Theory and Modern Applications
Table 4 The solution errors of Example 4 for \(\varepsilon=10^{-3}\) and \(\varepsilon=10^{-4}\) at \(h=0.001\)
From: Absolutely stable difference scheme for a general class of singular perturbation problems
\(x_{i}\) | ε = 10−3 | ε = 10−4 | ||||
|---|---|---|---|---|---|---|
CFD | ASCD | CFD | ASCD | |||
| Â | \(\delta_{i}=\frac{4}{\omega|p_{i}|}\) | \(\delta_{i}=\delta_{\mathrm{op}}\) | Â | \(\delta_{i}=\frac{4}{\omega|p_{i}|}\) | \(\delta_{i}=\delta_{\mathrm{op}}\) | |
0.000 | 2.4425e − 15 | 0.0000e + 00 | 2.4425e − 15 | 2.2204e − 15 | 0.0000e + 00 | 0.0000e + 00 |
0.500 | 2.4425e − 15 | 0.0000e + 00 | 2.4425e − 15 | 2.2204e − 15 | 0.0000e + 00 | 0.0000e + 00 |
0.700 | 2.4425e − 15 | 0.0000e + 00 | 2.4425e − 15 | 2.2204e − 15 | 0.0000e + 00 | 0.0000e + 00 |
0.900 | 2.4425e − 15 | 0.0000e + 00 | 2.4425e − 15 | 2.2204e − 15 | 0.0000e + 00 | 0.0000e + 00 |
0.950 | 2.4425e − 15 | 0.0000e + 00 | 2.4425e − 15 | 1.5683e − 09 | 0.0000e + 00 | 0.0000e + 00 |
0.980 | 1.7744e − 09 | 2.0612e − 09 | 1.7744e − 09 | 3.0073e − 04 | 0.0000e + 00 | 0.0000e + 00 |
0.990 | 2.8465e − 05 | 4.5400e − 05 | 2.8465e − 05 | 1.7342e − 02 | 0.0000e + 00 | 0.0000e + 00 |
0.991 | 7.2605e − 05 | 1.2341e − 04 | 7.2605e − 05 | 2.6012e − 02 | 0.0000e + 00 | 0.0000e + 00 |
0.992 | 1.8305e − 04 | 3.3546e − 04 | 1.8305e − 04 | 3.9018e − 02 | 0.0000e + 00 | 0.0000e + 00 |
0.993 | 4.5463e − 04 | 9.1188e − 04 | 4.5463e − 04 | 5.8528e − 02 | 0.0000e + 00 | 0.0000e + 00 |
0.994 | 1.1070e − 03 | 2.4788e − 03 | 1.1070e − 03 | 8.7791e − 02 | 0.0000e + 00 | 0.0000e + 00 |
0.995 | 2.6227e − 03 | 6.7379e − 03 | 2.6227e − 03 | 1.3169e − 01 | 0.0000e + 00 | 0.0000e + 00 |
0.996 | 5.9700e − 03 | 1.8316e − 02 | 5.9700e − 03 | 1.9753e − 01 | 0.0000e + 00 | 0.0000e + 00 |
0.997 | 1.2750e − 02 | 4.9787e − 02 | 1.2750e − 02 | 2.9630e − 01 | 9.3592e − 14 | 9.3592e − 14 |
0.998 | 2.4224e − 02 | 1.3534e − 01 | 2.4224e − 02 | 4.4444e − 01 | 2.0612e − 09 | 2.0612e − 09 |
0.999 | 3.4546e − 02 | 3.6788e − 01 | 3.4546e − 02 | 6.6671e − 01 | 4.5400e − 05 | 4.5400e − 05 |
1.000 | 0.0000e + 00 | 0.0000e + 00 | 0.0000e + 00 | 2.2204e − 16 | 0.0000e + 00 | 0.0000e + 00 |