First-Order Partial Differential Equations, Vol. 2

By Hyun-Ku Rhee, Rutherford Aris and Neal R. Amundson

Second volume of a highly regarded two-volume set, fully usable on its own, examines physical systems that can usefully be modeled by equations of the first order. Examples are drawn from a wide range of scientific and engineering disciplines. The book begins with a consideration of pairs of quasilinear hyperbolic equations of the first order and goes on to explore multicomponent chromatography, complications of counter-current moving-bed adsorbers, the adiabatic adsorption column, and chemical reaction in countercurrent reactors. Exercises appear at the end of most sections. Accessible to anyone with a thorough grounding in undergraduate mathematics — ideally including volume 1 of this set. 1989 edition. 198 black-and-white illustrations. Author and subject indices.

• Language: English
• Publisher: Dover Publications
• Release date: May 17, 2013
• ISBN: 9780486150369

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Copyright © 1989 by Hyun-Ku Rhee, Rutherford Aris, and Neal R. Amundson

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Library of Congress Cataloging-in-Publication Data

Rhee, Hyun-Ku, 1939-

Theory and application of hyperbolic systems of quasilinear equations / Hyun-Ku Rhee, Rutherford Aris, Neal R. Amundson.

p. cm.—(First-order partial differential equations ; v. 2)

Originally published: Englewood Cliffs, N.J. : Prentice-Hall, c1989, in series: Prentice-Hall international series in the physical and chemical engineering sciences.

Includes index.

9780486150369

1. Differential equations, Hyperbolic. 2. Quasilinearization. I. Aris, Rutherford. II. Amundson, Neal Russell, 1916–III. Title.

QA374 .R47 2001 vol. 2

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2001041183

Manufactured in the United States of America

Dover Publications, Inc., 31 East 2nd Street, Mineola, N.Y. 11501

To

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for their forbearance

Table of Contents

DOVER BOOKS ON MATHEMATICS

Title Page

Copyright Page

Dedication

Preface

1 - Pairs of Quasilinear Hyperbolic Equations of First Order

2 - Two-Solute Chromatography with the Langmuir Isotherm

3 - Hyperbolic Systems of First-Order Quasilinear Equations and Multicomponent Chromatography

4 - Wave Interactions in Multicomponent Chromatography

5 - Multicomponent Adsorption in Continuous Countercurrent Moving-Bed Adsorber

6 - More on Hyperbolic Systems of Quasilinear Equations and Analysis of Adiabatic Adsorption Columns

7 - Chemical Reaction in a Countercurrent Reactor

Author Index

Subject Index

Preface

This book is the continuation of First-Order Partial Differential Equations Volume I: Theory and Application of Single Equations, published in 1986. As before, there is a variety of physical systems, which are dominated by convective and exchange processes that can usefully be modelled by equations of the first order. The development of models demanding several equations was considered in the first chapter of Volume I and need not be repeated here. Many of the physical situations, such as adsorption, chromatography, chemical kinetics, sedimentation, ultracentrifugation, or oil recovery, call for extension to many components. As soon as there is any interaction between these components, the systems of equations become much more than simply the juxtaposition of single equations.

It is with the new features, introduced by the linking of these sets of partial differential equations, that we are concerned in the second volume. Simple waves and discontinuities will again appear and their interaction will be a major topic. Adsorption and desorption phenomena in flow through a fixed or moving bed again provide a most important example and the Riemann problem, with its piecewise constant boundary values, arises so naturally that it provides a suitable workhorse throughout this book. With the Langmuir isotherm the full range of nonlinear phenomena comes into play, yet there are certain simplifying characteristics that allow the geometry of the situation to be brought out rather clearly. At the same time we have not shunned more pedestrian situations where, to make any progress, one has to resort to numerical calculation at an early stage.

