Expansion shock waves in regularized shallow-water theory

Abstract

We identify a new type of shock wave by constructing a stationary expansion shock solution of a class of regularized shallow-water equations that include the Benjamin–Bona–Mahony and Boussinesq equations. An expansion shock exhibits divergent characteristics, thereby contravening the classical Lax entropy condition. The persistence of the expansion shock in initial value problems is analysed and justified using matched asymptotic expansions and numerical simulations. The expansion shock's existence is traced to the presence of a non-local dispersive term in the governing equation. We establish the algebraic decay of the shock as it is gradually eroded by a simple wave on either side. More generally, we observe a robustness of the expansion shock in the presence of weak dissipation and in simulations of asymmetric initial conditions where a train of solitary waves is shed from one side of the shock.

1. Introduction

In this paper, we consider a class of nonlinear, dispersive equations naturally arising in shallow-water theory, most concisely exemplified by a version of the Benjamin–Bona–Mahony (BBM) equation, also known as the regularized long-wave equation

$$u_t+u{u}_x=u_{xxt}.$$ 1.1

The original BBM equation, which contains an additional linear convective term u_x, is an important model for the description of unidirectional propagation of weakly nonlinear, long waves in the presence of dispersion. It first appeared in a numerical study of shallow-water undular bores [1], and later was proposed in [2] as an analytically advantageous alternative to the Korteweg–de Vries (KdV) equation

$$u_t+u{u}_x=-u_{xxx}.$$ 1.2

In the context of shallow-water waves, the BBM and KdV equations (1.1) and (1.2) are reduced, normalized versions of corresponding asymptotic models derived from the general Euler equations of fluid mechanics using small-amplitude, long-wave expansions. If δ≪1 is the ratio of the undisturbed depth to a typical wavelength and ϵ≪1 is the ratio of a typical wave amplitude to the undisturbed depth, then the asymptotic KdV and BBM equations occur under the balance ϵ∼δ² [3], and so can be used interchangeably within their common domain of asymptotic validity [4].

Despite asymptotic equivalence, the mathematical properties of the BBM and KdV equations are very different, which is acutely captured by their normalized versions (1.1) and (1.2). The KdV equation (1.2) is known to be integrable via the inverse scattering transform and to possess an infinite number of conservation laws. The BBM equation (1.1), by contrast, does not enjoy full integrability and has only three independent conservation laws. Nonetheless, well posedness of initial value problems for both equations has been established in the Sobolev spaces H^s (with s>0 for BBM [5], $s>-\frac{3}{4}$ for KdV [6]).

As a numerical and mathematical model, the BBM equation yields more satisfactory short-wave behaviour, owing to the regularization of the unbounded growth in frequency, phase and group velocity values present in the KdV equation. In particular, this enables less strict time-stepping in numerical schemes for the BBM equation. Indeed, linearizing (1.1) about a constant u=u₀: u(x,t)=u₀+a e^i(kx−ωt), we obtain the dispersion relation

$$\omega ={\omega }_0(k;u_0)=u_0\frac{k}{1+k^2}.$$ 1.3

The phase and group velocities are

$$c_p=\frac{{\omega }_0}{k}=\frac{u_0}{1+k^2}\mathrm{and}{c}_g={\mathrm{\partial }}_k{\omega }_0=u_0\frac{1-k^2}{{(1+k^2)}^2}.$$ 1.4

One can see that ω₀ as well as c_p and c_g are bounded as functions of the wavenumber k, in contrast to their counterparts for the KdV equation with dispersion relation ω₀=u₀k−k³. The rational form of BBM dispersion (1.3) indicates its non-local character. Moreover, the dynamics of linear dispersive equations with discontinuous initial data exhibit distinct qualitative structure depending upon bounded or unbounded dispersion behaviour for large k [7].

