The Poisson equation with local nonregular similarities

Moffatt and Duffy [1] have shown that the solution to the Poisson equation, defined on rectangular domains, includes a local similarity term of the form: r²log(r)cos(2θ). The latter means that the second (and higher) derivative of the solution with respect to r is singular at r = 0. Standard high-order numerical schemes require the existence of high-order derivatives of the solution. Thus, for the case considered by Moffatt and Duffy, the high-order finite-difference schemes loose their high-order convergence due to the nonregularity at r = 0. In this article, a simple method is outlined to regain the high-order accuracy of these schemes, without the need of any modification in the scheme's algorithm. This is a significant consideration when one wants to use a given finite-difference computer code for problems with local nonregular similarity solutions. Numerical examples using the modified scheme in conjunction with a sixth-order finite difference approximation are provided.