Theory and Modern Applications
Table 5 Solution errors of Example 5 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 | 4.2411e − 14 | 0.0000e + 00 | 4.2411e − 14 | 1.8874e − 14 | 0.0000e + 00 | 0.0000e + 00 |
0.500 | 5.0469e − 08 | 1.5141e − 04 | 5.0469e − 08 | 5.0537e − 08 | 1.2125e − 04 | 1.2125e − 04 |
0.700 | 5.7848e − 08 | 1.7354e − 04 | 5.7848e − 08 | 5.7926e − 08 | 1.3898e − 04 | 1.3898e − 04 |
0.900 | 6.0894e − 08 | 1.8267e − 04 | 6.0894e − 08 | 6.0976e − 08 | 1.4631e − 04 | 1.4631e − 04 |
0.950 | 6.1142e − 08 | 1.8341e − 04 | 6.1142e − 08 | 6.2717e − 08 | 1.4690e − 04 | 1.4690e − 04 |
0.980 | 5.9470e − 08 | 1.8361e − 04 | 5.9470e − 08 | 2.9484e − 04 | 1.4706e − 04 | 1.4706e − 04 |
0.990 | 2.8120e − 05 | 2.2859e − 04 | 2.8120e − 05 | 1.7169e − 02 | 1.4709e − 04 | 1.4709e − 04 |
0.991 | 7.1893e − 05 | 3.0595e − 04 | 7.1893e − 05 | 2.5779e − 02 | 1.4709e − 04 | 1.4709e − 04 |
0.992 | 1.8153e − 04 | 5.1643e − 04 | 1.8153e − 04 | 3.8708e − 02 | 1.4709e − 04 | 1.4709e − 04 |
0.993 | 4.5140e − 04 | 1.0892e − 03 | 4.5140e − 04 | 5.8119e − 02 | 1.4709e − 04 | 1.4709e − 04 |
0.994 | 1.1003e − 03 | 2.6476e − 03 | 1.1003e − 03 | 8.7266e − 02 | 1.4709e − 04 | 1.4709e − 04 |
0.995 | 2.6096e − 03 | 6.8880e − 03 | 2.6096e − 03 | 1.3103e − 01 | 1.4709e − 04 | 1.4709e − 04 |
0.996 | 5.9461e − 03 | 1.8426e − 02 | 5.9461e − 03 | 1.9674e − 01 | 1.4709e − 04 | 1.4709e − 04 |
0.997 | 1.2712e − 02 | 4.9822e − 02 | 1.2712e − 02 | 2.9541e − 01 | 1.4709e − 04 | 1.4709e − 04 |
0.998 | 2.4176e − 02 | 1.3525e − 01 | 2.4176e − 02 | 4.4356e − 01 | 1.4709e − 04 | 1.4709e − 04 |
0.999 | 3.4512e − 02 | 3.6770e − 01 | 3.4512e − 02 | 6.6605e − 01 | 1.0174e − 04 | 1.0174e − 04 |
1.000 | 0.0000e + 00 | 0.0000e + 00 | 0.0000e + 00 | 0.0000e + 00 | 0.0000e + 00 | 0.0000e + 00 |