NONLINEAR WAVE INTERACTIONS IN A MACROSCOPIC PRODUCTION MODEL?

MINHAJUL

Center for Applicable Mathematics,Tata Institute of Fundamental Research,Bangalore,Karnataka-560065,India

Department of Mathematics,Indian Institute of Technology Kharagpur,Kharagpur-2,India

E-mail:minhajulbgh63@gmail.com

T RAJA SEKHAR

Department of Mathematics,Indian Institute of Technology Kharagpur,Kharagpur-2,India

E-mail:trajasekhar@maths.iitkgp.ac.in

Abstract In this article,we study the exhaustive analysis of nonlinear wave interactions for a 2×2 homogeneous system of quasilinear hyperbolic partial differential equations(PDEs)governing the macroscopic production.We use the hodograph transformation and differential constraints technique to obtain the exact solution of governing equations.Furthermore,we study the interaction between simple waves in detail through exact solution of general initial value problem.Finally,we discuss the all possible interaction of elementary waves using the solution of Riemann problem.

Key words Elementary waves;wave interactions;Riemann problem;simple wave;differential constraints;Hodograph transformation

1 Introduction

In the recent past,the wave interactions problem for hyperbolic system of conservation laws has attracted the attention of the mathematicians and physicists because of its wide applications in various practical problems of engineering and science[1–9].The study of wave interactions is not only determine the qualitative properties of solution,but also provides some essential properties of solution with geometric structure.Therefore,it has its own significance in various practical problems.It is well known that inside the wave interaction regions,the different families of characteristic curves play an important role on the problems concerning the wave interactions for hyperbolic system of PDEs.Moreover,the solution for nonlinear hyperbolic systems in a closed form is not available in most of the cases.In fact,the analysis of nonlinear waves are usually carried out by asymptotic methods[10].Also,the wave dynamics of one-dimensional homogeneous system of hyperbolic PDEs is governed by the behavior of solutions along the associated characteristic fields.Simple waves[11,12]are one of the important class of solutions of hyperbolic system of PDEs which remain constant along the characteristic associated with it.Although the interaction of elementary waves for 2×2 system of hyperbolic PDEs are available in the literature[13,14],but for general initial value problem it is very difficult to study the wave interaction process for hyperbolic system of PDEs.In particular,simple wave interactions for 2×2 system of hyperbolic PDEs for the Cauchy problem is very interesting and challenging task.

It is known from the literature that using the classical hodograph transformation,the 2×2 system of homogeneous quasilinear strictly hyperbolic system of PDEs can be reduced to a linear system(canonical form)of PDEs.However,in most of the cases these resulting linear system of PDEs can not be integrated directly in a closed form[15].The authors in[16]characterized a second order hyperbolic PDE,where the solutions are obtained by integrating the hodograph system,and it is shown that the solutions to this system describe the waves these travel in the opposite directions and interact with each other those behave like solitons after interaction.Latter,the authors in[17]proposed a class of 2×2 system of quasilinear hyperbolic PDEs,including the model proposed in[16]which allowing soliton like wave interactions.In contrast to solitons,it is well known that the nonlinear waves for hyperbolic system of PDEs propagate and distort in time before interaction of the travelling pulses.Moreover,the solution is classical only upto the critical time[11,18]at which shock may occurs in the solution.However,the existence of global classical solution to the Goursat problem for reducible quasilinear hyperbolic system can be found in[19].

In[2],the authors proposed an idea to solve 2×2 system of strictly hyperbolic balance laws by using the combination of the hodograph transformation and differential constraints technique.If one can integrate the associated hodograph system in a closed form,then such an idea is very useful for 2×2 systems.Moreover,the exact wave-like solutions are obtained by integrating the hodograph system incorporating all the features of classical simple waves as well as Riemann invariants.Here,we consider a 2×2 system of PDEs governing the macroscopic production[20,21]based on the data fitting approach and study the simple wave interactions using the properties of Riemann invariants.Furthermore,we discuss the interactions of elementary waves of same family as well as different families using the solution of Riemann problem.In particular,we consider the system of PDEs in conservative form[22]

In order to know the production behavior and aggregate product flow,the system of PDEs(1.1)was derived from discrete simulations based on sampled data.The unknown variables u and ρ,respectively,denote the product velocity and density at time t and production stage x.Moreover,it is reasonable to assume that speed of propagation of information is bounded by velocity u,and that wave always travels with positive speed.

