On the numerical evaluation of a particular singular two-dimensional integral

Abstract:

We investigate the possibility of using two-dimensional Romberg integration to approximate integrals, over the square $0 \leqslant x$, $y \leqslant 1$, of integrand functions of the form $g(x,y)/(x - y)$ where $g(x,y)$ is, for example, analytic in x and y. We show that Romberg integration may be properly justified so long as it is based on a diagonally symmetric rule and function values on the singular diagonal, if required, are defined in a particular way. We also investigate the consequences of ignoring fhese function values (i.e. setting them to zero) in the context of such a calculation. We also derive the asymptotic expansion on which extrapolation methods can be based when $g(x,y)$ has a point singularity of a specified nature at the origin.