Briefly, Chapter 1 considers pairs of quasi-linear hyperbolic equations of the first order leading off with the example of the chromatography of two solutes. Through a discussion of reducible equations and simple waves, the characteristic initial value problem for two-solute chromatography is discussed and notions of compression waves and the formation of shocks are formulated. In a later section, this is also applied to polymer flooding. Riemann invariants and their applications and the development of singularities and weak solutions are considered. Chapter 2 continues the discussion of two-solute chromatography with Langmuir isotherms and introduces displacement chromatography and shock layers. The problems of two equations with two independent variables allow a very geometrical presentation, which is good preparation for the higher dimensional case. This is considered in Chapter 3, where multicomponent chromatography is discussed. The problem of wave interactions in this context is continued in Chapter 4. In Chapter 5, the additional complications of counter current moving-bed adsorbers are considered. Here, the solid moves in the opposite direction to the fluid phase and continuous separations can be achieved. The adiabatic adsorption column is the subject of Chapter 6, while Chapter 7 introduces the notion of chemical reaction in countercurrent reactors. In general, as has been shown in the previous volume, this means that the characteristics are no longer straight and a new range of interactions comes into play. In particular, it is found that it is possible for a discontinuity to cross the thermodynamic equilibrium line so that complete conversion can be achieved even for a reversible reaction.

The level, at which we have aimed, is again that of graduate study. The text should be accessible to anyone with a thorough grounding in undergraduate mathematics. The earlier volume on single equations is, of course, a good introductory preparation. We hope that it will be as useful to the practicing engineer as it is to the student and we have endeavored, at all points, to relate the mathematical developments to the physical background.

A word is essential about what parts will be useful as a teaching text. The materials in Chapter 1, Sec. 3.1 through Sec. 3.4, and Sec. 6.1 through Sec. 6.5 are general and may be given in any applied mathematics course. The remainder may serve as a source of applications and illustrations, which may prove helpful for the understanding of the subject. For engineering students, Secs. 1.10, 1.11, and 6.5 may be trying and may be omitted.

Those who are mainly interested in the theory of multicomponent chromatography may begin with Chapter 3. If, however, one aims for application to other systems as well, it is important to start from Chapter 1. For a one-semester course for graduate students in engineering, we suggest Sec. 1.1 through Sec. 1.9, Sec. 2.1 through Sec. 2.8, Sec. 3.1 through Sec. 3.9, Secs. 3.11 and 3.12, Sec. 4.1 through Sec. 4.5, Sec. 6.1 through Sec. 6.4, and Sec. 6.6 through Sec. 6.8.

Hyun-Ku Rhee

Rutherford Aris

Neal R. Amundson

1

Pairs of Quasilinear Hyperbolic Equations of First Order

In this chapter the mathematical theory for pairs of quasilinear equations is developed in a rather general fashion. Although we shall be dealing with some physical examples for illustration, extensive application is to be presented in Chapter 2. A number of important and interconnected ideas are introduced here, and since these are sufficiently complex subjects, we shall not avoid repetition of certain ideas but rather, attempt to elucidate the interconnected feature of these ideas in detail. The reader who finds the discussion here insufficient may go to the original sources, to which references are given at the end of the chapter.

Equilibrium chromatography of two solutes is employed as the representative example, for it gives rise to equations of general form. Other examples of conservation equations that are considered are those of polymer flooding, compressible fluid flow, and shallow-water wave theory.

Section 1.1 deals with formulation of the equations for two-solute chromatography and illustrates how a pair of first-order equations is realized in practice. It also discusses how to put the equations in a simplified form. In Sec. 1.2 we consider the general form of two quasilinear equations and introduce the method of characteristics. This then transforms the pair of differential equations into a system of four ordinary differential equations, which are the characteristic differential equations. The solution scheme is also discussed. Section 1.3 is concerned with reducible systems, special in a sense but typical of physical examples, and the hodograph transformation which is the key concept of the solution scheme is introduced. The notion of simple waves is discussed in detail since it assumes an important role in the development of the theory.

In the following three sections, equations of two-solute chromatography are treated in the light of the theoretical development of Secs. 1.2 and 1.3. In Sec. 1.4 the characteristic directions in both the hodograph plane and the (x, τ)-plane are formulated and the directional derivatives along Γ characteristics are introduced. We then proceed to treat the characteristic initial value problem as well as the Riemann problem in Sec. 1.5, showing the structure of the solution in detail and emphasizing the possibility of the necessary existence of discontinuities. In Section 1.6 we are concerned with compression waves and introduce the notion of genuine nonlinearity. The point of shock inception is determined by tracing the envelope of straight characteristics.

In Sec. 1.7 we consider discontinuities in solutions, for which the conservation law is formulated to give the compatibility relation and the jump condition. Important features are discussed, including the entropy condition which is required to ensure the uniqueness of the solution. Based on their different properties, discontinuities are classified into shocks, semishocks, and contact discontinuities. The nature of each discontinuity is fully explored. The equations of polymer flooding are treated in Sec. 1.8 to illustrate how discontinuities come into the solution in a physical system. A numerical example for the Riemann problem is analyzed in detail.

