an averaging method for the weakly unstable shallow water equations in a flat inclined channel

Fluid Dynamics Research 38 (2006) 469–488

An averaging method for the weakly unstable shallow waterequations in a flat inclined channel

Richard Spindler∗

Department of Mathematics, Bemidji State University, Bemidji, MN 56601, USA

Received 21 May 2005; received in revised form 31 January 2006; accepted 27 February 2006

Communicated by E. Knobloch

Abstract

The asymptotic behavior of small disturbances as they evolve in space from boundary conditions in a flat inclinedchannel under weakly unstable conditions is determined using a method of averaging. The shallow water equationsare first transformed to a normal form using a variation of parameters technique, and then a new method of averagingis defined. The method of averaging developed differs from averaging methods for ordinary differential equations andother partial differential equations by averaging along the characteristics of the linearized equations rather than alongthe time variable. It is found that the solution is dominated by the evolution of the solution along one characteristic.Both asymptotic and numerical results for periodic disturbances are presented. In addition, connections to dynamicalsystems concepts are described.© 2006 The Japan Society of Fluid Mechanics and Elsevier B.V. All rights reserved.

PACS: 47.20. − k; 47.35.+i; 47.60.+i

MSC: 34E10

Keywords: Averaging; Instability; Shallow water

1. Introduction and linear instability

Normalization and asymptotic averaging techniques are developed to analyze the semi-infinite bound-ary value problem of shallow water waves in an inclined constant-slope channel, where the evolution is in

∗ Tel.: +1 802 865 5245.E-mail address: [email protected].

0169-5983/$30.00 © 2006 The Japan Society of Fluid Mechanics and Elsevier B.V. All rights reserved.doi:10.1016/j.fluiddyn.2006.02.006

mailto:[email protected]

470 R. Spindler / Fluid Dynamics Research 38 (2006) 469–488

the spatial variable. The system is assumed to be in a “weakly unstable” state, meaning the Froude numberF is slightly greater than the neutrally stable value (Yu and Kevorkian, 1992). The methods developedhere could also work in the initial value problem with minor modifications.

The averaging technique created in this study differs from previous averaging methods in severalrespects. First, and most importantly, the averaging is performed over two variables, namely the charac-teristics of the reduced (linear) problem. It appears that in previous averaging applications, the averagingwas performed on all equations along one variable, namely the “time” variable. An important exceptionshould be noted. Chikwendu and Kevorkian (1972) developed a method of multiple scales for a classof nonlinear wave equations which in the final result is an average along characteristics. Hence, there issome precedent for this type of averaging. Second, the normalization process (where the equation(s) aretransformed to Lagrange standard form, to be defined in Section 2) in many, though not all, averagingmethods are analogous to the variation of parameters used for first order linear ordinary differential equa-tions with exponential solutions. A variation of parameters idea is used in this study, but it is not exactlyanalogous to first order linear ordinary differential equations.

In the shallow water equations, shallow means that the depth is small compared to some characteristiclength, such as the wavelength. In addition, it is assumed the depth is small compared to the breadth ofthe conduit. Using dimensionless variables, the shallow water equations are then

ht + uhx + hux = 0, (1)

ut + uux + hx = 1 − u2

hF 2 . (2)

See Kevorkian (1990, Section 5.1.1) or Spindler (2005, Chapter 1). Here h is the height of the free surfacenormal to the channel bottom and is normalized by the uniform height. Also, u is the flow speed parallel tothe channel bottom averaged over the cross-section of the channel, and is normalized by the uniform flow.See Fig. 1. (The uniform solutions are the constant solutions of the dimensional shallow water equationsessentially corresponding to equilibrium, smooth, stratified flow.) Note that this is an inviscid flow andyet has a quadratic drag law. This is due to the channel frictional effects modeled by a Chezý law. In thederivation of these equations, the Froude Number F is defined to be

F :=√

tan

C,

where is the angle of inclination and C is the coefficient of friction. It can also be written as

F =√

U2s

gHs cos ,

where Hs and Us are the uniform depth and flow speed, respectively. Thus, the Froude number is of theorder of the ratio of the uniform fluid velocity to the velocity

√gHs of the gravity wave. It represents a

balance between inertia and gravity. In general, the one-dimensional shallow water equations have beenanalytically insolvable.

The uniform solutions of (1) and (2) are hs := 1 and us := F . In dynamical systems terms, this is arest, or fixed, point. The next obvious question to ask is: under what conditions is the system locally stableor unstable near this rest point? There is only one parameter, the Froude number, in this model and hence

R. Spindler / Fluid Dynamics Research 38 (2006) 469–488 471

1

h (x,t)

u (x,t)

Bottom

θ

Free Surface

X

Fig. 1. Flat inclined channel.

the question of stability focuses on the Froude number. Linear instability occurs, by definition, at valuesof the parameter(s) of a given system where the solution to the linearized problem near the rest pointgrows without bound. Whitham (1974, pp. 85–86) showed that the flow is linearly unstable for F > 2and Spindler (2005) confirms this for the boundary value problem. Recall that the usual procedure is tosubstitute in traveling wave solutions ei(kx−t) with k real. Then the regions of instability are determinedby conditions under which the imaginary part of is positive. Here, the same procedure was executed, butbecause it is a boundary value problem, is assumed real. Then the regions of instability are determinedby conditions when the imaginary part of k is negative.

