Hyperbolic approximation of the BBM equation

Abstract

It is well known that the Benjamin–Bona–Mahony (BBM) equation can be seen as the Euler–Lagrange equation for a Lagrangian expressed in terms of the solution potential. We approximate the Lagrangian by a two-parameter family of Lagrangians depending on three potentials. The corresponding Euler–Lagrange equations can be then written as a hyperbolic system of conservations laws. The hyperbolic BBM system has two genuinely nonlinear eigenfields and one linear degenerate eigenfield. Moreover, it can be written in terms of Riemann invariants. Such an approach conserves the variational structure of the BBM equation and does not introduce the dissipation into the governing equations as it usually happens for the classical relaxation methods. The state-of-the-art numerical methods for hyperbolic conservation laws such as the Godunov-type methods are used for solving the ‘hyperbolized’ dispersive equations. We find good agreement between the corresponding solutions for the BBM equation and for its hyperbolic counterpart.

Perfectly Matched Layers Methods for Mixed Hyperbolic–Dispersive Equations

“…Thus, the relaxation system (46) conserves the variational structure of the KdV equation which turns out to be important for good approximation properties with respect to the original equations. Such a method of extended Lagrangian transforming a dispersive system admitting a variational formulation into the hyperbolic system was successfully used for the Serre-Green-Naghdi equations [12,27], NLS equation and Euler-Korteweg equations [16], and BBM equation [34]. The method of extended Lagranian was, in particular, justified in [19] for the Serre-Green-Naghdi equations.…”

“…Denote v g and v φ respectively the group velocity and phase velocity associated to the system (33). A necessary condition of stability of (34)…”

Hyperbolicity study of the modulation equations for the Benjamin–Bona–Mahony equation

“…Figure 4 shows the numerical results, at time $$t=100$$, observing qualitative agreement of the results on the wave envelopes and the average state between the BBM bore solution and the modulation simple wave solution. Numerical results obtained using the hyperbolized BBM model (BBMH) 18 are included also where $$\hat{c} = c = 10^{3}$$ is used in the computation (see Ref. 18 for details).…”

“…Numerical results obtained using the hyperbolized BBM model (BBMH) 18 are included also where $$\hat{c} = c = 10^{3}$$ is used in the computation (see Ref. 18 for details). Here, $$\hat{c}$$ and c are two parameters of the hyperbolized model.…”

“…Numerical results for the initial‐boundary value problem of the Benjamin–Bona–Mahony (BBM) equation and the BBM modulation system. In the graph, the solid red line is the solution obtained using the BBM equation (), the blue solid line is the solution obtained using the hyperbolized BBM equation, 18 and the black dashed line is the solution using the BBM modulation system ().…”