We analyze a fully discrete spectral method for the numerical solution of the initial- and periodic boundary-value problem for two nonlinear, nonlocal, dispersive wave equations, the Benjamin–Ono and the Intermediate Long Wave equations. The equations are

In this paper, we present several new algorithms for estimating the sensitivity of solutions of partial differential equations to certain parameters in the equations. There has been a great deal of work done on sensitivity analysis using direct linear algebra, but we are concerned with large-scale simulations so the use of direct linear algebra is not feasible. Therefore we have developed a new class of algorithms which use matrix-free methods to solve all the linear systems. We describe several of the direct algorithms and show how some of the ideas contained in them can be used to develop the matrix-free methods. Several numerical examples are given to compare the methods, and a convergence theory for the methods is presented.