P2) and the continuity of the map [ t (cid:55)→ ξ ( t )] we have that forany t > bp1 if ξ ( t ) < ρ L , then p ( ξ ( t )) ≡ p ( ρ L − );14 p2 if ρ L < ξ ( t ), then p ( ξ ( t )) ≡ p ( ρ L +).The case ξ ( t ) ≡ ρ L is somehow special and has to be studied separately for eachspecific case.Second, when the solution is nonclassical, due to the finite speed of propagationof the waves, the assumption (P2) and properties bp1 and bp2 , we have np1 if R [ ρ L , ˆ ρ (¯ p )] ( x ) ≡ ρ L for x <
0, then ¯ p = f ( ρ L ) ∈ [ p ( ρ L +) , p ( ρ L − )]; np2 if ¯ p (cid:54) = f ( ρ L ) and ρ L < ˆ ρ (¯ p ), then ¯ p = p ( ρ L +); np3 if ¯ p (cid:54) = f ( ρ L ) and ˆ ρ (¯ p ) < ρ L , then ¯ p = p ( ρ L − ); np4 if p is continuous in ρ L , namely p ( ρ L − ) = p ( ρ L +), then ¯ p = p ( ρ L ).Now we start the description of the possible cases and we proceed as follows.First, we show that for any initial datum satisfying (C i ) , i = 1 , . . . ,
5, the problemactually has a unique solution and that the solution is classical. Second, we takeinto consideration the corresponding case (N i ) , for which we prove that the classicalsolution is not suitable and that there exists a unique nonclassical solution.In general the solutions to the constrained Riemann problem 5 are not self–similar. All the cases listed below describe self–similar solutions because we let thesolutions evolve only on a small interval of time. (C1) In this case [( t, x ) (cid:55)→ R [ ρ L , ρ R ]( x/t )] performs a shock with negative speed σ ( ρ L , ρ R ) and satisfies 5b because f ( ρ R ) ≤ p ( ρ L +) and p ( ξ ( t )) ≡ p ( ρ L +)by bp2 . Assume that there exists a nonclassical solution of the form 11.Observe that the assumptions ρ L < ρ R and f ( ρ R ) < f ( ρ L ) together implythat ¯ ρ < ρ R . Then ˇ ρ (¯ p ) ≤ ¯ ρ < ρ R and R [ˇ ρ (¯ p ) , ρ R ] is given by a shock withnon negative speed if and only if ¯ p ≤ f ( ρ R ), or equivalently, ρ R ≤ ˆ ρ (¯ p ). Asa consequence, ¯ p ≤ f ( ρ R ) < f ( ρ L ), ρ L < ρ R ≤ ˆ ρ (¯ p ) and by np2 ¯ p coincideswith p ( ρ L +). In conclusion we have ¯ p ≤ f ( ρ R ) ≤ p ( ρ L +) = ¯ p , namely f ( ρ R ) = ¯ p and the nonclassical solution coincides with the classical one. (N1) In this case [( t, x ) (cid:55)→ R [ ρ L , ρ R ]( x/t )] does not satisfy 5b because f ( ρ R ) >p ( ρ L +), see case (C1) . Therefore, there does not exist any classical solutionand we can consider only nonclassical solutions of the form 11. If p iscontinuous in ρ L , then by np4 we have that ¯ p = p ( ρ L ). If p experiences ajump at ρ L then, one may wonder which value in [ p ( ρ L +) , p ( ρ L − )] has tobe chosen as ¯ p . As in the case (C1) , the assumptions imply that ¯ ρ < ρ R and then that ˇ ρ (¯ p ) < ρ R and ¯ p ≤ f ( ρ R ). Then ¯ p is strictly smaller than f ( ρ L ) and ˆ ρ (¯ p ) > ρ L . As a consequence, property np2 forces us to choosethe unique possible value of ¯ p , which is p ( ρ L +). (C2) In this case [( t, x ) (cid:55)→ R [ ρ L , ρ R ]( x/t )] performs a shock with non negativespeed σ ( ρ L , ρ R ) and it satisfies 5b because f ( ρ L ) ≤ p ( ρ L +). Assume thatthere exists a nonclassical solution of the form 11. Observe that the as-sumptions ρ L < ρ R and f ( ρ R ) ≥ f ( ρ L ) together imply that ¯ ρ > ρ L . Thenˆ ρ (¯ p ) ≥ ¯ ρ > ρ L and R [ ρ L , ˆ ρ (¯ p )] is given by a shock with non positive speedif and only if ¯ p ≤ f ( ρ L ). Thus ¯ p ≤ f ( ρ L ) ≤ p ( ρ L +) and this implies by 11cthat ¯ p = f ( ρ L ) = p ( ρ L +) and that the nonclassical solution coincides withthe classical one. (N2) In this case [( t, x ) (cid:55)→ R [ ρ L , ρ R ]( x/t )] does not satisfy 5b because f ( ρ L ) >p ( ρ L − ), see case (C2) . Therefore, there does not exist any classical solutionand we can consider only nonclassical solutions of the form 11. As in thecase (C2) , the assumptions imply ˆ ρ (¯ p ) ≥ ¯ ρ > ρ L . Furthermore, by 11c we15ave ¯ p ≤ p ( ρ L − ) < f ( ρ L ), and as a consequence, property np2 forces us tochoose ¯ p = p ( ρ L +). (C3) In this case [( t, x ) (cid:55)→ R [ ρ L , ρ R ]( x/t )] performs a possible null rarefactionon the right of the constraint and it satisfies 5b because f ( ρ L ) ≤ p ( ρ L +).Assume that there exists a nonclassical solution of the form 11. Since ρ L ≤ ¯ ρ ≤ ˆ ρ (¯ p ), R [ ρ L , ˆ ρ (¯ p )] is given by a shock that has non positive speed if andonly if ¯ p ≤ f ( ρ L ). Therefore ¯ p ≤ f ( ρ L ) ≤ p ( ρ L +) and this by 11c impliesthat ¯ p = f ( ρ L ) = p ( ρ L +) and that the nonclassical solution coincides withthe classical one. (N3) In this case [( t, x ) (cid:55)→ R [ ρ L , ρ R ]( x/t )] does not satisfy 5b because f ( ρ L ) >p ( ρ L − ), see case (C3) . Therefore, there does not exist any classical so-lution and we can consider only nonclassical solutions of the form 11. Byhypothesis and 11c we have f ( ρ L ) > p ( ρ L − ) ≥ ¯ p . Therefore ρ L ≤ ¯ ρ < ˆ ρ (¯ p )and by np2 we have ¯ p = p ( ρ L +). (C4) In this case [( t, x ) (cid:55)→ R [ ρ L , ρ R ]( x/t )] performs a rarefaction with speedsbetween λ ( ρ L ) < λ ( ρ R ) ≥ f (¯ ρ ) = p ( ρ L +)implies that p ( ρ ) = f (¯ ρ ) for all ρ ≤ ρ L . Moreover, it implies also that p is continuous in ρ L and therefore, by np4 , any nonclassical solution of theform 11 must have ¯ p = p ( ρ L ) = f (¯ ρ ), but in this case the nonclassicalsolution coincides with the classical one. (N4) In this case [( t, x ) (cid:55)→ R [ ρ L , ρ R ]( x/t )] does not satisfy 5b because f (¯ ρ ) >p ( ρ L − ), see case (C4) . Therefore, there does not exist any classical solutionand we can consider only nonclassical solutions of the form 11. (N4a) By assumption and 11c f ( ρ L ) < p ( ρ L +) ≤ ¯ p and therefore ˆ ρ (¯ p ) <ρ L and by np3 we have ¯ p = p ( ρ L − ). (N4b) By assumption and 11c f ( ρ L ) > p ( ρ L − ) ≥ ¯ p and therefore ˆ ρ (¯ p ) >ρ L and by np2 we have ¯ p = p ( ρ L +). (C5) In this case [( t, x ) (cid:55)→ R [ ρ L , ρ R ]( x/t )] performs a possible null rarefaction onthe left of the constraint and it satisfies 5b because f ( ρ R ) ≤ p ( ρ L − ) and p ( ξ ( t )) ≡ p ( ρ L − ) by bp1 . Assume that there exists a nonclassical solutionof the form 11. Since by assumption and 11c ¯ p ≥ p ( ρ L +) > f ( ρ L ), we haveˆ ρ (¯ p ) < ρ L and by np3 ¯ p = p ( ρ L − ), but in this case the nonclassical solutioncoincides with the classical one. (N5a) In this case [( t, x ) (cid:55)→ R [ ρ L , ρ R ]( x/t )] does not satisfy 5b because f ( ρ R ) >p ( ρ L − ), see case (C5) . Therefore, there does not exist any classical solutionand we can consider only nonclassical solutions of the form 11. (N5b) By assumption and 11c, f ( ρ L ) < p ( ρ L +) ≤ ¯ p and therefore ˆ ρ (¯ p ) <ρ L and by np3 we have ¯ p = p ( ρ L − ). (N5b) By assumption and 11c, f ( ρ L ) > p ( ρ L − ) ≥ ¯ p and therefore ˆ ρ (¯ p ) <ρ L and by np2 we have ¯ p = p ( ρ L +).16 .1.2 Cases in which uniqueness is violated Now we list the “pathological” cases, where we have more than one admissiblesolution. We stress once again that a necessary condition for non–uniqueness is p ( ρ L − ) (cid:54) = p ( ρ L +) and p ( ρ L − ) ≥ f ( ρ L ) ≥ p ( ρ L +). (CN2) If ρ L < ρ R , f ( ρ L ) ≤ f ( ρ R ) and p ( ρ L +) < f ( ρ L ) ≤ p ( ρ L − ), then