In this work,we combine the ideas of differential constraints technique[23–26]and hodograph transformation to study the interaction of simple waves for general initial value problem of(1.1).It is very difficult to study the simple wave interactions for general initial value problem unless the governing system of PDEs satisfies a certain structural condition.We observe that such condition is satisfied by this macroscopic production model and we are able to integrate the hodograph system of linear differential equations to obtain the wave pulses.Moreover,we use the Riemann solution of(1.1)to study the all possible interactions of elementary waves.

The rest of the paper is organized as follows:in Section 2,we express the hodograph differential equations and discuss the problem of integrating the hodograph differential equations using the differential constraints technique.In Section 3,we obtain the solution of linear hodograph system of equations in terms of the Riemann invariants.We investigate the simple wave interaction problem exhaustively in Section 4 by means of exact solution to the Cauchy problem.In Section 5,we obtain the solution of Riemann problem and discuss all possible elementary wave interactions using the phase plane analysis.Finally,Section 6 consists of brief conclusions.

2 Hodograph Transformation and Differential Constraints

In this section,we briefly outline the differential constraints technique and hodograph transformation.Let us consider a 2×2 strictly hyperbolic and homogeneous system of quasilinear first order PDEs given by

where U is a 2×1 column vector and A(U)is the 2×2 Jacobian matrix.Here,the independent variables x and t denote the space and time coordinates,respectively.As system(2.1)is assumed to be strictly hyperbolic,it admits two real and distinct eigenvalues λ1and λ2.Let us denote the corresponding left and right eigenvectors by l(1),l(2)and r(1),r(2),respectively.Without loss of generality,we normalize the eigenvectors which implies that

where δijis the Kronecker delta function.Let Π1(U)and Π2(U)denote the Riemann invariants corresponding to system(2.1).We use the standard procedure[11]to obtain the Riemann invariants Π1(U)and Π2(U)for system(2.1),which are

One can verify that the system of PDEs can be expressed in the characteristic form in terms of the Riemann invariants as follows:

By using the hodograph transformation,one can recast the independent variables x and t in terms of the Riemann variables Π1and Π2,which are given by

provided that the Jacobian of transformation isDifferentiating both sides of(2.5)with respect to x and t and using the characteristic form(2.4),system(2.1)can be reduced to the following pair of linear PDEs:

Eliminating x from(2.6),we obtain the following second order linear differential equation:

It can be noted that the linear second order PDE(2.7)and the quasilinear system of PDEs(2.1)are equivalent till t

Now,we consider the quasilinear PDEs(2.1)along with the differential constraints[3,31]as

where q is an unknown function which is to be determined.The consistency condition of(2.1)and(2.8)can be expressed as the following system of PDEs[25]:

Furthermore,using(2.3)and(2.6),the constraint equation(2.10)is reduced to

Finally,conditions(2.9)is reduced to

Now,we are in a position to state that we can characterize a class of 2×2 system of quasilinear hyperbolic system of PDEs,such that the solution of the hodograph equations can be expressed in a closed form.Specifically,we state the following theorem(see[1]):

Theorem 2.1Let us consider 2×2 quasilinear system of strictly hyperbolic PDEs

with eigenvalues λ1and λ2of the Jacobian matrix A(U)satisfying the following condition:

Then,the solution of the linear hodograph system of PDEs(2.6)can be expressed as

with Γ(Π1)and Z(Π2)being arbitrary functions.

3 Solution of Hodograph System

The quasilinear form of the system of PDEs(1.1)is given by

where the primitive variable U and the Jacobian matrix A(U)are,respectively,

where tr denotes the transposition.One can easily compute that the Riemann invariants corresponding to the system of PDEs(3.1)are

where Z(Π2)can be obtained from the initial data.In the next section,we use this solution of the hodograph system to study the simple wave interactions by considering the Cauchy problem for the system of PDEs(3.1).

