Studies on the accuracy of time-integration methods for the radiation–diffusion equations

The governing equations for the radiation–diffusion approximation to radiative transport are a system of highly nonlinear, multiple time-scale, partial-differential equations. The numerical solution of these equations for very large-scale simulations is most often carried out using semi-implicit linearization or operator-splitting techniques. These techniques do not fully converge the nonlinearities of the system so as to reduce the cost and complexity of the transient solution at each time step. For a given time-step size, this process exchanges temporal accuracy for computational efficiency. This study considers the temporal-accuracy issue by presenting detailed numerical-convergence studies for problems related to radiation–diffusion simulations. In this context a particular spatial discretization based on a Galerkin finite-element technique is used. The time-integration methods that we consider include: fully implicit, semi-implicit, and operator-splitting techniques. Results are presented for the relative accuracy and the asymptotic order of accuracy of the various methods. The results demonstrate both first-order and second-order asymptotic order of accuracy for the fully implicit, semi-implicit, and the operator-splitting schemes. Additionally a second-order operator-splitting linearized-diffusion method is also presented.