Theory and Modern Applications
Table 2 The \(L_{\infty }\), \(L_{2}\) errors and spatial convergence orders with \(\tau = 1/1000\) for Example 1
α | h | \(L_{\infty }\) error | Convergence order | \(L_{2}\) error | Convergence order |
|---|---|---|---|---|---|
0.1 | \(\frac{1}{10}\) | 3.4339e−006 | 2.5064e−006 | ||
\(\frac{1}{20}\) | 2.1541e−007 | 3.9947 | 1.5705e−007 | 3.9963 | |
\(\frac{1}{40}\) | 1.3497e−008 | 3.9964 | 9.8209e−009 | 3.9992 | |
\(\frac{1}{80}\) | 8.4315e−010 | 4.0007 | 6.1352e−010 | 4.0007 | |
0.3 | \(\frac{1}{10}\) | 3.1758e−006 | 2.3248e−006 | ||
\(\frac{1}{20}\) | 1.9975e−007 | 3.9909 | 1.4568e−007 | 3.9962 | |
\(\frac{1}{40}\) | 1.2499e−008 | 3.9983 | 9.1095e−009 | 3.9993 | |
\(\frac{1}{80}\) | 7.7988e−010 | 4.0024 | 5.6827e−010 | 4.0027 | |
0.5 | \(\frac{1}{10}\) | 2.8749e−006 | 2.1134e−006 | ||
\(\frac{1}{20}\) | 1.8151e−007 | 3.9854 | 1.3245e−007 | 3.9960 | |
\(\frac{1}{40}\) | 1.1350e−008 | 3.9993 | 8.2822e−009 | 3.9993 | |
\(\frac{1}{80}\) | 7.0704e−010 | 4.0048 | 5.1598e−010 | 4.0046 | |
0.7 | \(\frac{1}{10}\) | 2.5405e−006 | 1.8696e−006 | ||
\(\frac{1}{20}\) | 1.6044e−007 | 3.9850 | 1.172e−007 | 3.9957 | |
\(\frac{1}{40}\) | 1.0032e−008 | 3.9994 | 7.3288e−009 | 3.9993 | |
\(\frac{1}{80}\) | 6.2455e−010 | 4.0056 | 4.5634e−010 | 4.0054 | |
0.9 | \(\frac{1}{10}\) | 2.1719e−006 | 1.5897e−006 | ||
\(\frac{1}{20}\) | 1.3618e−007 | 3.9954 | 9.9689e−008 | 3.9952 | |
\(\frac{1}{40}\) | 8.5366e−009 | 3.9957 | 6.2348e−009 | 3.9990 | |
\(\frac{1}{80}\) | 5.3234e−010 | 4.0032 | 3.8885e−010 | 4.0031 | |
0.98 | \(\frac{1}{10}\) | 2.0086e−006 | 1.4663e−006 | ||
\(\frac{1}{20}\) | 1.2597e−007 | 3.9950 | 9.1972e−008 | 3.9948 | |
\(\frac{1}{40}\) | 7.8801e−009 | 3.9987 | 5.7531e−009 | 3.9988 | |
\(\frac{1}{80}\) | 4.9259e−010 | 3.9998 | 3.5943e−010 | 4.0006 |