The authors consider problems of nonlinear radiative transfer and their solutions obtained by the method of J. A. Fleck jun. and J. D. Cummings [ibid. 8, 313-342 (1971; Zbl 0229.65087)]. This method firstly consists in a time discretization; then in every time interval \(\Delta t_ i=t_{i+1}-t_ i\) the nonlinear transport process is approximated by a linear one, which is eventually handled by a standard linear transport Monte Carlo method.
It was recently proved that the solutions of these problems satisfy a maximum principle (that is, under suitable hypotheses, they satisfy, for every \(t>0\), some inequalities, if these are satisfied by the initial values). The authors observe that the approximate solutions obtained by the Fleck-Cummings method may violate this principle if the \(\Delta t_ i\) are not sufficiently small, and give an upper bound for the maximum \(\Delta t_ i\) so that the principle will be surely satisfied.
The solutions the authors consider are “ideal” solutions free from statistical errors and not any actual Monte Carlo solution. The paper contains also numerical applications from which it can be deduced that the theoretical bound is indeed much smaller than the value of max \(\Delta\) \(t_ i\) for which the violation of the principle may occur.