Theory and Modern Applications
Table 6 The solution errors of Example 6 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 | \(\delta_{i}=\frac{4}{\omega|p_{i}|}\) | \(\delta_{i}=\delta_{\mathrm{op}}\) | |
−1.00 | 2.2826e − 12 | 0.0000e + 00 | 2.2826e − 12 | 6.1286e − 13 | 0.0000e + 00 | 1.1351e − 13 |
−0.50 | 2.2826e − 12 | 0.0000e + 00 | 2.2826e − 12 | 6.1286e − 13 | 0.0000e + 00 | 1.1351e − 13 |
−0.10 | 5.3097e − 06 | 1.5654e − 03 | 5.3097e − 06 | 6.1286e − 13 | 3.0826e − 71 | 1.1351e − 13 |
−0.08 | 1.9540e − 05 | 1.1412e − 02 | 1.9540e − 05 | 6.1363e − 13 | 1.2212e − 15 | 1.1274e − 13 |
−0.06 | 4.4839e − 05 | 5.7780e − 02 | 4.4839e − 05 | 5.5249e − 10 | 1.9732e − 09 | 5.5177e − 10 |
−0.04 | 6.2362e − 05 | 2.0590e − 01 | 6.2362e − 05 | 4.5925e − 06 | 6.3342e − 05 | 4.5925e − 06 |
−0.02 | 4.6484e − 05 | 5.2709e − 01 | 4.6484e − 05 | 4.0526e − 04 | 4.5500e − 02 | 4.0526e − 04 |
−0.01 | 2.5503e − 05 | 7.5183e − 01 | 2.5503e − 05 | 6.0561e − 04 | 3.1731e − 01 | 6.0561e − 04 |
0.00 | 1.1580e − 12 | 0.0000e + 00 | 1.1580e − 12 | 3.1630e − 13 | 0.0000e + 00 | 4.9516e − 14 |
0.01 | 2.5503e − 05 | 7.5183e − 01 | 2.5503e − 05 | 6.0561e − 04 | 3.1731e − 01 | 6.0561e − 04 |
0.02 | 4.6484e − 05 | 5.2709e − 01 | 4.6484e − 05 | 4.0526e − 04 | 4.5500e − 02 | 4.0526e − 04 |
0.04 | 6.2362e − 05 | 2.0590e − 01 | 6.2362e − 05 | 4.5925e − 06 | 6.3342e − 05 | 4.5925e − 06 |
0.06 | 4.4839e − 05 | 5.7780e − 02 | 4.4839e − 05 | 5.5189e − 10 | 1.9732e − 09 | 5.5188e − 10 |
0.08 | 1.9540e − 05 | 1.1412e − 02 | 1.9540e − 05 | 4.8850e − 15 | 1.3323e − 15 | 1.1102e − 15 |
0.10 | 5.3097e − 06 | 1.5654e − 03 | 5.3097e − 06 | 3.5527e − 15 | 0.0000e + 00 | 2.2204e − 16 |
0.50 | 0.0000e + 00 | 0.0000e + 00 | 0.0000e + 00 | 8.8818e − 16 | 0.0000e + 00 | 0.0000e + 00 |
1.00 | 0.0000e + 00 | 0.0000e + 00 | 0.0000e + 00 | 0.0000e + 00 | 0.0000e + 00 | 0.0000e + 00 |