Nonlinear variants of the TR/BDF2 method for thermal radiative diffusion

We apply the Trapezoidal/BDF2 (TR/BDF2) temporal discretization scheme to nonlinear grey radiative diffusion. This is a scheme that is not well-known within the radiation transport community, but we show that it offers many desirable characteristics relative to other second-order schemes. Several nonlinear variants of the TR/BDF2 scheme are defined and computationally compared with the Crank–Nicholson scheme. It is found for our test problems that the most accurate TR/BDF2 schemes are those that are fully iterated to nonlinear convergence, but the most efficient TR/BDF2 scheme is one based upon a single Newton iteration. It is also shown that neglecting the contributions to the Jacobian matrix from the cross-sections, which is often done due to a lack of smooth interpolations for tabular cross-section data, has a significant impact upon efficiency.