Explicit construction of effective flux functions for Riemann solutions
aa r X i v : . [ m a t h . A P ] J a n Explicit construction of effective flux functionsfor Riemann solutions
Pablo Casta˜neda
Abstract
For a family of Riemann problems for systems of conservation laws, weconstruct a flux function that is scalar and is capable of describing the Riemannsolution of the original system.
We are interested in injection problems leading to flow in porous media, which aremodeled by systems of conservation laws; a survey of the mathematical theory forsuch flow may be found in [2, 6, 21] and references therein. In this work we focuson the Riemann problems and their solutions via the Wave Curve Method ( cf. [1, 8])and on the construction of effective flux functions (EFF) allowing to understand thewhole system as a single scalar conservation law, see [8, 18].The setting for such a construction is given for a fixed state in physical space.Thus we can develop the construction of a wave group the lifting of which willgive an effective flux function. Such functions can be treated as the flux functionsof a scalar conservation law in a certain parametrized coordinate. This lifting isthe crucial part in the construction. However, we will show that this function is notunique. We only have uniqueness as a class of functions, the representation of whichwill be the effective flux function for each state, each starting eigenvalue family and,the chosen coordinate system.Analogous effective flux functions have been used satisfactorily in many works.There are implicit uses in [1, 2, 3, 19] and, explicit constructions in [8, 10]. An-other potential application is modeling special flux functions in experimental datafor which classical known models are inappropriate ( e.g. [13] vs. [20]).
Pablo Casta˜nedaDepartment of Mathematics, ITAMR´ıo Hondo 1, Ciudad de M´exico 01080, Mexico.e-mail: [email protected] 1 Pablo Casta˜neda × Let us write as a system the conservation laws used along this work. We focus in asystem with two equations, since it is only important to take more than one equation.The extension to any number of equations would be natural.Let u ( x , t ) , u ( x , t ) to be the conserved quantities at distance x along the realaxis, at time t . Typically in one spatial dimension a set of equations governing thesystem ¶ U ¶ t + ¶ F ¶ x = , or ¶ u / ¶ t + ¶ f / ¶ x = ¶ u / ¶ t + ¶ f / ¶ x = , (1)for x ∈ IR, t ≥
0, representing the conservation of U = ( u , u ) . The flow functionscharacterize the system and are denoted as the vector F ( U ) = ( f ( u , u ) , f ( u , u )) .We denote as D the space of states U , in general we consider D ⊂ IR .Of special interest in applications and numerical calculations consists in the clas-sification of the solution structure of the system of PDE (1) with discontinuous data: U ( x , t = ) = ( U L if x < , U R if x > . (2)The Riemann problem consists of system (1) with initial Riemann data (2), whichwe will denote as RP ( U L , U R ) , with left and right values U L and U R , respectively.Strictly hyperbolic systems of conservation laws, i.e. , where the eigenvalues ofthe Jacobian matrix of the flux function are real and distinct, provide a relativelywell understood framework for the solution of Riemann problems [11]. Here weassume that the system is not necessarily strictly hyperbolic.A remarkable observation is that in many problems for flow in porous media,there is an extra conserved quantity or fluid, thus an extra equation for system (1).Typically the extra quantity and the phases in (1) add up to one. In the same way,there is an extra flux function that depends on these phases, see for example [6, 8, 19]where u , u , and u can represent the saturation of water, gas and oil. Here thestate space is the saturation triangle defined by U satisfying 0 ≤ u , u , u + u ≤ u + u + u =
1. For this model an extra function satisfies f = − f − f that gives a third redundant equation. This is an important fact since theparametrization of the EFF can be given in any of those coordinates. Equations (1) have solutions that propagate as nonlinear waves. Because of self-similarity of the data and the PDE, the solutions of a Riemann problem depend on x / t and consist of centered rarefaction waves, shock waves and sectors of constantstates, see e.g. [14, 15, 18]. The characteristic speeds are the two eigenvalues of theJacobian matrix xplicit construction of effective flux functions for Riemann solutions 3 J ( S ) : = ¶ ( f ( U ) , f ( U )) ¶ ( u , u ) = ¶ F ( U ) ¶ U . When the eigenvalues are distinct and real we say that the system is strictly hyper-bolic. Sometimes, it loses hyperbolicity at particular states, as the cases registeredin [4, 17, 19] for umbilic and quasi-umbilic points. For distinct eigenvalues, thesmaller and larger are called the slow- and the fast-family characteristic speed. Ac-tually, these eigenvalues can be equal on a curves or on larger sets, see [22].System (1) has smooth solutions called (slow- and fast-family) rarefaction waves.They arise by solving an ODE, namely, dU / d x = r k ( U ) , (3) k = s or f (slow or fast), defined by the eigenvectors of the Jacobian matrix J ( U ) ,with { J ( U ) − x I } r k ( U ) = , (4)where U ( x ) , for x = x / t , is the profile of the forward rarefaction, provided x ismonotone increasing; it is called backward for x monotone decreasing. The k -integral curve of a state U o , denoted by R k ( U o ) , consists of all states U for eachof which U ( x ) solves the initial value problem (3) for the given k and initial condi-tion U o , either backward and forward.This system also admits solutions in the form of moving jump discontinuities.In order to respect conservation of u and u , the fluxes in and out of the movingdiscontinuity must balance. In terms of the state U o = ( u o , u o ) on the left of thediscontinuity, the state U = ( u , u ) on the right of the discontinuity, and the propa-gation speed s , this balance is expressed as F ( U ) − s U = F ( U o ) − s U o , or f ( U ) − s u = f ( U o ) − s u o , f ( U ) − s u = f ( U o ) − s u o . (5)Eqs. (5) are the Rankine-Hugoniot (RH) conditions. The Rankine-Hugoniot locusof a state U o , denoted by H ( U o ) , consists of all states U for each of which thereexists a value s = s ( U o , U ) such that the RH conditions (5) are satisfied.The two former loci in the space of states will be the basis for the construction ofthe wave group and for the effective flux function. We recall that in practice we mustselect a criterion for the “physically admissible” discontinuities that appear in thesolutions. This will be Liu’s criterion including also the Lax’s admissibility criteria,namely: Shock admissibility.
A discontinuity with propagation speed s = s ( U L , U R ) be-tween a left state U L and a right state U R is admissible if it satisfies Liu’s admissi-bility criterion [16], whenever it is applicable. We use Lax’s admissibility criterion[14] in order to classify the propagation speed as follows: l s ( U R ) < s < l s ( U L ) , s < l f ( U R ) , for slow-family shocks , l f ( U R ) < s < l f ( U L ) , l s ( U L ) < s , for fast-family shocks . (6) Pablo Casta˜neda
Moreover, in the previous definitions we allow one of the inequalities to become anequality; hence our admissibility criterion. The nomenclature slow- and fast-familyoriginates from the 1-Lax and 2-Lax shock waves, [14].The following definition is inspired by the Welge-Ole˘ınik’s construction for asingle conservation equation. Let U o be a state in physical space; a state U is a slow (respectively fast ) extension of U o if U belongs to H ( U o ) and the shock speed s ( U o , U ) equals the slow characteristic speed l s ( U ) (resp. fast l f ( U ) ), i.e. , theshock is characteristic at U .The Bethe-Wendroff Theorem ( cf. [12]) guarantees that at an extension point oneof the rarefactions curves starting at U is tangent to the H ( U o ) at U . Solutions found by Wave Curve Method consist of rarefaction fans, shock dis-continuities and constant states, a survey is found in [1]. The classical construc-tion is guaranteed to succeed only when U L and U R are close. The Buckley-Leverett (BL) solution exhibits inflection points ( cf. [5, 18]), where equalities suchas (cid:209) l k ( U ) · r k ( U ) = k = s or f occur. The BL shows that rarefaction wavesand shock waves of the same family can be adjacent. For adjacency to occur, theshock speed must coincide with the same-family characteristic speed at an edge ofthe rarefaction wave, and one of the Lax’s inequalities in (6) becomes an equality.Of course, when traversing the solution by increasing x / t monotonically, the cor-responding wave speed must also increase. Consequently, fast-family waves followslow-family waves. This structural feature was first identified by Liu, [15], undertechnical restrictions and holds in general; see [21]. Wave sequences are concate-nated following certain rules.A wave group is a sequence of waves, all associated to the same family, which areadjacent, meaning that no two waves are separated by a constant state. As in the Laxconstruction (Fig. 1), if the beginning or end state of a wave group is prescribed, thenthe state on the opposite end of the wave group lies on a curve in state space calleda wave curve. In the language of wave groups, the Riemann solution consists of thefollowing wave sequence, from left to right: the left state U L , a slow-family wavegroup, an intermediate state U M , a fast-family wave group, and the right state U R .Notice that the slow-family wave curve from U L consists of all states U M attainablethrough a slow-family wave group; similarly, the backward fast-family wave curvefrom U R consists of all states U M attainable through a fast-family wave group. Wavecurves and an algorithm to construct the Riemann solution were described in [15].In constructing wave groups with two or more waves, in [15] Liu introduced ashock wave admissibility criterion that encompasses the criterion of Ole˘ınik, [18].The latter is a generalization of Welge’s construction, the one we use for construct-ing the wave group, hence we call Welge point to values where a shock wave ischaracteristic for a scalar flux function. The wave curve method is applicable to sys- xplicit construction of effective flux functions for Riemann solutions 5 u u U L U M U R u xu L u M u R Fig. 1