4 The Cauchy Problem and Simple Wave Interactions

In this section,we consider the Cauchy problem to study the interaction of simple waves.Usually,interaction of simple waves is investigated by considering the regions in which the Riemann invariants Π1and Π2are constants[15].However,one can study such type of wave interactions and exhaustive analysis of the problem[16,17]whenever the explicit form of the characteristic wavelets are available.Here,we use the procedure outlined in the preceding sections to discuss the interaction of simple waves.As λ1>λ2,the wave associated to λ1interacts with simple wave associated with λ2.Let us denote the characteristic wavelets corresponding to the characteristic curvesby a(x,t)=constant and b(x,t)=constant,respectively,and satisfying the following equations:

We know that the Riemann variables Π1(a)and Π2(b)are invariant along the associated characteristic curves.Hence,in terms of characteristic variables,solution(3.6)can be represented as follows:

where we use the notation f(a,b)=fΠ1(a),Π2(b)for any function f and

where the point x0is an arbitrary point.In order to study the Cauchy problem,we normalize the characteristic parameters a and b as

where the initial data for Π1and Π2can be represented by

Now,using(2.10)and the value of q given in(3.5),one can verify that the initial data Π1(x)and Π2(x)must satisfy the following condition:

If the function Γ(Π1)is specified,then using relation(3.3)in(4.10),one can easily verify that the governing system of PDEs(1.1)must be endowed with the following initial data:

Now,we describe interaction of two simple waves in the(x,t)-plane,where the two simple waves travel along the characteristic curves associate with different families.Here,we use the analysis developed in[1]to describe the wave interactions in(x,t)-plane.Accordingly,we assume that at t=0,the travelling pulseoccupies the region xl≤x≤?xfwhile thetravelling pulse occupies the region xf≤x≤xr(see Figure 1).Hence,at t=0,we have

Figure 1 Qualitative behavior of interaction between two simple waves in the(x,t)-plane travelling along different characteristic curves

where θ1(xl)=θ1(?xf)=Π10and θ2(xf)=θ2(xr)=Π20.

In the(x,t)-plane,we can identify six distinct regions,which are named as I,II,III,IV,V and VI.In these six regions,we compute the values of characteristic wavelets x?λ1t and x?λ2t explicitly to study the behavior of emerging simple waves.In fact,the results can be obtained from(4.2)as follows:

Region I(Constant state):

Therefore,in view of(4.15)and(4.22)it can be observed that the simple wave travelling alongtraverses the region II and interacts with thetraveling pulse in region IV which emerges into the region V as a simple wave.Moreover,it can be easily seen that the simple wave pulse in region V is identical with the simple wave in region II with the initial data given by

where Iris given by

Similarly,the simple wave travelling alongtraverses region III and interacts withtravelling pulse in region IV.After the interaction,it emerges as a simple wave with the initial data given by

where Ilis given by

Hence,after the interaction simple wave travelling alongC(λ1)emerges in region V and the simple wave travelling alongC(λ2)emerges in region VI as a simple wave.Here,the simple wave travelling along theC(λ1)curve is a contact discontinuity which corresponds to the situation in which each product in the production system follows the leading product at the same speed(as Π2=u=constant).But,the simple wave travelling along theC(λ2)characteristic curve describes more realistic situation(u(1+ρ)=constant)in which the product speed depends on the product density.

Now,we illustrate the interaction of two simple waves along with their behavior with the help of a numerical example.The numerical solution of(3.1)is considered to observe the behavior of simple waves and interaction process of two emerging simple waves in detail.Here,we choose the initial data in such a way that it simulate two simple waves travelling along different families of characteristic curves.The simulation of two regular pulses travelling along the positive x-direction(λ1>λ2>0)is shown in Figure 2.We observed the solution profile of density at different times to illustrate the detailed interaction process.It is observed that due to exceptional(linearly degenerate)characteristic field,the corresponding solution profile after the interaction remains same,but the wave pulse corresponding to the genuinely nonlinear characteristic field is changed after the interaction.Figure 2 clearly illustrates the situation,where it is observed that one peak of the density profile remains same but the other decreases with time and moves as a simple wave after the interaction.However,any kind of discontinuity in the solution profile is not observed.

