The radiative transport equation with heterogeneous cross-sections
Provides theoretical guarantees for numerical methods and solution properties in radiative transport, relevant to nuclear engineering and random media, but the results are incremental extensions of known techniques.
The paper proves explicit norm estimates for the integral operator in the radiative transport equation with heterogeneous cross-sections, yielding convergence rate bounds for source iteration and solution estimates in Bochner norms for random cross-sections. It also provides an elementary proof that eigenvalues in a nuclear reactor safety problem are real and positive.
We consider the classical integral equation reformulation of the radiative transport equation (RTE) in a heterogeneous medium, assuming isotropic scattering. We prove an estimate for the norm of the integral operator in this formulation which is explicit in the (variable) coefficients of the problem (also known as the cross-sections). This result uses only elementary properties of the transport operator and some classical functional analysis. As a corollary, we obtain a bound on the convergence rate of source iteration (a classical stationary iterative method for solving the RTE). We also obtain an estimate for the solution of the RTE which is explicit in its dependence on the cross-sections. The latter can be used to estimate the solution in certain Bochner norms when the cross-sections are random fields. Finally we use our results to give an elementary proof that the generalised eigenvalue problem arising in nuclear reactor safety has only real and positive eigenvalues.