Solution of RP ( U L , U R ) , U L determines the slow-family wave curve and U R the backwardfast-family wave curve. Their intersection determines U M . At the right figure, the solution profileat a fixed time, notice the value of both coordinate system at u . tems with any number of conservation laws. The effective flux function is given foreach wave group as we show soon. The main idea is the following: for any given fixed state R in D construct a basecurve G : I → D and an Effective Flux Function (EFF) f : I → IR. The base curve G is a parametrization of a wave group; for the 2 × f ( ℓ ) is the lifting of the base curve definedby the wave group on the physical space.The heart of our work resides in two facts: (1) when the state G ( ℓ ) is at a shockcurve, the RH condition (5) is satisfied, thus the shock speed can be given by any ofthose relations or a linear combination of them, (2) when the state G ( ℓ ) is at an inte-gral curve, the wave speed is given as l k (cid:0) G ( ℓ ) (cid:1) with k = s or f , the correspondingfamily of such a rarefaction.Now let us explain the construction of the base curve and how its lifting is found.As pointed out, we subdivide these constructions into two cases. This is done onlyfor an easier exposition because it always works in the same manner. GGG ((( ℓℓℓ )))
In Sec. 2.2 it is explained how to construct slow- and fast-family wave groups withintegral curves and Hugoniot loci on the state space. For a hyperbolic point, i.e.
Pablo Casta˜neda where both characteristic speeds are real and distinct, there are two linearly inde-pendent eigenvectors that describe the tangents to the slow and fast wave groups.Typically, for each family the respective eigenvalue increases in the directionof an eigenvector and decreases in the opposite direction. Therefore, for a forwardwave group construction, we have a rarefaction wave in one direction and a shockwave in the opposite one. There are cases where both directions have shock or rar-efactions as initial waves, such points are in the inflection manifold of the respectivefamily.The base curve G : I → D will be a parametrization of one wave group. Thechoice of the interval of interest I and its parametrization is essential in the con-struction of the EFF f ( ℓ ) , its lifting , the construction of which will be given soon.First of all, we need some guiding notation for the analysis that follows. As G ( ℓ ) belongs to D we take the coordinates as G ( ℓ ) = ( g ( ℓ ) , g ( ℓ )) . Thus for U ∈ G ( ℓ ) wehave u i = g i ( ℓ ) for i = ,
2. Actually, if one of the coordinates g or g is monotonic,say g , then it is possible to do a reparametrization ℓ = g ( ℓ ) . Thus we can assumethat G ( ℓ ) = ( ℓ, g ( ℓ )) holds at least locally. (In [8] the parametrization is given with ℓ as the oil saturation, the third implicit coordinate, and it is easy to see that it isalways possible to take ℓ as a linear combination of the coordinates g and g ; weassume that g ( ℓ ) = ℓ holds for an easier exposition.)In the following sections the lifting construction of the EFF f ( ℓ ) is described.Nonetheless, it is important to remark that such a lifting has as motivation to bea function that behaves as a single scalar flux function. Therefore, the Ole˘ınik E-criterion is applicable ( cf. [18]), moreover as such a construction is based on theenvelope of the flux function, the “mirror effect” given in [7] follows. Assume that the base curve G ( ℓ ) starts with a k -Lax shock wave at the referencestate R = ( u R , u R ) . As pointed out in Sec. 2.2 and in Fig. 1, the forward wave groupis relevant for k = s and the backward wave group for k = f . In the forward construc-tion we take the part of H ( R ) that satisfies (6) for the chosen k and such that Liu’scriterion holds for all points between R and U , respectively as left and right states,see [15, 16]. (For backward construction, U and R are the left and right states.) Thusfor each U = ( u , u ) in H ( R ) there exists g ( ℓ ) such that ( ℓ, g ( ℓ )) = U holds;see Sec. 3.1. Notice that the set of admissible shocks may stop at a Bethe-Wendroffpoint U ∗ where s ( R , U ∗ ) = l k ′ ( U ∗ ) occurs. (Actually k ′ = k may hold.)We set the interval I = [ u R , u ∗ ] assuming that u R < u ∗ . (Conversely, I = [ u ∗ , u R ] for u ∗ < u R .) Now we have the base curve G : I → D satisfying G ( ℓ ) = ( ℓ, g ( ℓ )) ∈ H ( R ) the position of which determines the corresponding shock speed (5). Thus,we construct the lifting as f : I −→ IR ℓ f ( ℓ ) : = f ( R ) + s (cid:0) R , G ( ℓ ) (cid:1) [ ℓ − u R ] , (7) xplicit construction of effective flux functions for Riemann solutions 7 thus, notice that the scalar shock speed satisfies s ( u R , ℓ ) = f ( ℓ ) − f ( R ) ℓ − u R = s (cid:0) R , G ( ℓ ) (cid:1) , for all ℓ ∈ I , (8)so the Rankine-Hugoniot condition (5) holds as desired. Remark 1.
