Theory and Modern Applications
Table 1 The \(L_{\infty }\), \(L_{2}\) errors and temporal convergence orders with \(h = 1/1000\) for Example 1
α | τ | \(L_{\infty }\) error | Convergence order | \(L_{2}\) error | Convergence order |
|---|---|---|---|---|---|
0.1 | \(\frac{1}{10}\) | 6.1855e−007 | 4.2944e−007 | ||
\(\frac{1}{20}\) | 7.6955e−008 | 3.0068 | 5.3427e−008 | 3.0068 | |
\(\frac{1}{40}\) | 9.5964e−009 | 3.0034 | 6.6625e−009 | 3.0034 | |
\(\frac{1}{80}\) | 1.2013e−009 | 2.9979 | 8.3402e−010 | 2.9979 | |
0.3 | \(\frac{1}{10}\) | 1.8656e−006 | 1.2958e−006 | ||
\(\frac{1}{20}\) | 2.2978e−007 | 3.0213 | 1.5960e−007 | 3.0213 | |
\(\frac{1}{40}\) | 2.8517e−008 | 3.0104 | 1.9807e−008 | 3.0104 | |
\(\frac{1}{80}\) | 3.5512e−009 | 3.0054 | 2.4666e−009 | 3.0054 | |
0.5 | \(\frac{1}{10}\) | 2.8278e−006 | 1.9657e−006 | ||
\(\frac{1}{20}\) | 3.4372e−007 | 3.0404 | 2.3893e−007 | 3.0404 | |
\(\frac{1}{40}\) | 4.2395e−008 | 3.0193 | 2.9470e−008 | 3.0193 | |
\(\frac{1}{80}\) | 5.2617e−009 | 3.0103 | 3.6576e−009 | 3.0103 | |
0.7 | \(\frac{1}{10}\) | 2.9898e−006 | 2.0810e−006 | ||
\(\frac{1}{20}\) | 3.5699e−007 | 3.0661 | 2.4846e−007 | 3.0662 | |
\(\frac{1}{40}\) | 4.3664e−008 | 3.0314 | 3.0389e−008 | 3.0314 | |
\(\frac{1}{80}\) | 5.4024e−009 | 3.0148 | 3.7599e−009 | 3.0148 | |
0.9 | \(\frac{1}{10}\) | 1.6262e−006 | 1.1342e−006 | ||
\(\frac{1}{20}\) | 1.8607e−007 | 3.1276 | 1.2972e−007 | 3.1282 | |
\(\frac{1}{40}\) | 1.8607e−007 | 3.1276 | 1.2972e−007 | 3.1282 | |
\(\frac{1}{80}\) | 2.7640e−009 | 3.0228 | 1.9267e−009 | 3.0229 | |
0.98 | \(\frac{1}{10}\) | 5.1376e−007 | 3.5984e−007 | ||
\(\frac{1}{20}\) | 4.4162e−008 | 3.5402 | 3.0819e−008 | 3.5455 | |
\(\frac{1}{40}\) | 5.2578e−009 | 3.0703 | 3.6683e−009 | 3.0706 | |
\(\frac{1}{80}\) | 6.4300e−010 | 3.0316 | 4.4857e−010 | 3.0317 |