Finite difference method for treating the Doppler broadening of neutron cross sections

It is well known that Doppler-broadened cross sections satisfy the usual heat equation of the form δ2F/δx2 = δF/δt, where x is the variable in either energy or momentum space and t is proportional to the physical temperature in question. Given the initial condition F(x,O) for -infinity < x < infinite and the boundary condition F(infinity,t) = F (infinity,O); F(-infinity,t) = F(-infinity,O), the function can be calculated by use of finite difference methods. One inherent problem in using such an approach is that excessive numbers of mesh points in both x and t are generally required near the peaks of the resonances where the maxima of the function and its higher order derivatives occur. The problem is inevitable where the explicit or implicit scheme is used. An explicit finite difference scheme that is believed to be more suitable than the implicit method in treating the Doppler broadening of the resonance cross sections is described. Pertinent issues and inherent limitations of the method are also addressed