Figure 2 Simulation of two interacting simple waves travelling in the positive x-direction.The initial data Π1(x)=180+20sech(0.09(x+90))and Π2(x)=20?4sech(0.06(x?90))are taken to obtain the numerical solutions of(3.1)

5 Interaction of Elementary Waves

In this section,we discuss the solution of Riemann problem for the system of PDEs(1.1)and study all possible interactions among the elementary waves.We have already seen that 1-characteristic field is linearly degenerate and 2-characteristic field is genuinely nonlinear.Therefore,the solution corresponding to 1-family is always a contact discontinuity and the solution corresponding to 2-family is either a shock wave or a rarefaction wave.Now,we express these three elementary waves,namely,contact discontinuity,shock wave and rarefaction wave,as follows:

5.1 Contact discontinuity

Let us denote the left-hand and right-hand states of an elementary wave by Ul=(ρl,ul)and U=(ρ,u),respectively.Since the characteristics on either side of the contact discontinuity are parallel,so we have

which implies that ul=τ=u,where τ denotes the speed of contact discontinuity.Therefore,we can represent the curve of contact discontinuity J as follows:

5.2 Shock wave

By using the properties of Rankine-Hugoniot jump conditions and Lax entropy conditions across the shock wave,we obtain the shock curve S which is given by

5.3 Rarefaction wave

Because the Riemann invariant Π1(U)=(1+ρ)u is constant across the rarefaction wave and characteristic speed increases from left to right,the rarefaction wave curve R can be expressed as

For a fixed left-hand state Ul,we draw all possible states which are connected to the right either by a contact discontinuity or a shock wave or a rarefaction wave(see Figure 3).If we draw these elementary waves through a given Ul,it divides the quarter phase plane into three regions which are given as follows:

Figure 3 Elementary waves in(u,ρ)plane

where u?=ul(ρl+1).Now we consider the Riemann problem and we use the solution of Riemann problem and properties of elementary waves to study the wave interactions.Let us consider the Riemann initial data

where Uland Urare constant vectors.Using the properties of elementary waves,one can easily prove the following theorem which states that initial data must satisfy certain condition to avoid the vacuum state.Specifically,we state the following theorem:

Theorem 5.1If ur≥(1+ρl)ul,then vacuum occurs.

In our problem,we choose the initial data in such a way that the solution does not consists of vacuum.Therefore,for a given Ulthe solution of Riemann problem(1.1)and(5.5)can be constructed globally depending upon the choice of Ur,where it lies in the quarter phase plane(see[32]for details).Therefore,if Ur∈A,the solution of the Riemann problem consists of contact discontinuity followed by shock wave.However,its solution consists of contact discontinuity followed by rarefaction wave if Ur∈B and vacuum occurs in the solution if Ur∈C.Now,we consider system(1.1)with initial data

where x1,x2∈R.Here,we choose the states Umand Urin terms of Ul.From(5.6),we have two local Riemann problems at x1and x2,respectively.It can be noted that an elementary wave associated with first Riemann problem may interact with an elementary wave associated with second Riemann problem.Moreover,if they interact,then a new Riemann problem is formed at the time of interaction.Here,we use the notation J1R2?J1S2to denote that a 1-contact discontinuity,J1,of first Riemann problem interacts with 2-rarefaction wave,R2,of second Riemann problem,which leads to a new Riemann problem whose solution consists of 1-contact discontinuity,J1,and 2-shock wave,S2,(that is,J1S2).

5.4 Interaction between contact discontinuity and rarefaction wave

In this case,for a given initial data Ul,we choose Umand Urin such a way that Ulis connected to Umby a contact discontinuity J1of the first Riemann problem and Umis connected to Urby a rarefaction wave R2of the second Riemann problem.Therefore,we have that um=ul,ur(1+ρr)=um(1+ρm),ρr≤ρmand ur>um.Since the speed τ1=umof contact discontinuity J1of the first Riemann problem is greater than the speed of tail of the rarefaction wave R2of the second Riemann problem,so J1overtakes R2.Since ur>um=ul,which implies that the right-hand state Ur∈B.Now,we show that the curve R2(Um)never intersects with R2(Ul),that is,R2(Um)always lies either above or below the curve R2(Ul).One can verify that rarefaction wave curve satisfies the following property:

which shows that the rarefaction wave curve is monotonically decreasing and convex in the quarter phase plane.Furthermore,ul=umimplies that either ρm>ρlor ρm<ρl.In order to prove that R2(Um)lies above or below the curve R2(Ul),it is enough to prove thatrespectively.As um=ul,we have

