Theory and Modern Applications
Table 4 Errors and convergence orders of scheme (45)–(46) with an optimal step size ratio for τ, \(h_{1}\), \(h_{2}\) and Δβ (Example 1)
τ | \(h_{1}=h_{2}\) | Δβ | \(\|e^{N} \|_{L^{2}}\) | Order | \(\|e^{N} \|_{L^{\infty}}\) | Order | CPU time |
|---|---|---|---|---|---|---|---|
1/16 | π/4 | 1/2 | 0.0913 | – | 0.0641 | – | \( 0.0234~\mbox{s} \) |
1/256 | π/16 | 1/4 | 0.0056 | 4.0271 | 0.0040 | 4.0023 | \( 0.0729~\mbox{s} \) |
1/4096 | π/64 | 1/8 | 3.3948e−04 | 4.0440 | 2.4093e−04 | 4.0533 | \( 30.6810~\mbox{s} \) |