The instability results say in effect that, for the linearized system, a transcritical bifurcation occursat F = 2. Here the system’s (linear) stability at this rest point changes from stable to unstable as F in-creases through 2. In addition, the original system is assumed stable for F < 2 and unstable for F > 2in some neighborhood of the rest point and in some neighborhood of F = 2. As described in slightlymore detail in Section 3.1, this is similar to the statement in ordinary differential equation dynamicalsystems theory that the 1-jet of the system is sufficient to determine local stability near the rest pointfor F = 2.

For the nonlocal situation, the nonlinear terms of the model come into play by modifying this instabilityin the far-field (large x). Weak instability, defined by 0 < F − 2>1, is the region of concern in this study.Assuming a small boundary disturbance to uniform flow (the inlet conditions)

h(0, t) = 1 + h(1)b (t), (3)

u(0, t) = F + u(1)b (t), (4)


P

Q

t

x

Characteristic of slope one x = t

Characteristics of slope one andt hree

Boundary Values

Ω

Fig. 2. Domain of interest: .

the method of averaging is developed and applied to this weakly unstable situation. In dynamical systemterms, we “start” at a point near the rest point where the above linear instability result applies and developa method of averaging to observe how the system evolves. Thus, inlet conditions perturbed from the reststate are being specified. To model the weak nonlinearity, set F = 2 + where > 0 is “small”. Note thatthis ties together the physical description of the system (the Froude number) with the boundary condition.

Before proceeding, the domain of interest is described. Recall that in a hyperbolic system the solutionat a point is determined by the backward characteristics to the Cauchy data, which in this case is theboundary data. The slopes of the characteristics of the shallow water equations are u + √

h and u − √h.

To a first approximation, these are F + 1 and F − 1, since u = F and h = 1 are the uniform solutions.In the case here, F ≈ 2, so the characteristic slopes are given approximately by 3 and 1. The relevantdomain for the boundary value problem is found by observing that the backward characteristics fromthe points in the domain := (t, x) ∈ R2 : 0 < x < t intersect the positive t axis on which the inletconditions are specified. In Fig. 2, P is in the domain because the solution there is determined solely bythe evolution from the boundary data. On the other hand, the backward characteristic of slope 1 from thepoints in the region (t, x) ∈ R2 : 0 < t < x intersect the positive x axis, on which data is not specified.Thus, to determine the solution at point Q in Fig. 2, initial data must also be specified. In this study, onlyboundary data is considered, and hence only the region is considered.

2. Introduction to averaging

The method of averaging has been used to derive asymptotic evolution equations and to prove uniformconvergence of asymptotic expansions over expanding intervals of order 1/ for ordinary differential


equations in Lagrange standard form (to be defined shortly). Sanders and Verhulst (1985) provide anexcellent account of this theory, as does Murdock (1991). In this study, a method of averaging that appliesto the shallow water equations is created.

Krol (1991) developed a method for an advection–diffusion equation by averaging along the timevariable. Buitelaar (1993) defines a time averaging method for systems of equations in abstract Banachspaces of the form

dw

dt+ Aw = f (w, t, ),

where A is the generator of a C0 group of unitary operators on a Hilbert Space. By Stone’s theorem,this implies iA is self-adjoint. In all of the applications considered by Buitelaar (e.g. nonlinear waveequations), this self-adjointness requires homogeneous boundary conditions. This study’s approach differsfrom that of Krol or Buitelaar in a couple of ways. First, the average is applied along the characteristicsof the linearized equations, and second, the averaging is applied to a hyperbolic system. As noted above,Chikwendu and Kevorkian (1972) developed a method of multiple scales for a class of nonlinear waveequations which in the final result is an average along characteristics.

The idea behind averaging methods is as follows. Assume an ordinary differential equation can betransformed into Lagrange standard form, which is defined to be an ordinary differential equation of theform

dx

dt= f (t, x, ),

where these could be vector quantities and where f is real analytic in . In addition, initial conditions arespecified at t = 0. In effect, this form implies that t is a slow variable. Now suppose we expand f in aTaylor’s series in to obtain

dx

dt= f (1)(t, x) + 2f (2)(t, x, ). (5)

The system is now in a normal form of order 1 (Murdock 2003, pp. 12–13). Usually f is required to beLipschitz continuous in x. Also, often the existence of the average quantity

f (1)(x) := limT →∞

1

T

∫ T

0f (1)(t, x) dt (6)

is imposed. In that case, Sanders and Verhulst (1985) call f a KBM-vectorfield (after Krylov, Bogoliubov,and Mitrolosky). An alternative assumption used is that f is periodic in time.

To understand the mathematical origin of averaging, perform what is sometimes called a near-identitytransformation

x = x + g(t, x) (7)

to transform Eq. (5) to the “autonomous” (in t) problem

dx

dt= p(1)(x) + 2p(2)(t, x, ). (8)


What makes this work is for p(1) to be the average of f (1) holding x constant, namely Eq. (6). To seewhy, differentiate Eq. (7) with respect to t

dx

dt= dx

dt+

xg · xt + g

t

=

p(1)(x) + g

t

+ O(2) (9)

using Eq. (8) and the fact that xt is O(). Now express xt in terms of x instead of x by expanding theright-hand side of Eq. (5) around x in a Taylor’s series. (Recall that x and x differ by an order of .) Equatethe order terms of Eqs. (5) and (9) to obtain

f (1)(t, x) = p(1)(x) + g

t.