When ℓ ∗ is a value of a Welge point of the EFF (always with left state u R ),then U ∗ = G ( ℓ ∗ ) is a Bethe-Wendroff point of H ( R ) , thus the eigenvector pointsparallel to the Hugoniot locus and the wave curve may be followed by a rarefactioncurve. (As stated by the Bethe-Wendroff theorem.) Lemma 1.
The EFF f ( ℓ ) in (7) is the first flux function over the base curve, i.e. , f ( ℓ ) = f (cid:0) G ( ℓ ) (cid:1) .Proof. The RHS of (8) is equivalent to [ f (cid:0) G ( ℓ ) (cid:1) − f ( R )] / [ ℓ − u R ] , equating withthe middle term in (8) proves that f ( ℓ ) = f (cid:0) G ( ℓ ) (cid:1) holds in (7). (cid:3) The Bethe-Wendroff point at the state U ∗ can be taken as a new starting referencepoint for a rarefaction wave. Just make sure that the starting family k now must betaken as k ′ ; they may not be the same. Assume that the base curve G ( ℓ ) starts with a k -rarefaction curve at the referencestate R = ( u R , u R ) . In the forward construction we take the R k ( R ) part which hasincreasing eigenvalue x = l k ( U ) for U ∈ R k ( R ) . (In the backward construction wetake the decreasing eigenvalue direction.) Thus for each U = ( u , u ) in R k ( R ) thereexists g ( ℓ ) such that ( ℓ, g ( ℓ )) = U holds; see Sec. 3.1.Recall that the rarefaction curve may stop at an inflection point U ∗ (where (cid:209) l k ( U ∗ ) · r k = I = [ u R , u ∗ ] assuming that u R < u ∗ .(Conversely, I = [ u ∗ , u R ] for u ∗ < u R .) Thus, the base curve G : I → D satisfies G ( ℓ ) = ( ℓ, g ( ℓ )) ∈ R k ( R ) , the position of which determines the correspondingeigenvalue. Thus, we construct the lifting as f : I −→ IR ℓ f ( ℓ ) : = f ( R ) + R ℓ u R l k (cid:0) G ( t ) (cid:1) dt , (9)and notice that f ′ ( ℓ ) = l k ( G ( ℓ )) holds as desired. Lemma 2.
The EFF f ( ℓ ) in (9) is the first flux function over the base curve, i.e. , f ( ℓ ) = f (cid:0) G ( ℓ ) (cid:1) .Proof. Direct differentiation in (9) shows that f ′ ( ℓ ) = l k ( G ( ℓ )) holds, notice alsothat (cid:209) G ( ℓ ) = ( , g ′ ( ℓ )) is parallel to r k ( G ( ℓ )) . Thus, the identity J ( G ( ℓ )) (cid:209) G ( ℓ ) = l k ( G ( ℓ )) (cid:209) G ( ℓ ) holds, see (4), the first coordinate indicates that Pablo Casta˜neda dd ℓ f ( G ( ℓ )) = (cid:18) ¶ f ¶ u , ¶ f ¶ u (cid:19) · ( , g ′ ( ℓ )) = l k ( G ( ℓ )) is satisfied. Then, as f ( ℓ ) and f ( G ( ℓ )) solve the same IVP with initial condition f ( R ) , the fluxes are the same. (cid:3) The inflection at the state U ∗ can be taken as a new starting reference point fora shock wave, just make sure that in (7) the reference state for determining theHugoniot locus and the shock speed is the original reference state R and not U ∗ . In previous sections we depicted the construction of a base curve G ( ℓ ) , see Sec. 3.1,and two ways for the lifting of f ( ℓ ) based on shock curves (Sec. 3.2) or on rarefactioncurves (Sec. 3.3). In Sec. 2.2 we pointed out that a wave curve is a composition ofthe former waves; it changes types at inflection or Bethe-Wendroff points. In thissection we show the construction of a complete EFF which is actually a smoothfunction. From shock to rarefaction curve.