theclassical solution [( t, x ) (cid:55)→ R [ ρ L , ρ R ]( x/t )], which consists of a shock with nonnegative speed, as well as the nonclassical solution 11, with ¯ p = p ( ρ L +), aredistinct solutions of 5. (CN3) If ρ R ≤ ρ L ≤ ¯ ρ and p ( ρ L +) < f ( ρ L ) ≤ p ( ρ L − ), then the classical solution[( t, x ) (cid:55)→ R [ ρ L , ρ R ]( x/t )], which consists of a possible null rarefaction on theright of the constraint, as well as the nonclassical solution 11, with ¯ p = p ( ρ L +),are distinct solutions of 5. (NNN4) If ρ R ≤ ¯ ρ < ρ L , p ( ρ L − ) (cid:54) = p ( ρ L +) and p ( ρ L +) ≤ f ( ρ L ) ≤ p ( ρ L − ),then the nonclassical solutions of the form 11 which corresponds to ¯ p in theset { p ( ρ L +) , f ( ρ L ) , p ( ρ L − ) } satisfy 5. This is the situation considered in theExample 2 in [3]. Observe that such solutions are distinct as far as theycorrespond to distinct constraint levels ¯ p , and that in any case there exist atleast two distinct nonclassical solutions. (CNN5) If ¯ ρ < ρ R ≤ ρ L , f ( ρ R ) ≤ p ( ρ L − ), p ( ρ L − ) (cid:54) = p ( ρ L +) and p ( ρ L +) ≤ f ( ρ L ),then the classical solution [( t, x ) (cid:55)→ R [ ρ L , ρ R ]( x/t )], which consists of a pos-sible null rarefaction on the left of the constraint, as well as the nonclassicalsolutions of the form 11 corresponding to ¯ p ∈ { p ( ρ L +) , f ( ρ L ) } satisfy 5. Ob-serve that the two nonclassical solutions are distinct as far as they correspondto distinct constraint levels ¯ p , and that in any case there exist at least twodistinct solutions, one classical and one nonclassical. (NNN5) If ¯ ρ < ρ R < ρ L , f ( ρ R ) > p ( ρ L − ) ≥ f ( ρ L ) ≥ p ( ρ L +) and p ( ρ L − ) (cid:54) = p ( ρ L +), then the nonclassical solutions of the form 11 corresponding to ¯ p ∈{ p ( ρ L +) , f ( ρ L ) , p ( ρ L − ) } satisfy 5. Observe that such solutions are distinctas far as they correspond to distinct constraint levels ¯ p , and that in any casethere exist at least two distinct nonclassical solutions. (R1) Any solution given by R (cid:63) coincides on each side of the constraint with asolution given by the classical Riemann solver R . Therefore it satisfies theRankine–Hugoniot jump condition along any of its discontinuities away fromthe constraint. Finally, by definition of ˆ ρ and ˇ ρ , it satisfies the Rankine–Hugoniot jump condition also along the constraint. (R2) It is clear by the proof of Proposition 2. (R3)
It is proved as in (R1) since any classical solution is in BV . (R4) As was proved in [7], R (cid:63) is continuous on C ∪ N . If ( ρ L , ρ R ) is not in C ∪ N then p experiences a jump at ξ = ρ L . Therefore, the local in time solutions ofthe Riemann problem for the initial conditions ( ρ L + ε, ρ R ) and ( ρ L − ε, ρ R )are different and only one of the two converges to R (cid:63) [ ρ L , ρ R ] as ε > (R5) We first stress once again that we can discuss the consistency property ofour Riemann solvers only locally in time because, in general, the solutions17ay be not even self–similar globally in time. However, locally in time, theefficiency of the exit can be assumed to be constant and it is thus sufficientto proceed as in [7]. (R6)
It is clear by the proof of Proposition 2.
All the authors are supported by French ANR JCJC grant CoToCoLa and Polonium2014 (French-Polish cooperation program) No.331460NC. The second author is alsosupported by the Universit´e de Frache-Comt´e, soutien aux EC 2014. The last authoris also supported by ICM, University of Warsaw, and by Narodowe Centrum Nauki,grant 4140.
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