In Sec. 1.9 there is a discussion of the Riemann invariants, whose existence is special to systems of two equations. Their application is illustrated with physical examples. In Sec. 1.10 Riemann invariants are used to show under general circumstances that singularities can develop in solutions after finite time. Then weak solutions are defined, the jump condition is formulated, and the entropy condition is discussed. In particular, Lax’s treatment of conservation equations with a convex extension is presented in some detail.

Finally, Sec. 1.11 is concerned with the existence and uniqueness proof for weak solutions as well as their structure and asymptotic behavior. A large number of papers have been published on these subjects and it seems just reasonable to make a survey of these developments rather than going into the detail of specific papers.

1.1 Equations for the Chromatography of Two Solutes

We start with a practical realization of the type of equation we wish to discuss in this chapter, the equilibrium chromatography of two interacting solutes. When two solutes, A1 and A2, are present in concentrations c1 and c2 in the fluid phase, their equilibrium concentrations n1 and n2 in the solid phase will be functions of both concentrations, i.e.,

(1.1.1)

the nature of which is essentially nonlinear because the mutual influence between the solutes A1 and A2 is taken into account. Just what form these functions may take is seen by considering the case of adsorption according to the Langmuir isotherm. If the number of sites at which adsorption can take place is limited, the total adsorbed concentration (n1 + n2) cannot exceed an upper bound, say N. In fact, (N–n1–n2) will be proportional to the number of vacant sites and the rate at which A1 is adsorbed might be expected to be proportional to the product of its concentration above the surface, c1, and the availability of adsorption sites on the surface, N–n1–n2. The rate of adsorption of A1 might be written as

But at equilibrium this is exactly balanced by the rate of desorption, which we may take to be kd1 n1, proportional to the adsorbed concentration, n1. Thus

or

(1.1.2)

where

(1.1.3)

Compare the rate expressions to those in Eq. (1.3.9) of Vol. I. Similarly, the equilibrium of A2 gives

(1.1.4)

where

(1.1.5)

Now Eqs. (1.1.2) and (1.1.4) can be solved to give

(1.1.6)

If K2 > K1, it is reasonable to speak of the solute A2 as being more strongly adsorbed than A1. Figure 1.1 shows two typical surfaces of n1 and n2 as functions of c1 and c2; the surface corresponding to the more strongly adsorbed solute A2 lies above that of A1 over most of the plane. Clearly, we can always take K2 > K1 by correctly numbering the two solutes; the case K1 = K2 is too special to merit attention. Another way of representing the two surfaces is by contours of n1/N and n2/N in the plane of K1c1 and K2c2. These are the families of straight lines shown in Fig. 1.2. The family of contours for n1/N radiates from the point K1c2 = 0, K2c2 =–1; and any line intersects the K1c1 axis at K1c1 = n1/(N–n1). Similarly, the contours for n2/N all emanate from the point K1c1 =–1, K2c2 = 0 and have intercepts n2/(N–n2) on the K1c2 axis.

Figure 1.1

We observe that

(1.1.7)

Figure 1.2

The general features of these expressions are shared by all isotherms representing the adsorption of two solutes that compete for the same sites. Thus we expect the derivatives to tend to zero as either c1 or c2 tends to infinity and their signs to be

(1.1.8)

We also expect to have the conditions

(1.1.9)

which imply that the variation of c1 (or c2) has a stronger influence on the adsorption of A1 (or A2) than on the adsorption of A2 (or A1). The Jacobian of n1 and n2 with respect to c1 and c2 will generally be positive and in the case of the Langmuir isotherm

(1.1.10)

In the chromatographic column with voidage ε and V as the interstitial velocity of the fluid phase we have the usual mass balance equations for Ai, i = 1, 2, in terms of ci(z, t) and ni(z, t), the two concentrations at position z and time t. These have been derived in Secs. 1.2 and 1.3 of Vol. 1. Thus

(1.1.11)

and

If we define the dimensionless parameter

(1.1.12)

and introduce the equilibrium relationships in Eq. (1.1.1), we obtain a pair of first-order equations for c1(z, t) and c2(z, t),

(1.1.13)

The natural initial data on these equations would be the specification of c1 and c2 at the inlet z = 0 and on the column at t = 0. These equations are entirely typical of the general pair of equations we shall study in Sec. 1.2, for under suitable conditions the coefficients could be functions of the independent as well as the dependent variables.

