Abstract
A generalized viscous Rosenau equation containing linear and nonlinear advective terms and mixed third- and fifth-order derivatives is studied numerically by means of an implicit second-order accurate method in time that treats the first-, second-, and fourth-order spatial derivatives as unknown and discretizes them by means of three-point, fourth-order accurate, compact finite differences. It is shown that the effect of the viscosity is to decrease the amplitude, curve the wave trajectory, and increase the number and width of the waves that emerge from an initial Gaussian condition, whereas the linear convective term pushes the wave front towards the downstream boundary. It is also shown that the effect of the nonlinear convective term is to increase the steepness of the leading wave front and the number of sawtooth waves that are generated behind it, while that of the first dispersive term is to increase the number of waves that break up from the initial condition as the coefficient that characterizes this term is decreased. It is also shown that, for reasons of stability, the second dispersion coefficient must be much smaller than the first one and its effects on wave propagation are relatively small.
Highlights
In his 1986 and 1988 papers, Rosenau [1, 2] developed a formalism to treat the dynamics of discrete dense systems that can deal with wave-wave and wave-wall interactions that cannot be treated with the Korteweg-de Vries (KdV) equation
As indicated at the beginning of this section, for the initial Gaussian conditions considered in this study, the nonlinear convective terms result in the formation of a shock wave for μ = δ = γ = 0 and a very steep front for small μ ≠ 0 and δ = γ = 0; it may be concluded that the sawtooth waves observed for small value of δ are caused by the fifth-order derivative term that appears in (6)
The linear advective term was found to push the waves towards either the downstream or upstream boundaries, depending on its direction, whereas the nonlinear advective terms cause a steepening of the leading front and the formation of sawtooth waves and a radiation tail
Summary
In his 1986 and 1988 papers, Rosenau [1, 2] developed a formalism to treat the dynamics of discrete dense systems that can deal with wave-wave and wave-wall interactions that cannot be treated with the Korteweg-de Vries (KdV) equation. To an exact solution of (3) and ux(0, x) is negative, the leading part of the initial condition steepens due to the nonlinear advective term and, in the absence of dispersion and diffusion, would result in the formation of a shock wave [16, 17]. A generalized viscous Rosenau equation that includes linear and nonlinear advective terms is studied numerically by means of a second-order accurate, linearized Crank-Nicolson method and three-point, fourth-order accurate, compact operator discretizations for the first-, second-, and fourth-order spatial derivatives. The fourth section presents an exhaustive numerical study of the effects of the linear and nonlinear advective, viscous, and dispersive terms and initial conditions on wave breakup, generation, and propagation. A short concluding section summarizes the most important findings reported in the paper
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