We remark that ‘engineering’ the dispersive properties of model equations was pioneered by Whitham [8] in the context of water waves. In addition to some already mentioned mathematical and numerical advantages, one may also achieve superior physical accuracy, when compared with standard asymptotic models, by incorporating full linear dispersion [9].

Equations (1.1) and (1.2) represent two different dispersive mechanisms to regularize the scalar conservation law, the inviscid Burgers equation

$$u_t+{(\frac{1}{2}{u}^2)}_x=0.$$ 1.5

Dispersive regularization of hyperbolic conservation laws is known to give rise to dispersive shock waves (DSWs), also known as undular bores [10,11], which are in many respects very different from their diffusive or diffusive–dispersive counterparts [12]. These DSWs have a distinct oscillatory structure and expand with time, so that the Rankine–Hugoniot (RH) relations are not applicable to them. Instead, DSW closure is achieved via an appropriate solution of the Whitham modulation equations obtained by a nonlinear wave-averaging procedure applied to the full dispersive equation [3,13]. DSWs are evolutionary if they satisfy causality conditions [13], and thus represent dispersive counterparts of classical, Lax shocks [14]. All shock solutions of the KdV equation are evolutionary DSWs [13]. In contrast, we show in this paper that the BBM equation (1.1) admits a family of stationary (non-propagating), non-oscillatory expansion shocks that (i) satisfy the RH jump condition and (ii) violate causality. BBM expansion shocks are very different from both classical shocks of the inviscid Burgers equation (1.5) and DSWs of the KdV equation (1.2).

Nonlinear partial differential equations of hyperbolic type, such as those modelling inviscid gas dynamics (e.g. (1.5)), can have discontinuous solutions. These weak solutions may or may not be physical, depending on whether they are stable or persist under small changes to initial conditions or the governing equations. Shock waves are physical, discontinuous solutions that typically satisfy side conditions associated with either a physical or mathematical notion of entropy. In gas dynamics, these conditions force shock waves to be compressive in that they compress the gas as they pass a fixed location. This kind of condition was expressed by Lax [14] in the 1950s in terms of characteristics, requiring that shock waves are evolutionary, i.e. they are uniquely determined from initial conditions.

In this paper, we show that a non-evolutionary stationary shock wave of the BBM equation (1.1) persists but decays algebraically in time. This example is surprising, because hyperbolic theory would suggest that the stationary shock would immediately give way to a continuous solution, namely a rarefaction wave. The persistence is explained through the interaction of the particular non-local nature of dispersion in the BBM equation and a length scale associated with the stationary shock that sets the time scale for decay.

Expansion shocks are not unique to the BBM equation. We show that they also persist in one of the versions of the classical bidirectional Boussinesq equations for dispersive shallow-water waves [15,3]. Similar to the BBM equation, these Boussinesq equations have the term u_xxt in the momentum equation. (Existence of weak solutions of initial value problems for Boussinesq equations was established in [16].) More broadly, we identify a large class of non-evolutionary partial differential equations—i.e. equations not explicitly resolvable with respect to the first time derivative [17]—that exhibit decaying expansion shock solutions, indicating the ubiquity of these new solutions.

2. Shocks and rarefactions

If the dispersive right-hand side of the BBM (1.1) or KdV (1.2) equation is deleted, then we are left with the inviscid Burgers equation (1.5), a scalar conservation law that admits shock wave weak solutions

$$u(x,t)=\{\begin{array}{ll}{u}_{-},& x<st,\\ u_{+},& x>st,\end{array}$$ 2.1

provided the speed s is the average of the characteristic speeds u_± on either side of the shock: $s=\frac{1}{2}(u_{+}+u_{-})$. Such shocks are stable provided that characteristics enter the shock from each side, u₊<u₋, a condition known as the Lax entropy condition [14]. In this case, the shock is called an entropy shock, or, by analogy with gas dynamics, a compressive shock.