To simplify the equations, however, we introduce the equilibrium column isotherms, defined as

(1.1.14)

[see Eq. (7.1.5) of Vol. 1] and, as usual, make the independent variables dimensionless; i.e.,

(1.1.15)

where Z denotes the characteristic length of the system. Thus we can rewrite Eq. (1.1.11) as

(1.1.16)

or put it in the form

(1.1.17)

We shall return to this pair of equations in Sec. 1.4.

In the case of Langmuir isotherms we notice that K1c1 and K2c2 generally keep together so that, setting

(1.1.18)

and

(1.1.19)

we have the equations

or in the form

(1.1.20)

This will be the form of the equations that we will use to illustrate the general case in the following sections.

Although we shall not pursue it, another way of simplifying equations may be of particular interest. This is accomplished by applying the transformation of independent variables which is essentially the same as the one we treated in Eq. (0.2.8) of Vol. 1. Let

(1.1.21)

then

or

(1.1.22)

Thus Eq. (1.1.11) becomes

(1.1.23)

For the case of the Langmuir isotherm we shall again use u and v from Eq. (1.1.18), so that the equations are

This suggests that we put

(1.1.24)

and then

(1.1.25)

This pair of equations is equivalent to Eqs. (1.1.20). Although the dependent variables are the same, the independent variables are defined in a different manner. Both the independent variables here have the dimensions of time, but because of their origin, we can think of x as space-like and y as time-like. It involves the fewest possible number of parameters, the only one being κ, 0 < κ < 1.

Concerning the initial and boundary conditions, the line x = 0 clearly represents the inlet of the feedstream entering the column at z = 0 and thus

(1.1.26)

the superscript f standing for feed. Again the situation on the line y = 0 gives the condition at z = Vt. Now t = z/V is the instant at which an element of the carrier fluid entering the column at t = 0 first reaches the point z. Hence the feed conditions cannot have any influence until y = 0 and it has been commonly supposed in the theory of chromatography that u and v can be specified at y = 0, say

(1.1.27)

This argument implies that the system is passive until the adsorption front arrives and is quite valid for an initially constant state, but it would be more accurate to consider initial conditions specified on the line t = 0, i.e.,

(1.1.28)

In this case we would have

(1.1.29)

in which the superscript i stands for initial. We also observe that if we wish to recover the distribution of solutes in the column from the solution u(x, y), v(x, y) at any instant t, we must take sections by lines

(1.1.30)

The geometry of variables is shown in Fig. 1.3.

Exercise

1.1.1

When a single solute adsorbs in an adiabatic adsorption column, we have the usual mass balance equation (1.4.1) of Vol. 1 with i = 1 and the energy balance equation (1.4.9) of Vol. 1 with m = 1, where c2 and n2 are defined by Eqs. (1.4.8a) and (1.4.8b) of Vol. 1, respectively. Consider the Langmuir isotherm for a single solute, i.e., Eq. (1.2.7) of Vol. 1 with c = c1 and n = n1, and note that the parameter K = K1 is expressed as a function of the temperature T by Eq. (1.4.10) of Vol. 1 with i = 1. Rewrite the pair of conservation equations in terms of c1 and T, and arrange them in the form of Eq. (1.1.20) and also in the form of Eq. (1.1.25).

Figure 1.3

1.2 Hyperbolic Systems of Two First-Order Equations

We shall consider the general quasilinear system of first-order equations for two dependent variables, u and v, with two independent variables, x and y; i.e.,

(1.2.1)

where A1, A2, B1, . . . , E1, and E2 are given functions of x, y, u, and υ and continuous with as many continuous derivatives as may be required. We shall assume that nowhere in the domain do A1/A2 = B1/B2 = C1/C2 = D1/D2, which asserts that the two equations are independent. These equations are quasilinear since the derivatives all enter linearly; they would be called strictly linear if the coefficients plus E1 and E2 were all functions of x and y only, linear if dependence on u and υ only appeared in E1 and E2 in a linear fashion, and semilinear if dependence on u and υ only appeared in E1 and E2 in some nonlinear manner. Frequently, the equations are called simply linear if the coefficients A1, A2, . . . , D2 are all dependent on x and y only.

