Table 4 Computed maximum order of convergence for Example 2 by the proposed method
\(\varepsilon \downarrow N \to\) | 64 | 128 | 256 | 512 | 1024 |
|---|---|---|---|---|---|
\(\alpha = \frac{1}{12},\,\,\beta = \frac{5}{12}\) | |||||
\(10^{ - \,1}\) | 3.5665 | 3.3988 | 3.2498 | 3.1433 | 3.0779 |
\(10^{ - \,3}\) | 2.9718 | 3.1592 | 3.2940 | 3.4012 | 3.5009 |
\(10^{ - \,6}\) | 2.8274 | 2.2105 | 2.3784 | 2.6767 | 3.0134 |
\(10^{ - \,9}\) | 2.6857 | 2.0409 | 2.0323 | 2.0363 | 2.0569 |
\(10^{ - \,12}\) | 2.6810 | 2.0354 | 2.0206 | 2.0123 | 2.0080 |
\(\alpha = \frac{1}{3},\,\,\beta = \frac{1}{6}\) | |||||
\(10^{ - \,1}\) | 1.9962 | 1.9991 | 1.9996 | 1.9999 | 2.0000 |
\(10^{ - \,3}\) | 1.4451 | 1.5150 | 1.5898 | 1.6559 | 1.7009 |
\(10^{ - \,6}\) | 1.4462 | 1.3090 | 1.8062 | 1.6579 | 1.7026 |
\(10^{ - \,9}\) | 1.4462 | 1.5215 | 1.5941 | 1.6580 | 1.7026 |
\(10^{ - \,12}\) | 1.4462 | 1.5215 | 1.5941 | 1.6580 | 1.7026 |