Sloan iteration and Richardson extrapolation for Walsh series solutions of integral equations

Abstract

Algorithms have been developed by the authors and co-workers for the solution of both Volterra and Fredholm integral equations by using the discontinuous wavelet packets known as the Walsh functions. These Walsh function methods are typically globally convergent of order one (and locally of order two). The usual Walsh function method with m = 2 n terms approximates the solution with a piecewise constant function (constant on each sub-interval of width 1/m). For Fredholm integral equations, Sloan iteration of the Walsh series solution is globally convergent of order two. For linear Fredholm integral equations of the second kind, we show in this paper that the Sloan iterates for Walsh series solution with m and 2m terms can be extrapolated by the Richardson method to give a function which approximates the solution with global fourth order convergence. This is very easy and efficient to implement. A second Richardson iteration for smooth problems results in sixth order convergence. With Kulkarni's example (where the solution is only twice differentiable), a second Richardson iteration results in fifth order convergence. Our new method outperforms Kulkarni's method: it is easier implement, more accurate with 64 or more terms and the order of convergence is one higher. As a further example, we use the nonlinear Chandrasekar integral equation for which a new solution method was recently proposed by the authors. Here, our numerical experiment shows that the Richardson extrapolation of the Sloan iteration functions is third order convergent. The Walsh function methods are intuitively simple and robust. The straightforward implementation of Walsh series leads to schemes with low order convergence. However the implementation of the acceleration of convergence techniques described here is easy and efficient and provides schemes of higher order convergence.