The equations above are homogeneous if E1 = E2 = 0 and for reasons that will appear later are called reducible if they are homogeneous and the coefficients A1, . . . , D2 are functions of u and v only. Comparison of Eq. (1.2.1) and Eq. (1.1.17) or (1.1.20) shows that we shall be dealing with a reducible pair of equations in the equilibrium theory of two-solute chromatography.

There are two ways of working toward ordinary differential equations such as we have seen to be characteristic of single equations, and these are extensions of two of the features of characteristics that we discussed previously. The first way is to look for directional derivatives that will simplify the equations; the second is to ask under what conditions the initial data would fail to specify the solution properly. We shall follow each way in turn.

The combination A1 ∂u/∂x + B1 ∂u/∂y represents the derivative of u in a direction such that dy/dx = B1/A1 (see Sec. 0.9 of Vol. 1). Similarly, C1 ∂v/∂x + D1 ∂v/∂y is the derivative of v in the direction dy/dx = D1/C1. Unless B1C1 happens to equal A1D1, these directions will not be the same, but we may ask whether it is possible to 2 = 0, in which these derivatives are taken in the same direction. Let us thus consider the linear combination

(1.2.2)

The derivatives of both u and v = 0 will be in the same direction dy/dx if

(1.2.3)

Suppose that at a point (x, y, u, υ) on the solution surface, λ1 and λ2 are chosen to satisfy these equations; then

(1.2.4)

But if these equations are to give nontrivial values of λ1 and λ2, the determinant of the coefficients must vanish, i.e.,

(1.2.5)

This may be written as a quadratic,

(1.2.6)

where

(1.2.7)

If the discriminant

(1.2.8)

this quadratic has two real roots and there will be two directions in which it is true that the directional derivatives of u and υ 2 = 0 are aligned. Such systems are called hyperbolic, and it is with these that we will be concerned. If b²–ac = 0, the system is said to be parabolic, and if b²–ac < 0, it is elliptic—an obvious classification that we will not pursue further.

–. Since they are real, they will both be positive if a, b, and c are all of the same sign, both negative if b is of the opposite sign to a and c, and of contrary sign if the signs of a and c are different. If we imagine for a moment that the solution surfaces u(x, y), v(x, y) are known, then Eq. (1.2.6) could be solved at every point and two families of curves C+ and C– could be drawn in the (x, y)-plane, respectively, satisfying at each point

(1.2.9)

These two directions will be called the characteristic directions and the two families of curves the net of characteristic curves or simply characteristics, C+ and C– in the (x, y)-plane, belonging to the solution u(x, y), υ(x, – would be functions only of x and y, and the family of solutions of Eqs. (1.2.9) would form a net of characteristic ground curves.

The two families of characteristics C+ and C– may be represented in the form β(x, y) = constant and α(x, y) = constant, respectively, and form a curvilinear coordinate net. We can then introduce new parameters α and β instead of x and y in such a way that β is constant along the C+ characteristics and α is constant along the C– characteristics. In other words, α and β are the parameters running along the C+ and C– characteristics, respectively, as shown in Fig. 1.4 and we shall call them characteristic parameters. To see how these parameters can be specified, let us consider any curve I, x = x(ξ) and y = y(ξ), that is nowhere tangent to a characteristic, i.e.,

(1.2.10)

on the curve I. Through any two points ξ = α and ξ = β that are located sufficiently close on I, we draw the C– and C+ characteristics, respectively, up to the point of intersection P(x, y) as illustrated in Fig. 1.5. The pair (α, β) is then the set of characteristic parameters for the point P(x, y). Obviously, we can use any monotone functions α′ = A(α) and β′ = B(β) as characteristic parameters since such a transformation leaves Eq. (1.2.9) unchanged.

Figure 1.4

Figure 1.5

Now we can write from Eq. (1.2.9)

(1.2.11)

where ∂/∂α or ∂/∂β denotes differentiation with respect to α or β along a C+ or C– characteristic. In general, we have

(1.2.12)

or

(1.2.13)

+ in the first equation of Eq. (1.2.4) and multiply this equation by ∂x/∂α. Then we have

(1.2.14)

Similarly, multiplying Eq. (1.2.2) by ∂x/∂α gives

+ to eliminate the terms in the B’s and D’s yields

Equation (1.2.12) can now be used to express the derivatives with respect to α; then

But this can be rearranged to give

(1.2.15)

If we had multiplied Eq. (1.2.2) by ∂y/∂α and eliminated the terms in A’s and C’s by using Eq. (1.2.3), we would have had the equation

(1.2.16)

which can be used as well instead of Eq. (1.2.15).

