Nonlinear stability and convergence of finite-difference methods for the ?good? Boussinesq equation

Abstract

Summary. The "good" Boussinesq equation utt = -u .... + Uxx + (u2)~x has recently been found to possess an interesting soliton-interaction mechanism. In this paper we study the nonlinear stability and the convergence of some simple finite-difference schemes for the numerical solution of problems involving the "good" Boussinesq equation. Numerical experiments are also reported. I IntroductionIt has recently been discovered [7] that the interactions of solitary-wave solutions of the "good" Boussinesq equationobey a highly interesting mechanism. The analytic expression of such solutions iswhere 40 and P> 0 are free real parameters and the amplitude A and velocity c of the wave are related to P through the formulasc= +I//I-P 2.Note that 40 determines the initial position of the wave, and that, due to the square root in (1.2), the parameter P can only take values in 0<P=< 1. Thus, the solitary waves (1.1) only exist for a finite range of velocities -l<c<l. Of course, a positive (respectively negative) velocity corresponds to a wave moving to the right (respectively to the left). When two solitary waves with parameters P1 and P2 are initially well separated and approach each other, a nonlinear interaction takes place. If PI and P2 are of moderate size, the incoming waves emerge of the interaction without Offprint requests to: T. Ortega

Pseudospectral method for the ‘‘good” Boussinesq equation

“…-j<p<j Differentiation of (2.5) with respect to x and evaluation at the mesh points lead to the following definition of the standard second difference pseudospectral operator D mapping Zh into itself: The weights 1/4, 1/2, 1/4 of 7>V+1, flV, D4Vn~x in (2.8) have been chosen for well-known accuracy reasons, but clearly other choices of weights are possible and can be analyzed by using the techniques below (cf. [11]). Also note that the time-continuous version of (2.8) is appealing when combined with the use of a standard ODE package.…”

“…In [11] two of the present authors have shown the nonlinear stability and convergence of a family of finite difference schemes for the numerical solution of the GB equation. While these schemes may provide a useful integration method when high accuracy is not required, finite difference algorithms are often (see, e.g., [15,17]) judged not to be competitive with their spectral and pseudospectral counterparts.…”

“…We employ the technique of the proof of Proposition 3.1 of [11]. The first inequality in (3.7) is obvious.…”

“…For the implications of this localized definition of stability, see [7,8,3]. For applications of the notion of stability threshold to other discretizations, see [13,4,11].…”

“…To simplify the structure of this matrix, (5.2) was implemented with the boundary conditions u = ux = 0 at x = xL, xR, rather than with periodicity conditions. This is a standard practice in this sort of problem (cf., e.g., the experiments in [11]). The method (5.2) is clearly second-order accurate in space and time.…”