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

Abstract

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 largescale 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, semiimplicit, 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 operatorsplitting linearized-diffusion method is also presented.

A computationally-efficient, semi-implicit, iterative method for the time-integration of reacting flows with stiff chemistry

“…However, the associated efficiency gain is not significant and the loss of accuracy might be problematic given the large range of timescales in a reacting flow simulation. Alternatively, the non-linear chemical source terms may be expanded using some type of low-order expansion [19][20][21]. However, it has been shown that these methods may be subject to severe time step size restrictions and stability issues when applied to combustion simulations [19], since they do not fully account for the non-linearity of the system [20].…”

Coupling Schemes for Multiphysics Reactor Simulation

“…Some thoughts regarding time adaptation by these different methods and future endeavors necessary are discussed below. 4 …”

“…This test problem was previously introduced in [4]. Then, by assuming a particular solution for T (x), a suitable forcing function can be obtained by substituting in (9), a method more commonly known as the Method of Manufactured Solutions (MMS).…”

Stability of semidiscrete formulations for elastodynamics at small time steps

“…Indeed, procedures for time step selection often advise against reducing time steps far below the critical values for temporal stability [5, p. 510]. Nevertheless, small time steps are often necessary in practice, for example in problems of radiative transport [6] and fluid-structure interaction [7]. As the time step is reduced with a fixed mesh size, the deleterious effects of higher modes are inevitably admitted into the computation, even with algorithmic damping [8].…”