Consider now Eq. (1.2.15) along with Eq. (1.2.14) as a pair of equations to be solved for λ1 and λ2. Since these equations are homogeneous, their determinant must vanish if we are not to have trivial values for λ1 and λ2. Thus

or

(1.2.17)

where

(1.2.18)

– and the parameter β to obtain a similar equation,

(1.2.19)

where M– and N+. Then the four equations

(1.2.20)

are called the characteristic differential equations. They will be satisfied on the solution surfaces u(x, y), υ(x, y– are roots of the quadratic (1.2.6), and they allow us to learn a good deal about the way the solution can be developed from the initial data. We may think of them as four coupled equations for x, y, u, and υ as functions of two parameters α and β. When Eqs. (1.2.20) are solved to give x and y as functions of α and β, x = x(α, β), y = y(α, β), these functions may be inverted to give α and β in terms of x and y. Finally, when these are substituted in u(α, β) and v(α, β), one obtains u and υ as functions of x and y. In principle this can be done provided that

(1.2.21)

but this is ensured by the criterion for hyperbolicity.

Before seeing what these equations tell us about the structure of the solution, let us look at the initial data and ask when it seems to be suitably given. Suppose that u and υ are given along a curve I in the (x, y)-plane. If ξ is a parameter running along this curve, we may specify the curve and the initial data by giving four functions,

(1.2.22)

To move off this initial curve I it is essential to be able to calculate the partial derivatives of u and υ at any point ξ. These will satisfy the differential equations

(1.2.23)

in which A1, . . . , E2 have been made functions of ξ by substituting from Eqs. (1.2.22), and also the strip conditions [see Eq. (0.12.2) of Vol. 1].

(1.2.24)

Thus the four equations (1.2.23) and (1.2.24), as four simultaneous equations for ∂u/∂x, ∂u/∂y, ∂v/∂x, and ∂v/∂y, must be solvable. This will only be the case if

(1.2.25)

This implies that the partial derivatives of u and υ can be calculated if the initial curve I is nowhere characteristic in the sense defined above. Conversely, this is to say that we shall not be able to determine the partial derivatives uniquely along the curve I if it is characteristic, i.e., if Y‘/X’ –. We have observed this feature with the single equation in Sec. 2.5 of Vol. 1. Another way of putting it is to say that two different solutions can intersect only along a characteristic. Thus the derivatives across a characteristic can be discontinuous, or a sharp edge in a solution surface must be a characteristic. This also is true of a single equation (see Sec. 5.2 of Vol. 1).

On the other hand, if the initial curve I is characteristic, we can only get a nontrivial solution of Eqs. (1.2.23) and (1.2.24) when the matrix of coefficients and its augment,

are of the same rank. But by setting the determinant of the first three columns and the last equal to zero, we recover Eq. (1.2.17) for Y′/X′ +, ξ = α, and Eq. (1.2.19) for Y′/X′ = –, ξ = β. Thus if data are given on a characteristic curve, they must satisfy the full characteristic differential equations, which is to say that the solution must satisfy the full characteristic differential equations.

Consider next the noncharacteristic data given on the initial curve I shown in Fig. 1.6. We shall show how a step-by-step procedure will allow us to calculate u and υ given their values on I. Now α is a parameter running along the characteristics C+ and β along C—. Hence two points such as A and E will have the same value of β but different values of α, and points such as B and E the same α but different β’s. Let us assume that these points are close together and write the parametric coordinates of the three points as follows (see the diagram on the right of Fig. 1.6):

A is (α, β); B is (α + δα, β - δβ); E is (α + δα, β)

Since we know x, y, u, and υ at A and B+A –B. Then if ξA and ξB are the values of ξ on I at A and B and if we put xA = X(ξA), yA = Y(ξA), xB = X(ξB), and yB = Y(ξB), the coordinates of E lying at the intersection of the two characteristics must satisfy

(1.2.26)

where the suffixes A and B denote that the quantities must be evaluated at these points. Hence

(1.2.27)

Figure 1.6

Another way of getting these equations is to multiply the first two equations of Eq. (1.2.20) by δα and δβ, respectively, and to note that yE–yA = (∂y/∂α) δα, yE–yB = (∂y/∂β)δβ, etc. Similarly, multiplying the last two equations of Eq. (1.2.20) by δα and δβ, respectively, and noting that uE–uA = (∂/u/∂α)δα, uE–uB = (∂u/∂β)δβ, etc., we obtain

(1.2.28)

This provides a pair of equations to be solved for uE and υE. Thus, from values of u and υ on the initial curve I, the values of u and υ at a series of points such as E, F, G can be determined and the location of these points fixed. From this row of points we can go to a row of points such as H, K, L, and so on.

