A Frequency-Domain Approach for the $$\hbox {P}_1$$ P 1 Approximation of Time-Dependent Radiative Transfer

Abstract

We propose a new frequency-domain method to solve the simplified \(\text {P}_1\) approximation of time-dependent radiative transfer equations. The method employs the Fourier transform and consists of two stages. In the first stage the equations are transformed into an elliptic problem for the frequency variables. The numerical solutions of this problem are approximated using a Galerkin projection method based on the tensor-product B-spline interpolants. In the second stage a Gauss–Hermite quadrature procedure is proposed for the computation of the inverse Fourier transform to recover the numerical solutions of the original simplified \(\text {P}_1\) problem. The method avoids the discretization of the time variable in the considered system and it accurately resolves all time scales in radiative transfer regimes. Several test examples are used to verify high accuracy, effectiveness and good resolution properties for smooth and discontinuous solutions.