On Solution Regularity of the Two-Dimensional Radiation Transfer Equation and Its Implication on Numerical Convergence

Figures & data

Figure 1. One-dimensional mesh.

Figure 2. Flux distribution on the mesh $160\times 160.$

Figure 3. Flux error distribution on the mesh $160\times 160$ (DD).

Figure 4. Flux error distribution on the mesh $160\times 160$ (SD).

Table 1. Scalar flux $L^1$ errors and convergence rates of DD.

Mesh ($\mathit{N}\times \mathit{N}$)	Case 1		Case 2		Case 3		Case 4
	Error^1	Rate^2	Error	Rate	Error	Rate	Error	Rate
$10\times 10$	2.87E-03		3.59E-03		2.31E-03		9.29E-03
$20\times 20$	7.95E-04	1.85	1.01E-03	1.83	8.12E-04	1.51	2.56E-03	1.86
$40\times 40$	2.90E-04	1.45	3.73E-04	1.44	2.31E-04	1.82	5.89E-04	2.12
$80\times 80$	1.14E-04	1.35	1.44E-04	1.37	5.19E-05	2.15	1.37E-04	2.10
$160\times 160$	5.04E-05	1.17	6.32E-05	1.19	1.32E-05	1.97	3.53E-05	1.96
$320\times 320$	2.46E-05	1.03	3.06E-05	1.04	3.61E-06	1.87	9.39E-06	1.91
$640\times 640$	1.31E-05	0.91	1.63E-05	0.91	1.11E-06	1.71	2.70E-06	1.80
$1280\times 1280$	6.26E-06	1.07	7.76E-06	1.07	3.87E-07	1.51	8.51E-07	1.66

1. The $L^1$ error on each mesh grid is defined by averaging the absolute differences between the numerical scalar flux and reference solution on each cell.

2. The convergence rate $=\ln \left(\frac{\mathrm{Error}\left(N\times N\right)}{\mathrm{Error}\left(2N\times 2N\right)}\right)/\ln 2.$

Table 2. Scalar flux $L^1$ errors and convergence rates of SD.

Mesh ($\mathit{N}\times \mathit{N}$)	Case 1		Case 2		Case 3		Case 4
	Error^1	Rate^2	Error	Rate	Error	Rate	Error	Rate
$10\times 10$	1.40E-02		2.24E-02		2.58E-03		5.74E-02
$20\times 20$	7.83E-03	0.84	1.19E-02	0.92	1.88E-03	0.45	3.92E-02	0.55
$40\times 40$	4.18E-03	0.91	6.20E-03	0.94	1.08E-03	0.81	2.17E-02	0.85
$80\times 80$	2.20E-03	0.93	3.20E-03	0.96	5.95E-04	0.85	1.16E-02	0.90
$160\times 160$	1.14E-03	0.95	1.65E-03	0.96	3.14E-04	0.92	5.99E-03	0.95
$320\times 320$	5.90E-04	0.95	8.50E-04	0.95	1.62E-04	0.96	3.05E-03	0.98
$640\times 640$	3.06E-04	0.95	4.40E-04	0.95	8.23E-05	0.98	1.54E-03	0.99
$1280\times 1280$	1.60E-04	0.94	2.29E-04	0.94	4.15E-05	0.99	7.71E-04	0.99