Now, although this is a crude scheme and we would have to indulge in some rather tricky limiting processes to justify it, it does show how the solution can in principle be built up. It shows, for example, that in determining the values of u and υ at a point P, all the data on the segment AR of the initial curve I between the C+ and C– characteristics through P will be needed (see also Fig. 1.7). In this sense the segment AR is called the domain of dependence of the point P. Similarly, the values of u and υ at the point A are needed in the calculation of u and υ at any point between the two characteristics emanating from A. The angular region PAQ is therefore called the range of influence of the point A. In other words, the range of influence of the point A consists of all points whose domains of dependence contain the point A. Compare these definitions here with those for the case of a single equation shown in Fig. 5.5 of Vol. 1.

The existence of such domains of dependence and ranges of influence, which is typical of hyperbolic systems, makes the solution relatively easy to obtain since it allows the construction of the solution by pieces. This further implies that the solution is not necessarily analytic and thus it may consist of analytically different portions in different regions of the (x, y)-plane.

Figure 1.7

Numerical techniques based on this way of discretizing the variables are indeed used and can be refined beyond the crude scheme outlined here. For example, once uE and υE +, L, M+, and N+ between the two points A and E+A, LA, M+A, and N+A–, L, M–, and N— can be determined between the two points B and E—B, LB, M—B, and N—B. Then Eqs. (1.2.27) and (1.2.28) are solved again to give new values of uE and υE, and this procedure can be repeated until a desired convergence is obtained. Various schemes are discussed in the book edited by Ralston and Wilf (1960) and listed at the end of this chapter. It is clear that linearity has the great advantage of making it possible to lay down a network of C+ and C– characteristics once and for all.

Before leaving this description we should note that data are sometimes given on a noncharacteristic curve that lies between the two characteristics, as in arc J in Fig. 1.8. Such a curve is called a time-like boundary, in contrast to I, which is space-like, and on J only one of the dependent variables can be specified. To see this let us take a triad of points A, B, and E just as before, but locate B at the intersection of the curves I and J. The values of x, y, u, and υ at E can certainly be determined as before, but now we should ask what can be said of them at D.

Figure 1.8

Since D lies on the C+ characteristic through A, we have

(1.2.29)

But if SB is the slope of the curve J at B, we also have

(1.2.30)

This determines the coordinates of D. On the other hand, since D lies on the C+ characteristic through A, we can adapt the first equation of Eq. (1.2.28) by replacing E by D; then

(1.2.31)

This relation between uD and υD means that not both of them can be specified independently on J. Of course, the argument applies only to the infinitesimal triangle ABD, but it can be used again on DFR, etc., and so shows that on a time-like curve only one of the dependent variables can be specified. This becomes even clearer if we consider the picture in the characteristic coordinate system as shown on the right-hand side of Fig. 1.8. The point D on J has the same value of β as the point A on I, which has already been specified, and thus we can specify only α there. This is true for every point on J so that we have one less degree of freedom along the curve J. This implies that we can specify only one of the dependent variables on the time-like boundary J.

Exercises

1.2.1.

Consider the spherical isentropic flow of a compressible fluid with spherical symmetry. If x is the distance from the origin, p the density of fluid, and u the radial velocity component, show that the governing equations at time t are

where c² is a given function of ρ. By considering a linear combination of these equations, determine the four characteristic differential equations.

1.2.2.

For steady irrotational isentropic flow in three dimensions with cylindrical symmetry, let x be the abscissa along the axis and y the distance from the axis, and show that the governing equations are

where u and υ are, respectively, the x and y components of the velocity and c² is a given function of u² + υ². Under what conditions is this system hyperbolic? When the system is hyperbolic, find the four characteristic differential equations.