By contrast, a shock wave solution (2.1) of (1.5) is called expansive if u₋<u₊. Expansion shocks are thought to be unstable and to violate causality, because characteristics leave rather than approach the shock. Instead of an expansion shock, a self-similar rarefaction wave resolves the discontinuity between u₋ and u₊:

$$u(x,t)=\{\begin{array}{ll}{u}_{-},& x<u_{-}t,\\ \frac{x}{t},& u_{-}t<x<u_{+}t,\\ u_{+},& x>u_{+}t.\end{array}$$ 2.2

When u₊=−u₋, the shock wave (2.1) is stationary, and hence is also a weak solution of the BBM equation (1.1) (because the shock is time-independent). With u₋>0, the stable case, the stationary shock persists. However, for u₋<0, the unstable case, hyperbolic theory would suggest that the jump is immediately replaced by the self-similar rarefaction wave (2.2) or some approximation to it. However, we find that the dispersive regularization resulting from the BBM equation sustains solutions in which a smoothed stationary shock persists but decays algebraically, as shown in figure 1. More precisely, we study the initial value problem with initial data being a smoothed stationary shock, with width ε>0.

Figure 1.

Numerical(solid, blue) and asymptotic (dashed, red) solutions of the initial value problem for equation (1.1) with initial data equation (3.1) where ε=0.1 and A=1. (Online version in colour.)

3. The expansion shock

To see the effect of dispersion on a stationary shock, we pose initial data

$$u(x,0)=A\tanh \frac{x}{\epsilon },-\mathrm{\infty }<x<\mathrm{\infty },$$ 3.1

with amplitude A>0 for the BBM equation (1.1). Thus, as ε→0, the initial data converge to a jump from u=−A to u=A, representing a stationary expansion shock solution to the inviscid Burgers equation (1.5). The numerical solution of (1.1) and (3.1) is shown in figure 1. We observe the development of a rarefaction wave on either side of a stationary but decaying shock. We analyse the solution by matched asymptotics using ε as the small parameter. First note that the initial function u(x,0) is an odd function, and the solution u(x,t) should therefore be an odd function of x for each t>0.

(a). The inner solution

To capture the inner solution, we introduce into equation (1.1) the short space ξ=x/ε and long time T=εt scalings of the independent variables x and t, yielding

$$\epsilon u_T+\frac{1}{\epsilon }u{u}_{\xi }=\frac{1}{\epsilon }{u}_{\xi \xi T}.$$ 3.2

Expanding the dependent variable

$$u=u_0(\xi ,T)+\epsilon u_1(\xi ,T)+\cdots$$ 3.3

and substituting this ansatz into (3.2) yields the leading-order equation

$$u_0{\mathrm{\partial }}_{\xi }{u}_0={\mathrm{\partial }}_{\xi \xi T}{u}_0.$$

This equation admits separated solutions u₀(ξ,T)=f(ξ)a(T), leading to

$$f{f}^{\prime }{a}^2=f^{″}\dot{a},$$

where ′, $\dot{\hspace{0.17em}}$ denote derivatives with respect to ξ and T, respectively. Introducing a separation constant K>0

$$\frac{\dot{a}}{a^2}=\frac{f{f}^{\prime }}{f^{″}}=-K,$$

we obtain the solution

$$a(T)=\frac{a_0}{1+a_{0}KT}$$

and

$$f(\xi )=\sqrt{c}\tanh (\frac{\sqrt{c}}{2K}(\xi -{\xi }_0)).$$

In these equations, K,a₀,c and ξ₀ are arbitrary constants. To agree with the initial data (3.1), we set $c=1,K=\frac{1}{2},a_0=A$ and ξ₀=0. Thus, the leading-order inner solution is

$$u\sim u_{in}(\xi ,T)=\frac{A}{1+(1/2)AT}\tanh \xi .$$ 3.4

The inner solution reveals the smoothed structure of the dispersively regularized expansion shock and its algebraic temporal decay.

