Theory and Modern Applications
Table 8 The \(L_{\infty }\), \(L_{2}\) errors and spatial convergence orders with \(\tau = 1/1000\) for Example 4
α | h | \(L_{\infty }\) error | Rate | \(L_{2}\) error | Rate |
|---|---|---|---|---|---|
0.2 | \(\frac{1}{10}\) | 1.5852e−006 | 1.2379e−006 | ||
\(\frac{1}{20}\) | 9.9075e−008 | 4 | 7.5769e−008 | 4.0301 | |
\(\frac{1}{40}\) | 6.1922e−009 | 4 | 4.6847e−009 | 4.0156 | |
\(\frac{1}{80}\) | 3.8703e−010 | 3.9999 | 2.9120e−010 | 4.0079 | |
0.4 | \(\frac{1}{10}\) | 1.6877e−006 | 1.3212e−006 | ||
\(\frac{1}{20}\) | 1.0548e−007 | 4 | 8.0871e−008 | 4.0301 | |
\(\frac{1}{40}\) | 6.5926e−009 | 4 | 5.0005e−009 | 4.0155 | |
\(\frac{1}{80}\) | 4.1219e−010 | 3.9995 | 3.1094e−010 | 3.9995 | |
0.6 | \(\frac{1}{10}\) | 1.8132e−006 | 1.4238e−006 | ||
\(\frac{1}{20}\) | 1.1333e−007 | 3.9999 | 8.7164e−008 | 4.0299 | |
\(\frac{1}{40}\) | 7.0834e−009 | 3.9999 | 5.3902e−009 | 4.0153 | |
\(\frac{1}{80}\) | 4.4314e−010 | 3.9986 | 3.3537e−010 | 4.0065 | |
0.8 | \(\frac{1}{10}\) | 1.9644e−006 | 1.5484e−006 | ||
\(\frac{1}{20}\) | 1.2278e−007 | 3.9999 | 9.4810e−008 | 4.0296 | |
\(\frac{1}{40}\) | 7.6745e−009 | 3.9999 | 5.8639e−009 | 4.0151 | |
\(\frac{1}{80}\) | 4.8065e−010 | 3.9970 | 3.6525e−010 | 4.0049 | |
0.98 | \(\frac{1}{10}\) | 2.1217e−006 | 1.6799e−006 | ||
\(\frac{1}{20}\) | 1.3261e−007 | 4 | 1.0288e−007 | 4.0293 | |
\(\frac{1}{40}\) | 8.2897e−009 | 3.9997 | 6.3642e−009 | 4.0148 | |
\(\frac{1}{80}\) | 5.1997e−010 | 3.9948 | 3.9697e−010 | 4.0029 |