Now average over t,

limT →∞

1

T(g(t, x) − g(0, x)) = lim

T →∞1

T

∫ T

0f (1)(t, x) dt − p(1)(x).

If g is uniformly bounded, the left-hand side is zero, and p(1) is found to be equal to the average of f (1).Thus,

dx

dt= f (1)(x) + 2p(2)(t, x, )

is the new differential equation. In effect, the t dependence has been “pushed” up to an order of . It canbe shown that the solution to the averaged system

dz

dt= f (1)(z)

is an o(1) approximation to the solution to the original problem on expanding intervals of 1/, i.e. onintervals 0 t L/ for some constant L (Sanders and Verhulst, 1985).

Murdock (2003) calls the method of using a near-identity transformation to simplify, or normalize, theright-hand side of Eq. (5) format 1a, or the direct/iterative format. To see how normal form theory worksin this context, append a zeroth component to x and f (i) as follows:

x0 = t ,

f(i)0 = 0, i1

and define a zeroth order column vector f (0) = [1 0 · · · 0]T so now Eq. (5) can be written as

dx

dt= f (0) + f (1)(t, x) + 2f (2)(t, x, ). (10)

Define the homological operator

Lu := Duf (0) − Df (0)u,


where u is a vector field as a function of the vector x and D indicates the total derivative with respect tox. (Thus, these derivatives are matrices.) Then, for format 1a, Eq. (10) transforms to

dx

dt= f (0) + [f (1)(t, x) − (Lg)(t, x)] + 2f (2)(t, x, ).

From the perspective of normal form theory, the idea then is to choose p(1) so that f (1) − p(1) is inthe range of L (which then determines g) and in such a way as to make the problem simpler. Theway in which this is done is called the style. In the case here, Lg = g/x0 = g/t . For the style,p(1) is chosen to be the projection of f (1) into the kernel of L, i.e. p(1) is the time average of f (1) asfound above.

The approach in this study is similar to the procedure outline above. Note that the progression throughthe above derivation showed how the method of averaging comes about. If one knows how to averagealready, then just define

g(t, z) =∫ t

0[f (1)(s, z) − f (1)(z)] ds

in the near-identity transformation and work with that (Verhulst, 1996). This also arises naturally fromthe normal form theory discussed above. In addition, note that if := t (slow time), then the averagedsystem is

dz

d= f (1)(z),

which indicates the connection to multiple scales. Finally, if f (1) is periodic with period T, the samemethod is used except the integration is over a period. (In that case g turns out to be periodic.)

3. The method of averaging applied to the weakly unstable shallow water equations

A method of averaging for the weakly unstable boundary value shallow water system is now de-veloped, applying similar reasoning as used for ordinary differential equations described above. Indynamical system problems for ordinary differential equations, the systems are often assumed to bein Lagrange standard form. Transforming a system of partial differential equations into this form isnot trivial, and that is true in this case too. One common method of attack, mentioned earlier andapplied in the next section, is to use a variation of parameters procedure. In that approach, the re-duced (linear) problem is solved, which results in some “constants” of integration. Then the full prob-lem is transformed by letting these “constants” of integration depend on all independent variables.Once this is finished, the method of averaging can be defined. Note that all of the calculations re-lated to this work can be found in Spindler (2005, Chapter 3), and also are available by contactingthe author.

3.1. The reduced problem and Lagrange standard form

The first task is to transform the shallow water equations (1) and (2) into Lagrange standard form. Todo this, the reduced problem is solved first. Center the shallow water equations (1) and (2) around the


uniform values by the trivial change of variables

h = 1 + h(1),

u = F + u(1), (11)

which define h(1) and u(1) for = 0, and substitute into the shallow water equations. (This is not aperturbation expansion.) Divide by , multiply the second equation through by 1 + h(1), and simplify theright-hand side of the second equation to arrive at

h(1)t + Fh(1)

x + u(1)x = −u(1)h(1)

x + h(1)u(1)x ,

u(1)t + h(1)

x + Fu(1)x −

(h(1) − 2

Fu(1)

)=

− 1

F 2 u(1)2 − [u(1)u(1)x + h(1)u

(1)t + h(1)h(1)

x

+ Fh(1)u(1)x ] − h(1)u(1)u(1)

x

. (12)

Of course, this change of variables centers the rest point at the origin, but it also in effect “lowers” theorder by 1 compared to the original shallow water equations. Note the time derivative on the right-handside of the second equation.

Before inserting F = 2 + , the reduced problem is determined and solved. Set = 0, so that Eqs. (12)reduce to the linearized shallow water equations with F = 2. Define h(1) and u(1) to be the solutions ofthese linearized equations. Add and subtract the two equations (12) (with = 0) and define

S := h(1) + u(1) and R := h(1) − u(1)

to obtain

St + 3Sx − R = 0,

Rt + Rx + R = 0.