1.3 Reducible Equations and Simple Waves

When the coefficients A1, A2, . . ., D2 in the homogeneous equations

(1.3.1)

are functions only of u and υ, the equations are said to be reducible. The reason for this is that their quasilinearity can be reduced to strict linearity by interchanging the roles of dependent and independent variables. This transformation, known as the hodograph transformation, can be applied whenever the Jacobian

(1.3.2)

does not vanish or become infinite. Clearly, if we can obtain equations for x = x (u, υ) and y = y(u, υ) as functions of u and υ, they will serve just as well to generate a solution. These functions will answer the question, "Where can a pair of values of u and υ be found?", just as the functions u = u(x, y) and υ = υ (x, y) answer the question, "What are the values of u and υ at (x, y)?"

If x = x(u, υ), y = y(u, υ) are the inverse functions of u(x, y), υ(x, y), then the equations

(1.3.3)

and

(1.3.4)

must be identities. Thus differentiating Eqs. (1.3.3) and (1.3.4) with respect to x keeping y constant, we have

These may be solved for ∂u/∂x and ∂υ/∂x to give

(1.3.5)

Similarly, partial differentiation with respect to y gives

and thus

(1.3.6)

Another way of putting this is to say that the Jacobian matrices

are inverses of each other. Now substituting from Eqs. (1.3.5) and (1.3.6) into Eq. (1.3.1) and dividing through by j, which we must presume does not vanish, we have

(1.3.7)

These equations are linear in the hodograph plane of u and υ, in which we can therefore lay down characteristic ground curves.

If we repeat the analysis at the beginning of Sec. 1.2 on the system (1.3.7), we see that

(1.3.8)

will be a characteristic direction if

(1.3.9)

where

(1.3.10)

It is a matter of algebra to check that the values of ζ given by Eq. (1.2.6), by the equation

(1.3.11)

But this is just what we should expect from the last two equations of the four characteristic differential equations (1.2.20). For in the reducible case N± ≡ 0, so that

(1.3.12)

or

(1.3.13)

where L, M+, and M– are given by Eq. (1.2.18).

Since all the coefficients are functions of only u and υ, the equations

(1.3.14)

and

(1.3.15)

are two separate ordinary differential equations and can be solved to give two one-parameter families of characteristic curves, Γ+ and Γ–, in the hodograph plane of u and υ. The advantage of reducibility is that these characteristics can be laid down once and for all and do not depend on the particular solution involved. These two families of Γ+ and Γ– may be represented in the form β(u, υ) = constant and α(u, υ) = constant, respectively, and form a curvilinear coordinate net in the hodograph plane. The curvilinear coordinates α and β of the point (u, υ) are then the characteristic parameters and there exists a one-to-one correspondence between the set (α, β) and the set (u, υ) since ξ+ and ζ– do not vanish at the same time.

It is now natural to introduce the characteristic coordinate system (α, β) instead of (u, υ) in such a way that β is constant along each of the Γ+ characteristics and α is constant along each of the Γ– characteristics. In other words, α and β are two new variables running along the Γ+ and Γ– characteristics, respectively. These characteristic parameters can be specified in the same way as we have discussed with the C-characteristics in the (x, y)-plane in the preceding section. Another choice for β and α in this case is the pair of the integration constants from Eqs. (1.3.14) and (1.3.15). More specifically, if we take a curve J given by u = g(υ), which nowhere has a characteristic direction, i.e., g′ ≠ ζ+ and g′ ≠ ζ–, the pair of integration constants can be expressed in terms of the information u = g(υ) given along the curve J. This pair certainly meets the requirement for the characteristic parameters. As before, any monotone functions α′ = A(α) and β′ = B(β) may be used as characteristic parameters since such a transformation leaves Eqs. (1.3.14) and (1.3.15) invariant and the characteristics Γ+ and Γ– unchanged. In Chapter 2 we shall see an example for which this transformation is conveniently used.

If u and υ are known as functions of α and β± in terms of α and β, i.e.,

(1.3.16)

and then solve the first two characteristic differential equations of Eq. (1.2.20),

(1.3.17)

(1.3.18)

for x and y as functions of α and β. This is just the sort of process we have outlined in Section 1.2, but it takes place in the hodograph plane of u and υ and the initial data provided take the form of the values of x and y for given u and υ.

In fact, the solutions of Eqs. (1.3.17) and (1.3.18) will give two families of