For every two independent non-negative identically distributed random variables X and Y with pdf equal to f(x) the  pdf of Z = abs(X-Y) equals 0 when x < 0 and equals 

2 \int_0^\infty f(y)f(x+y) dy 

when x > 0.

To the best of my knowledge, for log-normal random variables this integral cannot be calculated in a closed form, so in practice one needs to approximate it by some quadrature formula.

Thank you Vladimir Kadets . Could you please mention types of quadrature formulas?

I am not an expert in this. To the best of my knowledge the standard quadrature formulas like trapezoidal rule (see http://en.wikipedia.org/wiki/Trapezoidal_rule ) are applicable to integrals over finite intervals, so one must first find an appropriate approximation of the integral over the semi-axis by the integral over large interval [0, R] and then to apply a quadrature formula. Just a minute ago I have made a Google search with words

``quadrature formulas for improper integral'' and it gives a lot of references. For example,

 S.Haber, O Shisha:  Improper integrals, simple integrals, and numerical quadrature, Journal of Approximation Theory Volume 11, Issue 1, May 1974, Pages 1–15

http://www.sciencedirect.com/science/article/pii/0021904574900355

Thanks