The characteristic slopes of this system are 1 and 3 and so the characteristics of this system are x − t=constant and x − 3t= constant. Change variables along these characteristics, that is

(x, t) := x − t and (x, t) := x − 3t , (13)

which transforms the first order to

S(, ) = 12 R(, ), (14)

R(, ) = 12 R(, ). (15)

In effect, the linearized system has been transformed to normal form in the sense of hyperbolic partialdifferential equations (Forsythe and Wasow, 1960). The solution of Eq. (15) is found to be

R(, ) = r()e/2, (16)

where r() is in effect a “constant” of integration. Using this in the Eq. (14) above gives

S(, ) = R()e/2 + s(), (17)


Ω

= 3

Fig. 3. Domain of interest: .

where s() is in effect another “constant” of integration and

R() := 1

2

∫

r(′) d′. (18)

Note that the primes here do not designate differentiation. The ′ variable is just a dummy integrationvariable.

The domain will be needed in the new (, ) coordinate system. When x =0, Eqs. (13) are =−t and=−3t . Eliminating t determines the boundary =3. Similarly, x = t implies =0. Also, x0 maps tothe region 3, and x < t maps to the region < 0 under the transformation (13). (The boundary t 0maps to the region which is less restrictive than 3.) Thus, = (, ) : 30. See Fig. 3.

Note that using these transformations, the concept of jet sufficiency of ordinary differential equationscan be related to the situation here. Applying the change of variables S := h(1) + u(1), R := h(1) − u(1), := x − (F − 1)t , and := x − (F + 1)t , Eqs. (12) are then transformed to a normal form

S = 1

2

[(1

2− 1

F

)S +

(1

2+ 1

F

)R

]+ m1,

R = 1

2

[(1

2− 1

F

)S +

(1

2+ 1

F

)R

]+ m2. (19)

The remainder terms m1 and m2 terms are quadratic polynomials in S, R, and their derivatives. Becauseonly linear transformations have been applied to the system, its stability properties are not changed. Inordinary differential equation dynamical systems theory, a system in the form of (19) where the right-handside is expanded into a Taylor’s series in the dependent variables with F = 2, is 1-jet sufficient since thelinear polynomial terms determine the stability locally around the rest point. If F =2, the quadratic terms


would need to be examined to determine if the system is 2-jet sufficient, and so on. See Murdock (2003,p. 8) for a definition and discussion of sufficiency of jets in the context of ordinary differential equations.

The reason for determining the solution to the reduced problem is to transform the original problem(12) using this reduced solution by the method of variation of parameters. First, to determine the fulldependence of the full equations on , substitute F = 2 + into the second equation of (12) and multiplyboth sides by F 2 to obtain

4(u(1)t + h(1)

x ) + 8u(1)x − 4(h(1) − u(1)) + 4(u

(1)t + h(1)

x ) + (12u(1)x − 2h(1)) − 2(h(1) − u(1))

= −u(1)2 − 4[u(1)u(1)x + h(1)u

(1)t + h(1)h(1)

x + 2h(1)u(1)x ] + O(2).

All 2 terms are summarized in O(2). (Note that here the O(2) refers to the terms in this equation,of course, which would imply O(3) in the original shallow water equations.) Moving terms to theright-hand side, the equations become

h(1)t + 2h(1)

x + u(1)x = −(u(1)h(1)

x + h(1)u(1)x + h(1)

x ),

u(1)t + h(1)

x + 2u(1)x − (h(1) − u(1)) = −(1 + h(1))(u

(1)t + h(1)

x + 2u(1)x )

− (1 + u(1))u(1)x + h(1) − 1

2u(1) − 14u(1)2 + O(2). (20)

Observe that (20) is recursive in the sense that part of the left-hand side is in the right-hand side, namelyu

(1)t + h

(1)x + 2u

(1)x . Substitute this in for itself on the right-hand side, moving higher order terms into

O(2) to obtain

h(1)t + 2h(1)

x + u(1)x = −(u(1)h(1)

x + h(1)u(1)x + h(1)

x ), (21)

u(1)t + h(1)

x + 2u(1)x − (h(1) − u(1)) = −(1 + u(1))u(1)

x + 12u(1) − (h(1) − 1

2u(1))2+ O(2). (22)

As in the reduced problem, transform the left-hand side of (21) and (22) into normal form (in the senseof hyperbolic partial differential equations), so let

S := h(1) + u(1) and R := h(1) − u(1)

and eliminate h(1) and u(1) in the equations. (Add and subtract the equations). Again, transform along thecharacteristics of the linearized equations, (x, t) = x − t and (x, t) = x − 3t , to obtain

S − 12R = 1

2 −(1 + 34S − 1

4R)(S + S) + 14(S + R)(R + R)

+ 14(S − R) − 1

16(S + 3R)2 + O(2), (23)

R − 12R = 1

2 (1 + 14S − 3

4R)(R + R) + 14(S + R)(S + S)

+ 14(S − R) − 1

16(S + 3R)2 + O(2). (24)

Next, change variables using the reduced solutions, Eqs. (16) and (17), by the method of variationof parameters. Recall that s and r are “constants” of integration for the reduced equations with the first


depending on and the second on . Now let s depend also on and r also on . Therefore, define

r(, ) := R(, )e−/2,

s(, ) := S(, ) − R(, )e/2,

where R is similar to Eq. (18):

R(, ) = 1

2

∫

r(′, ) d′. (25)

Determine the derivatives of R and S and substitute these into the equations for S and R, (23) and (24),to obtain

s = 12 −[1 + 3

4(s + Re/2) − 14re/2](s + 1

2re/2 + s + Re/2 + 12Re/2)

+ 14(s + Re/2 + re/2)(re/2 + re/2 + 1

2re/2) + 14(s + Re/2 − re/2)

− 116(s + Re/2 + 3re/2)2 + O(2)

and

r = 12 (1 + 1

4s + Re/2 − 34re/2)(re/2 + re/2 + 1

2re/2)

+ 14(s + Re/2 + re/2)(s + 1

2re/2 + s + Re/2 + 12Re/2) + 1

4(s + Re/2 − re/2)

− 116(s + Re/2 + 3re/2)2e−/2 + e−/2O(2).