Therefore,R2(Um)lies above the curve R2(Ul)when ρm>ρl,while R2(Um)lies below the curve R2(Ul)when ρm<ρl.Hence,in either of the cases R2(Um)never intersects with R2(Ul)and Ur∈B.Hence,after the interaction Uland Urcan be connected by a contact discontinuity J1followed by a rarefaction wave R2,that is,J1R2?J1R2(see Figure 4).

Figure 4 Collision of J1 R2

5.5 Interaction between contact discontinuity and shock wave

For a given initial data Ul,we choose Umand Ursuch that Ulis connected to Umby a contact discontinuity J1of the first Riemann problem and Umis connected to Urby a shock wave S2of the second Riemann problem.Therefore,um=uland ur(1+ρr)=um(1+ρm)satisfying ρr>ρm,ur

Therefore,S2(Um)lies above the curve S2(Ul)when ρm>ρl,while S2(Um)lies below the curve S2(Ul)when ρm<ρl.Hence,in either case,S2(Um)never intersects with S2(Ul)and Ur∈A.

Therefore,after the interaction Uland Urcan be connected by a contact discontinuity J1and a shock wave S2,that is,J1S2?J1S2(see Figure 5).

Figure 5 Collision of J1 S2

5.6 Interaction between shocks of same family

Here,we choose the states Umand Urin such a way that the given left-hand state Ulis connected to Umby a shock wave S2of the first Riemann problem and Umis connected to Urby a shock wave S2of the second Riemann problem.So we have um(1+ρm)=ul(1+ρl)and ur(1+ρr)=um(1+ρm)satisfying ρr>ρm>ρl,ur

Since Ur∈A,after the interaction the solution of the Riemann problem consists of contact discontinuity followed by a shock wave.Therefore,the result of interaction is given by S2S2?J1S2(see Figure 6).

Figure 6 S2 overtakes S2

Figure 7 R2 overtakes S2

5.7 Interaction between rarefaction and shock waves of same family

In this case,for a given initial data Ul,we choose Umand Ursuch that Ulis connected to Umby a rarefaction wave R2of the first Riemann problem and Umis connected to Urby a shock wave S2of the second Riemann problem.Therefore,um(1+ρm)=ul(1+ρl),ur(1+ρr)=um(1+ρm),um≥uland ur

where u

Now,we have the following three possibilities:

? When ur

? When ur=ul:In this case,Urlies on J1and therefore the interaction result is R2S2?J1.

? When ur>ul:Here,we have Ur∈B and the interaction result can be represented as R2S2?J1R2,which shows that 2-shock is weaker compare to 2-rarefaction wave.

6 Conclusions

We have used the differential constraints technique and hodograph transformation to obtain exact solution in terms of Riemann invariants for the governing system of PDEs which describe macroscopic production model.We discussed the simple wave interactions using the solution obtained from the hodograph differential equations.It is observed that after the interaction of simple waves the solution profile remains similar(soliton-like);that is,it propagates as a simple wave and it does not produces any kind of discontinuity in the solution profile after the interaction.Furthermore,we discussed all possible elementary wave interactions of same family as well as from different families by considering the solution of Riemann problem.

AcknowledgementsThe fist author(Minhajul)is highly thankful to Ministry of Human Resource Development,Government of India,for the institute fellowship(grant no.IIT/ACAD/PGS&R/F.II/2/14MA90J08)from IIT Kharagpur.The second author(TRS)would like to thank SERB,DST,India(Ref.No.MTR/2019/001210)for its financial support through MATRICS grant.We thank Prof.N.Manganaro and Prof.C.Curro(University of Messina)for their valuable suggestions and comments.