Once we start a base curve with shock waves,this is a curve along H ( R ) and would change to an integral curve within the samewave only at a Bethe-Wendroff point U ∗ . Notice that at such a point the equality f ( u ∗ ) = l k ( U ∗ ) holds for certain k , therefore the continuity for the derivatives of thelifting between both expressions (7) and (9) holds.The continuity of the EFF itself holds because of the adding of f ( R ) in bothliftings (7) and (9): from Lemma 1 we have that f ( ℓ ) = f (cid:0) G ( ℓ ) (cid:1) holds in particularat U ∗ , and since from (9) we have that f ( u ∗ ) = f ( U ∗ ) holds, also the EFF continuity. From rarefaction to shock curve.
The continuity at this transition do not seems sonatural; the actual value of f ( ℓ ) in (9) is not know a priori for a given ℓ . Howeveras the transition must occur at a point U ∗ belonging to both R k ( R ) and H ( R ) , thusthe values l k ( U ∗ ) and s ( R , U ∗ ) agree and from the latter and (7), we notice that f ( u ∗ ) = f ( U ∗ ) holds; hence the smoothness at such transitions.For a complete construction of an EFF, we notice that transitions from shockwaves to rarefaction waves and vice-versa may occur many times. This is perfectlycontrolled by our way of doing the liftings; recall that from a shock to a rarefaction,the family k may have changed to k ′ and that from a rarefaction to a shock, thereference point R is the original one. We have proven our main result: Theorem 1.
Let R to be a fixed state in D . Construct a wave curve through R. Selecta coordinate ℓ , let us say ℓ = u , to parametrize the curve as G : I → D satisfying G ( u Ri ) = R. Therefore, an EFF is the respective flux function along G ( ℓ ) as f : I −→ IR ℓ f (cid:0) G ( ℓ ) (cid:1) . xplicit construction of effective flux functions for Riemann solutions 9 Proof.
See Lemmas 1 and 2. (cid:3)
Just recall that such an EFF is constructed based on R , thus even for R ′ ∈ G ( ℓ ) ,the respective EFF may be distinct. In the next section we show some examples. In this section we explain two examples. The first one based in the satisfactory useof EFFs in [8, 10], constructed along the so-called separatrix as a crucial wave groupfor the Riemann solution. The second one shows how choosing the parametrizationcoordinate is important for understanding an EFF.
In Enhanced Oil Recovery (EOR) the proposed conservation laws are given by frac-tional flux functions due to physical features as rock and fluid permeabilities (see e.g. [6, 19]). Here we take a schematic model with flow functions for (1) given by: f ( U ) = Au Au + Bu + Cu , f ( U ) = Bu Au + Bu + Cu , (10)where constants A , B and C depend on several physical quantities. It was pointed outthat u and u are related to water and gas saturations, the oil saturation is related to u = − u − u . The flow functions (10) are related to the flux function for waterand gas, which came from the so-called quadratic permeability Corey model; anextra implicit flow function for oil f ( U ) = Cu / ( Au + Bu + Cu ) is sometimesuseful. The domain is the saturation triangle given by the constraints 0 ≤ u , u , u and u + u + u = A classical problem in EOR is the Water alternating Gas (WAG) injection, which hasa direct relation to the Riemann problem RP ( U L , O ) where the left Riemann datum U L represents a mixture of water and gas, and the left Riemann datum O = ( , ) represents a virgin reservoir state containing solely oil.In the saturation triangle there are three base wave curves reaching O ( cf. [2]).They are lines which can be parametrized by ℓ ∈ I as the third (implicit) coordi-nate u for the interval I = [ , ] . Let G ( ℓ ) = ( − ℓ, ) , G ( ℓ ) = ( , − ℓ ) and, G ( ℓ ) = (( − ℓ ) B / D , ( − ℓ ) A / D ) be the base curves for such parametrization withthe denominator D = A + B , see Fig. 2. Thus, simple computations lead to the EFFs Fig. 2