Re-arrange and combine terms. Also, because s and r are equal to times terms on the right, move s,r, and R terms on the right up into O(2) and e−/2O(2) to arrive at

s =

2

[−(

1 + 3

4s

)s + 1

4s − 1

16s2]

+[−1

4(3R − r)s +

(−5

8r − 1

2R + 1

4r

)s +

(−1

4R − 3

4r

)]e/2

+[− 5

16r2 − 1

2rR − 7

16R2 + 1

4r(r + R)

]e

+ O(2) (26)

and

r =

2

[1

4ss + 1

4s − 1

16s2]

e−/2 +[r + 1

4r + 1

4R

]+[(

1

4r − 1

8r

)s + 1

4(r + R)s

]

+[−13

16r2 + 1

16R2 +

(1

4R − 3

4r

)r

]e/2

+ e−/2O(2). (27)

Basically these independent variables introduced into s and r correspond to slowly varying variables fortheir respective function, meaning that corresponds to a slowly varying independent variable for s and corresponds to a slowly varying independent variable for r. Observe that everything has been exact sofar. (Although the higher order terms are hidden in the O(2) term.) Various transformations and algebraicoperations have just been applied to the problem. Note that for = 0 the reduced solutions are recovered.Also observe that, at least up to order O(), s =0 and r =0 are rest points as they should be, correspondingto h(1) = 0 and u(1) = 0.


3.2. Near-identity transformation

The above system of equations, (26) and (27), is now expressed in normal form in the sense of partialdifferential equations. This is similar to the normal form as used in dynamical systems for ordinarydifferential equations (Murdock, 2003), but there are differences. For example, on the left-hand side,the derivatives are in terms of two independent variables instead of one, of course, since these arepartial differential equations. Also, derivatives appear on the right-hand side. Despite these differences,an averaging theory can be developed along the lines as described in Section 2. This will also be analogousto the transformation of normal forms (Murdock, 2003).

Now that the equations are in normal form, the method of averaging along characteristics arises formallyby applying the idea of a near-identity transformation defined by[

s

r

]=[

s

r

]+

2

[g1(, s, s, r, r)

e−/2g2(, s, s, r, r)

]. (28)

The goal is to define this transformation such that the Lagrange standard form equations (26) and (27)are “autonomous” and uncoupled from each other

s =

2p1(s, s) + O(2), (29)

r =

2p2(r, r) + e−/2O(2) (30)

solving the same boundary conditions, namely s(, 3) = s(, 3) and r(, 3) = r(, 3). This is similarto the example in Section 2 for ordinary differential equations where the independent variable t waseliminated. The difference here is that dependent variables are being eliminated from the right-hand sidefunctions instead of an independent variable. For convenience, a factor e−/2 of g2 is inserted in thesecond equation.

Assume that gi functions and their derivatives are uniformly bounded. Also, by definition of change ofvariables, the near-identity transformation must be invertible in a neighborhood of =0 which is assumed.To proceed, differentiate (28) to obtain[

s

r

]=[s

r

]

+

2

[g1 + g1s s + g1s s + g1r r + g1r r

e−/2g2 − 12 e−/2g2 + e−/2(g2s s + g2s s + g2r r + g2r r + g2)

]. (31)

Recall that s =O() and r = e−/2O(). Thus, those terms are moved up into higher orders. Also assumethe same is true of higher derivatives of these terms. Then[

s

r

]=[s

r

]+

2

[g1 + g1r r + g1r r

e−/2g2 − 12 e−/2g2 + e−/2(g2s s + g2s s + g2)

]+[

O(2)

e−/2O(2)

]

=[s

r

]+

2

⎡⎢⎢⎣

d

dg1

∣∣∣∣s,s constant

d

d(e−/2g2)

∣∣∣∣r ,r constant

⎤⎥⎥⎦+

[O(2)

e−/2O(2)

]. (32)


Observe that by the near-identity transformation (28), s and s differ byO() and r and r differ by e−/2O().Expand the right-hand side terms of Eqs. (26) and (27) (as functions of s and r and their derivatives) ina Taylor’s series around s and r and their derivatives, to simply replace s with s and r with r , with theremainder terms of the Taylor’s Series moved up into the O(2) and e−/2O(2) terms. (Note that to applythis Taylor Series, g1 and g2 and their derivatives must be uniformly bounded.)

