Theory and Modern Applications
Table 1 The solution errors of Example 1 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 | 6.0951e − 14 | 0.0000e + 00 | 6.0951e − 14 | 1.0592e − 13 | 0.0000e + 00 | 0.0000e + 00 |
0.001 | 2.1803e − 02 | 2.3236e − 01 | 2.1803e − 02 | 4.2100e − 01 | 1.1841e − 04 | 1.1841e − 04 |
0.002 | 1.5273e − 02 | 8.5511e − 02 | 1.5273e − 02 | 2.8036e − 01 | 1.4708e − 04 | 1.4708e − 04 |
0.003 | 8.0307e − 03 | 3.1542e − 02 | 8.0307e − 03 | 1.8672e − 01 | 1.4708e − 04 | 1.4708e − 04 |
0.004 | 3.7564e − 03 | 1.1708e − 02 | 3.7564e − 03 | 1.2436e − 01 | 1.4708e − 04 | 1.4708e − 04 |
0.005 | 1.6486e − 03 | 4.4190e − 03 | 1.6486e − 03 | 8.2822e − 02 | 1.4708e − 04 | 1.4708e − 04 |
0.006 | 6.9512e − 04 | 1.7401e − 03 | 6.9512e − 04 | 5.5160e − 02 | 1.4708e − 04 | 1.4708e − 04 |
0.007 | 2.8515e − 04 | 7.5553e − 04 | 2.8515e − 04 | 3.6736e − 02 | 1.4708e − 04 | 1.4708e − 04 |
0.008 | 1.1466e − 04 | 3.9370e − 04 | 1.1466e − 04 | 2.4466e − 02 | 1.4708e − 04 | 1.4708e − 04 |
0.009 | 4.5397e − 05 | 2.6072e − 04 | 4.5397e − 05 | 1.6295e − 02 | 1.4707e − 04 | 1.4707e − 04 |
0.010 | 1.7743e − 05 | 2.1185e − 04 | 1.7743e − 05 | 1.0852e − 02 | 1.4707e − 04 | 1.4707e − 04 |
0.020 | 5.9987e − 08 | 1.8343e − 04 | 5.9987e − 08 | 1.8638e − 04 | 1.4705e − 04 | 1.4705e − 04 |
0.050 | 6.1017e − 08 | 1.8322e − 04 | 6.1017e − 08 | 6.2155e − 08 | 1.4689e − 04 | 1.4689e − 04 |
0.100 | 6.0767e − 08 | 1.8247e − 04 | 6.0767e − 08 | 6.0963e − 08 | 1.4629e − 04 | 1.4629e − 04 |
0.300 | 5.7716e − 08 | 1.7332e − 04 | 5.7716e − 08 | 5.7913e − 08 | 1.3896e − 04 | 1.3896e − 04 |
0.500 | 5.0343e − 08 | 1.5119e − 04 | 5.0343e − 08 | 5.0524e − 08 | 1.2123e − 04 | 1.2123e − 04 |
1.000 | 0.0000e + 00 | 0.0000e + 00 | 0.0000e + 00 | 0.0000e + 00 | 0.0000e + 00 | 0.0000e + 00 |