(b). The outer solution

The outer solution has a different long space and time scaling,

$$X=\epsilon x\mathrm{and}T=\epsilon t.$$

This leads to the scaled equation

$$\epsilon u_T+\epsilon u{u}_X={\epsilon }^3{u}_{XXT}.$$ 3.5

With the expansion $u(X,T)={\tilde{u}}_0(X,T)+\epsilon {\tilde{u}}_1(X,T)+\cdots$, we have the leading-order conservation law

$${\mathrm{\partial }}_T{\tilde{u}}_0+{\tilde{u}}_0{\mathrm{\partial }}_X{\tilde{u}}_0=0.$$

We write the general, implicit solution by characteristics in the form

$${\tilde{u}}_0(X,T)=f(T-\frac{X}{{\tilde{u}}_0}).$$

Matching to the inner solution, we have, for x>0,

$$\underset{X\to 0+}{lim}{\tilde{u}}_0(X,T)=f(T)=\underset{\xi \to \mathrm{\infty }}{lim}{u}_0(\xi ,T)=\frac{A}{1+(1/2)AT}.$$

The matching for x<0 is similar, giving an odd function for the outer solution,

$${\tilde{u}}_0=\frac{A\hspace{0.17em}\mathrm{sgn}(X)}{1+(1/2)A(T-X/{\tilde{u}}_0)}.$$

Solving for ${\tilde{u}}_0$, we find the leading-order outer solution

$$u\sim u_{\mathrm{out}}(X,T)=\frac{A((1/2)X+\mathrm{sgn}(X))}{1+(1/2)AT},|X|<AT.$$ 3.6

Continuous matching to the constant, far-field conditions, we obtain

$$u_{\mathrm{out}}(X,T)=\mathrm{sgn}(X)A,|X|\ge AT.$$ 3.7

(c). Uniformly valid asymptotic solution

Using the standard technique from asymptotics, we can formulate a composite solution that is asymptotically valid over the entire range of x. Based on the outer solution (3.6) and (3.7), we define

$$F(x,t,\epsilon )=\{\begin{array}{ll}-A,& x<-At,\\ \frac{A((1/2)\epsilon x+\mathrm{sgn}(\epsilon x))}{1+(1/2)A\epsilon t},& |x|<At,\\ A,& x>At.\end{array}$$

Then, the uniformly valid asymptotic solution is

$$u(x,t)=\frac{A}{1+(1/2)A\epsilon t}(\tanh \frac{x}{\epsilon }-\mathrm{sgn}(x))+F(x,t,\epsilon ).$$ 3.8

A comparison of the uniform asymptotic expansion and the numerical solution is shown in figure 1. The two solutions are hardly distinguishable. The insets of figure 1 reveal the smoothed, non-oscillatory nature of the dispersively regularized expansion shock. In contrast, typical dispersive shock waves are characterized by their oscillatory structure [13]. Figure 2 displays the absolute error. Note that the largest error occurs at the outermost edges of the rarefaction wave where the asymptotic solution has a weak discontinuity. The error in the inner solution is approximately ε³=10⁻³, which can be formally identified by going to higher-order terms in equation (3.2). In figure 3b, we show characteristics calculated from the outer solution (3.6) and (3.7) with ε=0.1 and A=1.

Figure 2.

Pointwise error between the uniform asymptotic expansion and numerical solution of figure 1 at t=100 with ε=0.1 and A=1.

Figure 3.

Characteristics for: (a) the compressive shock solution of equation (1.5) and (b) the expansion shock solution of equation (1.1). (Online version in colour.)