Next substitute Eqs. (26) and (27) (with s and r , etc., instead due to the discussion in the last paragraph)into the left-hand side of Eq. (32), and Eqs. (29) and (30) into the right-hand side of Eq. (32). Aftercancelling an from both sides and re-arranging,

p1(s, s) + d

dg1


=[

−(

1 + 3

4s

)s + 1

4s − 1

16s2]

+[−1

4(3R − r)s +

(−5

8r − 1

2R + 1

4r

)s +

(−1

4R − 3

4r

)]e/2

+[− 5

16r2 − 1

2rR − 7

16R2 + 1

4r(r + R)

]e

+ O(), (33)

p2(r, r) + d

d(e−/2g2)


=[

1

4s s + 1

4s − 1

16s2]

e−/2

+[r + 1

4r + 1

4R

]+[(

1

4r − 1

8r

)s + 1

4(r + R)s

]

+[−13

16r2 + 1

16R

2 +(

1

4R − 3

4r

)r

]e/2

+ e−/2O(). (34)

(Note that the cancellation of lowers the order 1 more relative to the shallow water equations.) Here, Ris defined similar to R except using r instead of r in the integral defining R.

Examine Eq. (33) first. Move p1 to the other side of Eq. (33) and integrate as follows:

g1 =∫ [

−(

1 + 3

4s

)s + 1

4s − 1

16s2]

+ e/2[−1

4(3R − r)s +

(−5

8r − 1

2R + 1

4r

)s +

(−1

4R − 3

4r

)]

+e[− 5

16r2 − 1

2rR − 7

16R

2 + 1

4r(r + R)

]− p1(s, s)

d′∣∣∣∣s,s constant

+ O()


or

g1 =[−(

1 + 3

4s

)s + 1

4s − 1

16s2]

+ e/2[−1

4s

∫

(3R − r) d′

+s

∫ (−5

8r − 1

2R + 1

4r

)d′ +

∫ (−1

4R − 3

4r

)d′]

+ e∫ [

− 5

16r2 − 1

2rR − 7

16R2 + 1

4r(r + R)

]d′

− p1(s, s) + c1(, s, s) + O().

Next divide both sides by and let → −∞ and continue to treat s and s as constant. This limit couldconceivably be taken in a number of ways, as long as t > x0, that is 3 < 0 (to remain in the domainas discussed in the first section). However, another requirement for choosing the curve along which to takethe limit is that singularities (shocks) must not be encountered in the limit. The boundary data curve x =0certainly satisfies this requirement, and hence the limit is defined along this curve. Since g1 is assumeduniformly bounded, the left-hand side tends to zero. Suppose also there exists a positive constant C(x)0for each fixed x such that

|r|, |r|, |R|, |R|C(x)e−3/2 (35)

uniformly in t. (It will be seen in Section 4 that the solution is consistent with this assumption.) By thisassumption, each of the terms above, except the first and the p1 and c1 terms, will be of the form

e1/2(−3)

= e−x

or

e(−3)

= e−2x

,

which approaches zero uniformly as → −∞ since x is fixed. Assuming the integration “constant” c1 isuniformly bounded, all that remains is

p1(s, s) = −(1 + 34 s)s + 1

4 s − 116 s2 + O().

Substitute this into (29) to obtain one evolution equation.Process Eq. (34) in a similar manner. First, integrate (34) holding r and r constant,

g2 = e/2∫ [1

4s s + 1

4s − 1

16s2]

e−′/2 +[r + 1

4r + 1

4R

]

+[(

1

4r − 1

8r

)s + 1

4(r + R)s

]

+[−13

16r2 + 1

16R2 +

(1

4R − 3

4r

)r

]e′/2 − p2(r, r)

d′∣∣∣∣r ,r constant

+ e/2c2(, r, r) + O().


Then re-write the equation as

g2 =∫

e/2[

1

4s s + 1

4s − 1

16s2]

e−′/2

+[R + 1

4R + 1

4RR

]+[(

1

4R − 1

8R

)s + 1

4(R + RR)s

]

+[−13

16R2 + 1

16R

2R +

(1

4RR − 3

4R

)R

]e(′−)/2 − p2(R, R)

d′∣∣∣∣R,R constant

+ e/2c2(, r, r) + O()

using R = e/2r and where RR := e/2R. (In the first equation above, r and r have been held constantin the integral, which is equivalent to holding R = e/2r and R = e/2r constant.) Also, it has beenassumed that p2 is homogeneous in its arguments (meaning yp2(x1, x2) = p2(yx1, yx2)). Factor the R

terms out of the integrals to obtain

g2 = e/2∫ [1

4s s + 1

4s − 1

16s2]

e−′/2 d′ +[R + 1

4R + 1

4RR

]

+[(

1

4R − 1

8R

)∫

s d′ + 1

4(R + RR)s

]

+ 2

[−13

16R2 + 1

16R

2R +

(1

4RR − 3

4R

)R

]− p2(R, R) + e/2c2(, r, r) + O().

Divide both sides by , treat R and R as constant, and let → −∞. Again, choose the curve x =0 alongwhich to take this limit. Since g2 is assumed uniformly bounded, the left-hand side tends to zero. Alsosuppose that the various integrals involving s and s in this last equation are uniformly bounded as wellas the integration “constant” e/2c2(, r, r). In effect the solutions have been restricted to the class offunctions satisfying these specifications. Then

p2(R, R) = [R + 14 R + 1

4RR] + O()

or

p2(r, r) = [r + 14 r + 1

4R] + e−/2O().

