1 . Nonlinearity , numerics and propagation of information

1. Introduction. In the study of evolution equations that describe the dynamics of natural and man-made systems, it is always useful to determine the way in which information is propagated by the said equations. In other words, the manner in which different scales present in the solution of an evolution equation travel and decay through space and time. The ideal tool to determine the propagation properties of (continuous or discrete) evolution equations is Fourier or harmonic analysis. In the case of continuous systems, the study of propagation properties allows the understanding of their stability. On the other hand, much insight regarding the behavior of discrete approximations of partial differential equations may be gained by comparing the propagation properties of a continuous equation and its corresponding discrete analogue. Thus, so-called amplitude and phase portraits that respectively depict the ratio of numerical and analytical amplification factor amplitudes and the difference between analytical and numerical phases, both as functions of wavenumber, may be developed (see, for example, Abbot [1] and Vichenevsky and Bowles [17]). These portraits show in a very objective way the effects of " numerical diffusion " and " numerical dispersion " associated to each wave number. Furthermore, the determination of the stability of numerical approximations may be viewed as a by-product of their amplitude propagation properties. Interestingly enough, a similar approach may be applied to study of the convergence properties of iterative schemes for the solution of systems of equations, a fact that has been exploited by the champions of the multigrid approach (see, for instance, [9]). The author and his collaborators have demonstrated the power of Fourier techniques in the study of the propagation properties of non-orthodox approximations of the linear transport equation, via least-squares collocation (Bentley et al., [10]) and the Eulerian-Lagrangian localized adjoint method (Aldama and Arroyo, [6]). Moreover, they have established the existence of an ordinary differential analogy that simplifies the determination of the stability conditions for high order time discretizations of the linear transport equation (Aldama, [3], and Aldama and Aparicio, [5]). Finally, they have studied the convergence properties of a semi-iterative scheme for the solution of a coupled diffusion-reaction system that describes the decay of argon in rocks and minerals (Lee and Aldama, [15]). Unfortunately, the application of Fourier methods is limited to linear and constant coefficient equations, subject to periodic boundary conditions or to linear and constant coefficient pure initial value problems occurring …