(d). Boussinesq equations

The Boussinesq equations, formulated in the 1870s [15], can take a variety of asymptotically equivalent forms [3]. While having the same level of accuracy as the KdV and BBM equations in reproducing dispersive shallow-water dynamics, the Boussinesq equations have the advantage of bidirectionality. The system considered here,

$$\begin{array}{rl}& h_t+{(uh)}_x=0,\\ & u_t+u{u}_x+h_x-\frac{1}{3}{u}_{xxt}=0,\end{array}\}$$ 3.9

is a reduced, normalized version of an equation that appeared in [3] (see also [18]), which includes a u_x term in the dynamical equation for h. The non-dimensional variables, h,u, represent the height of the water free surface above a flat horizontal bottom and the depth-averaged horizontal component of the water velocity, respectively. A stationary shock solution of this system

$$h(x,t)=\{\begin{array}{ll}{h}_{-},& x<0,\\ h_{+},& x>0,\end{array}u(x,t)=\{\begin{array}{ll}{u}_{-},& x<0,\\ u_{+},& x>0,\end{array}$$ 3.10

will satisfy RH jump conditions derived from the time-independent equations

$$h_{+}{u}_{+}=h_{-}{u}_{-};h_{+}+\frac{1}{2}{u}_{+}^2=h_{-}+\frac{1}{2}{u}_{-}^2.$$ 3.11

The RH conditions (3.11) can be expressed as the two-parameter loci of states

$$u_{\pm }=h_{\mp }{\left(\frac{2}{h_{-}+h_{+}}\right)}^{1/2},$$ 3.12

with arbitrary, positive total water depths h_±. In figure 4, we show the result of a numerical simulation demonstrating the persistence of a stationary shock wave for the Boussinesq system (3.9).

Figure 4.

Evolution of h (a) and u (b) in an expansion shock for the Boussinesq system with h₋=2, h₊=1 and u₋, u₊ given by equation (3.12). Jump initial data are smoothed by $\tanh (x/\epsilon ),$ as in equation (3.1), with ε=0.1. (Online version in colour.)

The characteristic speeds for the dispersionless system ((3.9) with u_xxt→0) are ${\lambda }_{\pm }(h,u)=u\pm \sqrt{h}$. Therefore, if u_±≥0 as in the loci (3.12), then the characteristics with speed λ₊ pass through the stationary shock from left to right. However, the characteristics with speed λ₋ leave the shock on both sides if $u_{-}<\sqrt{h_{-}}$ and $u_{+}>\sqrt{h_{+}}$. This is the case for the choices of h_±,u_± in figure 4. These choices also satisfy the RH conditions for a stationary shock (3.11). To see that similar data with u<0 can make a stationary shock expansive in the λ₊ characteristic family, note that the system (3.9) is unchanged under the transformation x→−x, u→−u.

We remark that the analytical treatment of the Boussinesq expansion shock appears to be more challenging than it was for BBM. For example, there is no clear means to separate variables in an inner solution owing to the non-zero mean values of h and u.

(e). Discussion

The expansion shock solutions we have discovered here do decay slowly in time, but their persistence in the face of the usual rules of causality is a surprise. For the BBM expansion shock, we can identify further robustness to perturbation by considering the asymmetric initial condition passing through zero

$$u(x,0)=\frac{1}{2}((u_{+}-u_{-})\tanh \left(\frac{x}{\epsilon }\right)+u_{+}+u_{-}),$$

where u₋<0<u₊. The numerical simulation of equation (1.1) with this asymmetric data is shown in figure 5. As t increases, the solution quickly develops a stationary expansion shock with initial amplitude $A=min\{u_{+},|u_{-}|\}$ that decays according to the inner solution (3.4). However, the solution also sheds a train of rank-ordered solitons.

Figure 5.

Expansion shock and solitons as components of the initial value problem u(x,0)=0.55 tanh(x/ε)−0.45 for equation (1.1) where ε=0.1. (Online version in colour.)