This result provides a second evolution equation by substituting into Eq. (30). Also observe that p2, atleast up to order , is homogeneous and so that assumption is consistent.

From the work above, the averaging operation that appears to transform the equations as desired is toaverage Eq. (26) by

lim→−∞

(1

∫

[· · ·] d′)∣∣∣∣

s,s constant


along the line x = 0, where s and s are treated as constants in the limit. Average Eq. (27) by

e−/2 lim→−∞

(e/2

∫

[· · ·] d′)∣∣∣∣∣

R,R constant

along the line x = 0, where R, R, and RR are treated as constants in the limit.The near-identity transformations gi are

g1 = e/2[−1

4s

∫

(3R − r) d′ + s

∫ (−5

8r − 1

2R + 1

4r

)d′ +

∫ (−1

4R − 3

4r

)d′]

+ e∫ [

− 5

16r2 − 1

2rR − 7

16R2 + 1

4r(r + R)

]d′

and

g2 = e/2∫ [1

4s s + 1

4s − 1

16s2]

e−′/2 d′ +[(

1

4R − 1

8R

)∫

s d′ + 1

4(R + RR)s

]

+ 2

[−13

16R2 + 1

16R

2R +

(1

4RR − 3

4R

)R

].

These are uniformly bounded if the assumptions on s and r hold, as required.To summarize, the equations in Lagrange standard form (26) and (27) transform to the evolution

equations

s =

2

[−(

1 + 3

4s

)s + 1

4s − 1

16s2]

+ O(2) (36)

and

r =

2

[r + 1

4r + 1

4R

]+ e−/2O(2) (37)

under the near-identity transformation. In ordinary differential equations, the system of equations retainingthe higher order O(2) terms is sometimes called the transformed system. Of course, the O(2) terms aredropped to approximate the actual solution. When these higher order terms are dropped,

s =

2

[−(

1 + 3

4s

)s + 1

4s − 1

16s2]

(38)

and

r =

2

[r + 1

4r + 1

4R

](39)

are sometimes called the averaged system. If = is defined, then the averaged Eq. (38) becomes

s = 12 [−(1 + 3

4 s)s + 14 s − 1

16 s2]and, if = is defined, then the second averaged (39) becomes

r = 12 [r + 1

4 r + 14R].


In effect, the new variables define far-field scales for each equation. In ordinary differential equations,this system of equations is sometimes called the guiding system (Murdock, 1991).

The development of the averaging method can be placed in the context of normal forms theory ofdynamical systems. Here, the homological operator is defined to be

Lg :=

⎡⎢⎢⎣

d

dg1


d

d(e−/2g2)


⎤⎥⎥⎦ .

Let f (1) be the order term on the right-hand side of Lagrange standard form equations (26) and (27).Eqs. (33) and (34) result from transforming the normal form system (26) and (27) into format 1a byapplying the near-identity transformation. The abstract form of Eqs. (33) and (34) is

Lg = f (1) − p(1).

The average operation and the final evolution equations then arise by defining p(1) to be the projectionof f (1) into the kernel of L.

4. A comparison

The solutions to the averaged evolution equations (38) and (39) are determined, assuming a sine waveboundary condition, namely

h(0, t) = 1 + sin(t) and u(0, t) = F .

It is hoped that the solutions to the averaged equations are a good approximation to the solution of thetransformed Eqs. (36) and (37). This means that the solutions are within o(1) over expanding intervals of xof order 1/ as in the ordinary differential equations case.Also, recall that s=s+O() and r=r+e−/2O().Hence R = R + O() (where R := e/2r) and S = S + O(). Thus, up to order , R and S, and hence h(1)

and u(1), are determined by R and S. In this section, the solution of Eq. (39) is solved explicitly for r ,while Eq. (38) is solved numerically. Finally, the shallow water equations are numerically integrated andcompared to the solution of the evolution equations, at a perturbation of = 0.1.

The averaged r-evolution equation (39) is linear in r with constant coefficients, so it can be solvedanalytically. The boundary condition for r can be written

r(, 3) = − sin()e−3/2

using the definitions of r and R. Assume a solution of the form

r(, ) = −e−(3/2)−(/2)a(3−) sin( +

2b(3 − )

), (40)

where the parameters a and b are to be determined. Substitute the proposed solution into the r-evolutionequation (39) to solve for a and b to obtain

a = −5

4− 3/4

9 + 42 and b = /2

9 + 42 − . (41)


37 38 39 40 41 42 43-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

Time t

DirectAverage

h(1)

Fig. 4. Height h1 of wave at x = 30.

Note that a is less than zero. Also observe that r is consistent with assumption (35). Transforming backto the R variable

R(, ) = − e−(1/2)(1+a)(3−) sin( +

2b(3 − )

)= − e−(1+a)x sin

( +

2b(3 − )

),

where of course R = e/2r . This implies that the part of the solution from the characteristic decaysexponentially in the x direction as long as < 1/|a|. Thus, with this restriction on , the contribution ofR to h(1) and u(1) is negligible as x becomes large. The far-field evolution arises from the contributionof S which is essentially determined by s (since the R contribution to S decays exponentially in the xdirection also). This also implies that h(1) and u(1) become nearly identical rather quickly.

