Investigations of nonstandard, Mickens‐type, finite‐difference schemes for singular boundary value problems in cylindrical or spherical coordinates

Abstract

Abstract It is well known that standard finite‐difference schemes for singular boundary value problems involving the Laplacian have difficulty capturing the singular (𝒪(1/ r ) or 𝒪(log r )) behavior of the solution near the origin ( r = 0). New nonstandard finite‐difference schemes that can capture this behavior exactly for certain singular boundary value problems encountered in theoretical aerodynamics are presented here. These schemes are special cases of nonstandard finite differences which have been extensively researched by Professor Ronald E. Mickens of Clark Atlanta University in their most general form. Several examples of these “Mickens‐type” finite differences that illustrate both their accuracy and utility for singular boundary value problems in both cylindrical and spherical co‐ordinates are investigated. The numerical results generated by the Mickens‐type schemes are compared favorably with solutions obtained from standard finite‐difference schemes. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 380–398, 2003.