The expansion shock also persists in the presence of weak dissipation in the BBM–Burgers equation

$$u_t+u{u}_x=u_{xxt}+\nu u_{xx},$$ 3.13

where ν>0 is the dissipation coefficient. If we consider the initial data (3.1) for (3.13), then the inner solution exhibits exponential temporal decay

$$u_{in}(\xi ,T)=\frac{A{e}^{-\nu T/\epsilon }}{1+(A\epsilon /2\nu )(1-e^{-\nu T/\epsilon })}\tanh \xi .$$ 3.14

Thus, for $\nu =\mathcal{O}(\epsilon )$, the expansion shock decays exponentially as $t\to \mathrm{\infty },$ rather than the algebraic decay in the absence of dissipation. In fact, if ν≪ε, then (3.14) is asymptotically equivalent to (3.4).

The construction presented here can be generalized to higher-order nonlinearity f(u) and higher-order positive differential operators $\mathcal{L}$ in the form

$$\mathcal{L}[u_t]+f{(u)}_x=\nu u_{xx},$$ 3.15

so long as the non-evolutionary dispersive character is maintained. For example, $\mathcal{L}=1-{\mathrm{\partial }}_{xx}$ and f(u)=u⁴ or $\mathcal{L}=1+{\mathrm{\partial }}_{xxxx}$ and f(u)=u² admit expansion shock solutions that can be approximated with matched asymptotic methods.

Recalling that the original formulation of the BBM equation (1.1) was as a numerically advantageous shallow-water wave model [1] it is important to stress that ‘engineering’ the dispersion for mathematical or numerical convenience can lead to new, unintended phenomena, e.g. expansion shocks.

(f). Conclusion

We have identified decaying expansion shocks as robust solutions to conservation laws of non-evolutionary type that naturally arise in shallow-water theory. These models include versions of the well-known BBM and Boussinesq equations, which are weakly nonlinear models for unidirectional and bidirectional long-wave propagation, respectively, although, as written here, they are not asymptotically resolved. The requisite non-local dispersion in these models is not peculiar to shallow-water theory, occurring, for example, in a Buckley–Leverett equation with dynamic capillary pressure law [19], describing flow in a porous medium. Expansion shocks represent a new class of purely dispersive and diffusive–dispersive shock waves. An important open question is whether expansion shock solutions can be physically realized.

Appendix A. Numerical method

The numerical methods used here for both the BBM (1.1) and Boussinesq (3.9) equations incorporate a standard fourth-order Runge–Kutta timestepper (RK4) and a pseudo-spectral Fourier spatial discretization, similar to the method described in [12]. We briefly describe the method for BBM here.

We are interested in solutions u(x,t) to (1.1) that rapidly decay to the far-field boundary conditions $u(\pm \mathrm{\infty },t)=u_{\pm }$. The derivative v=u_x therefore rapidly decays to zero and satisfies

$$(1-{\mathrm{\partial }}_{xx})v_t+{(uv)}_x=0,$$ A 1

where $u(x,t)={\int }_{-L}^{x}v(y,t)\hspace{0.17em}\mathrm{d}y+u_{-}$. The Fourier transform (written $\hat{f}(k)$ with wavenumber k for a function f(x)) of equation (A.16) can therefore be written

$$\frac{\mathrm{d}}{\mathrm{d}t}\hat{v}=-\frac{\mathrm{i}k}{1+k^2}\widehat{uv}.$$ A 2

The term $\widehat{uv}$ is well defined, because the function uv is rapidly decaying. Suitable truncation of the spatial and Fourier domains turn equation (A.17) into a nonlinear system of ordinary differential equations, which we temporally evolve according to RK4. The computation of the nonlinear term in (A.17) is efficiently implemented using the fast Fourier transform (see [12] for further details). For BBM, (L,N,Δt)=(200,2¹⁵,0.01) (figures 1 and 2), (300,2¹⁵,0.01) (figure 5). For Boussinesq (L,N,Δt)=(100,2¹⁴,0.005).

Authors' contributions

All authors contributed equally to problem formulation, solution and manuscript writing.

Competing interests

The authors have no competing interests.

Funding

This work was supported by the Royal Society International Exchanges Scheme IE131353 (all authors), NSF CAREER DMS-1255422 (M.A.H.) and NSF DMS-1517291 (M.S.)