A basic forward–backward (FB) method is used to numerically integrate the s-evolution equation. Inaddition, the differences are designed to satisfy the CFL condition. The boundary condition in terms ofs is

s

(

3,

)= − sin

(

3

)− e/2R

(

3,

).

From Eq. (25) for R and the solution for r

s

(

3,

)= − sin

(

3

)− 1

9 + 42

(3 sin

(

3

)+ 2 cos

(

3

)). (42)

Eq. (42) is the boundary condition for the s-evolution equation (38) used in the numerical integration.The shallow water equations are numerically integrated by applying the artificial viscosity method

(Lapidus and Pinder, 1999, Section 6.2.3).Again, the differences are designed to satisfy the CFL condition.


0 5 10 15 20 25 30 35-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

Distance x

DirectAverage

h(1)

Fig. 5. Height h1 of wave at t = 30.

Finally, all of the programs were written in MatLab 6.5. Fig. 4 is an overlay of the two solutions at x =30and = 0.1. Observe that the waves are periodic in t as expected, and the formation of a bore occurs. Thewaves are within about 2 of each other up to (and past) x ≈ 1/ as expected from perturbation theory.Fig. 5 is an overlay of the two solutions at t = 30 and = 0.1. Notice the gradual exponential growth ofthe wave in the x direction, as the linear stability theory predicted. Also observe that the waves steepento a bore as x increases. Finally, the solutions from the two methods at t = 30 are still within about 2 ofeach other up to (and past) x ≈ 1/.

5. Conclusion

The averaging method introduced in this study for the shallow water equations indicates a methodof averaging for more general two-dimensional systems of hyperbolic partial differential equations. Byaveraging along the characteristics of the linearized equations, an approximate solution may be found inthe far-field. It is hoped that this method can be extended to that more general situation.

Note that Yu and Kevorkian (1992) analyzed the initial value problem for a flat inclined constant-slopechannel using a multiple scales method, while Spindler and Yu (2006) analyzed the boundary valueproblem using multiple scales in conjunction with the Fredholm alternative theorem. The s-evolutionequations found for mutliple scales there and for averaging here differ by one term, but the numericalsolutions of these equations are both within order 2 of each other. These results suggest that the twosolutions may be asymptotically equivalent, although this question of equivalence merits further study. Forordinary differential equations, Murdock (1991) shows that the first order two-scales method (using onefar-field scale) is identical to averaging. In fact, Perko (1969) shows that the two methods for ordinarydifferential equations are asymptotically equivalent, though not necessarily identical, up to all ordersof .


Although deriving the averaging technique is more involved than applying multiple scales, once theaveraging operation is known, it is no more difficult to apply than determining the secular terms inmultiple scales. However, the asymptotic validity of multiple scales for partial differential equations is anopen mathematical question as well as its relationship to averaging. The advantage of the normalizationand averaging technique is that it may provide a more tractable framework for deriving rigorous errorestimates for both techniques applied to partial differential equations or systems of partial differentialequations, just as with ordinary differential equations.

Acknowledgments

I would like to thank Ferdinand Verhulst for his helpful comments on this project. I am also gratefulfor the many suggestions the reviewers made to improve the manuscript.

References

Buitelaar, R.P., 1993. The method of averaging in Banach spaces: theory and applications. Ph.D. Thesis, Rijksuniversiteit Utrecht,Netherlands.

Chikwendu, S.C., Kevorkian, J., 1972. A perturbation method for hyperbolic equations with small nonlinearies. SIAM J. Appl.Math. 22 (2), 235–258.

Forsythe, G.E., Wasow, W.R., 1960. Finite-Difference Methods for Partial Differential Equations. Wiley, New York.Kevorkian, J., 1990. Partial Differential Equations: Analytical Solution Techniques. Wadsworth and Brooks/Cole, Pacific Grove,

CA.Krol, M.S., 1991. Nearly time-periodic advection–diffusion problems. SIAM J. Appl. Math. 51 (6), 1622–1637.Lapidus, L., Pinder, G., 1999. Numerical Solution of Partial Differential Equations in Science and Engineering. Wiley-

Interscience, New York.Murdock, J.A., 1991. Perturbation Theory and Methods. Wiley-Interscience, New York.Murdock, J., 2003. Normal Forms and Unfoldings for Local Dynamical Systems. Springer, New York.Perko, L.M., 1969. Higher order averaging and related methods for perturbed periodic and quasi-periodic systems. SIAM

J. Appl. Math. 17, 698–724.Sanders, J.A., Verhulst, F., 1985. Averaging Methods in Nonlinear Dynamical Systems. Springer, New York.Spindler, R., 2005. The one-dimensional shallow water boundary value problem and instability. Ph.D. Thesis, University of

Vermont, Burlington, VT.Spindler, R., Yu, J., 2006. Nonlinear spatial evolution of small disturbances from boundary conditions in flat inclined-channel

flow. Stud. Appl. Math. 116, 173–200.Verhulst, F., 1996. Nonlinear Differential Equations and Dynamical Systems, second ed. Springer, Berlin.Whitham, G.B., 1974. Linear and Nonlinear Waves. Wiley, New York.Yu, J., Kevorkian, J., 1992. Nonlinear evolution of small disturbances into roll waves in an inclined open channel. J. Fluid Mech.

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an averaging method for the weakly unstable shallow water equations in a flat inclined channel

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