X-factor: A modified relaxation factor to accelerate the convergence rate of the radiative transfer equation with high-order resolution schemes using the Normalized Weighting-Factor method: A modified relaxation factor to accelerate the convergence rate of the radiative transfer equation with high-order resolution schemes using the Normalized Weighting-Factor method

In this paper the high computational cost problem by using high-order (HO) and high-resolution (HR) schemes is addressed and for that, we propose the incorporation of a modified relaxation factor to accelerate the numerical solution of the radiative transfer equation (RTE) using the Normalized Weighting-Factor (NWF) method to implement several high-order resolution schemes. The modified relaxation factor to accelerate the convergence rate is based on the artificial incorporation of a semi-implicit X-factor. This procedure is denoted as the X-factor method. The X-factor method is compared, in terms of computer time needed to obtain a converged solution, with the widely used deferred-correction (DC) method for the calculations of a two-dimensional cavity with emitting–absorbing–scattering gray media using the discrete ordinates method. Four parameters are considered to evaluate the purpose of this paper: the absorption coefficient, the emissivity of the boundary surface, the under-relaxation factor, and the scattering albedo. In general, the results showed that using the X-factor procedure there is superiority over the DC procedure for reducing the CPU time when the DIAMOND, QUICK, SMART and WACED schemes are used. Additionally, the results showed that the CPU time for the MUSCL scheme can be smaller than that obtained with the DC method. The absorption coefficient effect showed that the X-factor method provided a reduction of CPU time between 20 and 211%. The results of emissivity effect showed that the computational time decreases between 2 and 162% by using the X-factor procedure. Regarding the effect of the under-relaxation factor, the results showed that X-factor method provided reductions of the CPU time from 52 to 181%. Analogously, the results for the scattering albedo showed that the X-factor method reduced the CPU time by a factor ranging between 4 and 219%. Additionally, a second test case was presented and the results showed that our code produces the same solution considering the use of DC method, X-factor method and several high-order resolution schemes. Results clearly demonstrated the effectiveness of the X-factor method to reduce the CPU time and therefore, the X-factor method can potentially be used in commercial software and in-house codes due to the substantial reduction of the computational cost.