Solitary waves generated by bell-shaped initial conditions in the inviscid and viscous GRLW equations

Abstract

The objective of this paper is three-fold. First, the accuracy of three methods of lines that employ either piecewise analytical integration and yield exponential methods or three-point, fourth-order accurate finite difference approximations for the spatial derivatives, is assessed by comparing the numerical solutions with the exact one of the generalized regularized long wave (GRLW) equation. Second, the three-point, fourth-order accurate, compact method is used to determine the disintegration or breakup of bell-shaped initial conditions for the GRLW and viscous GRLW equations. Third, the dynamics of the GRLW and viscous GRLW equations is studied as a function of the amplitude and width of Gaussian and triangular initial conditions. It is shown that the compact operator method yields slightly more accurate results than an exponential method that treats the source terms with fourth-order accuracy. It is also shown that the disintegration of the bell-shaped initial conditions is characterized by an initial transient, steepening of the leading front and the formation of solitary waves in the absence of viscosity. The number, speed and distance between successive solitary waves are shown to increase as the dispersion parameter is decreased in accord with a linear stability analysis. The number and speed of solitary waves are also functions of the amplitude and width of the bell-shaped initial conditions, whereas the effect of the linear advective terms is to push the solitary waves towards the downstream boundary. In the presence of viscosity, it is shown that the amplitude of the leading solitary wave that results from the disintegration of the initial conditions, decreases as a function of time and the waves exhibit a curvature which is a function of the linear and nonlinear advective and dispersion terms, and the viscosity coefficient. For large viscosities, it is shown that the initial flow adjustment results in a steep leading front whose slope decreases as time increases and waves analogous to the ones observed in the viscous Burgers’ equation are formed. It is also shown that, for the GRLW equation, the number and speed of solitary waves that are formed after the break-up of the initial profile increase as the amplitude and width of the initial conditions is increased, and that Gaussian initial conditions result in faster and larger amplitude solitary waves than triangular ones. The long-term dynamics of the viscous GRLW equation is found to be nearly independent of the initial conditions.