The shadowed regionrepresents the saturation tri-angle. The vertex O , W and G are related to pure oil, waterand gas, respectively. Thebase curves G i are the linesconnecting O to W , G and B respectively for i = , H ( R ) ,both branches of a hyperbola.The thin continuous curveis a slow-family rarefactionconnecting J to S ∗ . O R WS ∗ B JG u u f i ( ℓ ) = f ( G i ( ℓ )) for i = , , f ( ℓ ) = C ℓ A ( − ℓ ) + C ℓ , f ( ℓ ) = C ℓ B ( − ℓ ) + C ℓ , f ( ℓ ) = C ℓ AB ( − ℓ ) / D + C ℓ . The EFFs above satisfy the BL solution, see [5], with S-shaped flux functions.Their Welge points can be calculated by the values ℓ ∗ = − p C / ( A + C ) , ℓ ∗ = − p C / ( B + C ) , ℓ ∗ = − p CD / ( AB + CD ) , with relative positions satisfying ℓ ∗ < ℓ ∗ , ℓ ∗ . These inequalities guarantee that theoptimal injection mixture for oil production occurs within the separatrix, see [8, 10].For Welge points, their values ℓ ∗ i also indicate the locations of Bethe-Wendroffpoints U ∗ i : = G ( ℓ ∗ i ) . It is possible to verify that s ( U ∗ , O ) = l s ( U ∗ ) holds, so therest of the base curve G is a slow rarefaction. The analog Bethe-Wendroff points U ∗ , U ∗ , show that along G , G the rarefactions are of the fast family. The complete solution for the WAG injection needs a slow wave group. It can befound in forward direction from any point representing mixture of water and gas.However, intermediate states lie over one of the base curves of Sec. 4.2, here weshow the construction of EFF over backward slow curves for states R = ( m , ) .The RH relation (5) lead to the RH locus H ( R ) for all states U in the sat-uration triangle satisfying f ( U ) − f ( R ) = [ f ( U ) / u ]( u − m ) , since the shockspeed is f ( U ) / u , see (5.b). A simple manipulation shows that H ( R ) is the edge xplicit construction of effective flux functions for Riemann solutions 11 U = ( u , ) and, with f R = f ( R ) , all U satisfying ( A − A f R − C f R ) u − ( B + C f R ) u u − ( B + C ) f R u + C f R u + ( C f R + Bm ) u − C f R = , which is a hyperbola, see also [2]. Of course, for the parametrization of the hyper-bola in Fig. 2, the u coordinate seems to be a good choice, the u coordinate havea detour. (The third coordinate u is also a good choice.)The base curve G ( ℓ ) = ( g ( ℓ ) , ℓ ) is parametrized with g ( ℓ ) = − b + √ b − ac a , where a = A − ( A + C ) f R , b = C f R − ( B + C f R ) ℓ and c = − ( B + C ) f R ℓ +( C f R + Bm ) ℓ − C f R hold. Thus, from the flux function (10.b), the EFF is f ( ℓ ) = f ( G ( ℓ )) = B ℓ A g ( ℓ ) + B ℓ + C ( − ℓ − g ( ℓ )) , at the value ℓ ∗ for the Welge point it is actually satisfied l s ( G ( ℓ ∗ )) = f ( G ( ℓ ∗ )) /ℓ ∗ ,as proven in [9]. (The related Bethe-Wendroff point S ∗ = G ( ℓ ∗ ) belongs to the exten-sion boundary.) From S ∗ the slow-family rarefaction follows until the third (implicit)coordinate vanish at J , see Fig. 2. This work contributes to the applicability of EFFs for systems of conservation laws.The remarkable feature resides on restricting each wave group of a Riemann prob-lem, into a problem with a single scalar equation. Of course, via the Wave CurveMethod, the system determines the base curve which supports the EFF, thereforethe Riemann problem is satisfied both for the system and for the restricted scalarconservation law. This contribution is a first step, this engine will be important inproving conjectures we have as well as new emerging results.
Acknowledgments
I’m very grateful to Prof. Dan Marchesin (IMPA) for his friendship and, the manyenlightening discussions, comments and suggestions for this work. This work waspartially supported by grants CNPq 402299/2012-4, 170135/2016-0 and FAPERJE-26/210.738/2014. I gratefully acknowledge the financial support by Asociaci´onMexicana de Cultura A.C. and by the Hyp2016 conference.
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