Theory and Modern Applications
Table 2 Errors and convergence orders of scheme (43)–(44) with \(h_{1}=h_{2}=\frac{\pi}{500}\), and an optimal step size ratio for τ and Δβ (Example 1)
τ | Δβ | \(\|e^{N} \|_{L^{2}}\) | Order | \(\| e^{N} \|_{L^{\infty}}\) | Order | CPU time |
|---|---|---|---|---|---|---|
1/100 | 1/10 | 0.0133 | – | 0.0093 | – | 3.7889 s |
1/400 | 1/20 | 0.0034 | 1.9678 | 0.0024 | 1.9542 | 44.8796 s |
1/1600 | 1/40 | 8.6411e−04 | 1.9762 | 6.0481e−04 | 1.9885 | 654.4187 s |
1/6400 | 1/80 | 2.1076e−04 | 2.0365 | 1.4751e−04 | 2.0357 | 2 h 48 m 8 s |