Theory and Modern Applications
Table 2 \(L^{\infty }\)-norm errors and convergence orders of Ṽ obtained by the fourth-order scheme for Example 1
τ | h | α = 1.2 | α = 1.5 | α = 1.8 | α = 2.0 | ||||
|---|---|---|---|---|---|---|---|---|---|
\( \Vert v^{n}-\widetilde{V}^{n} \Vert _{\infty }\) | Order | \(\Vert v^{n}-\widetilde{V}^{n} \Vert _{\infty }\) | Order | \(\Vert v^{n}-\widetilde{V}^{n} \Vert _{\infty }\) | Order | \(\Vert v^{n}-\widetilde{V}^{n} \Vert _{\infty }\) | Order | ||
1/32 | 1/32 | 3.42e−07 | – | 5.19e−07 | – | 1.08e−06 | – | 1.29e−06 | – |
1/64 | 1/64 | 1.96e−08 | 4.12 | 3.04e−08 | 4.09 | 4.62e−08 | 4.55 | 5.69e−08 | 4.50 |
1/128 | 1/128 | 1.16e−09 | 4.08 | 1.78e−09 | 4.09 | 2.79e−09 | 4.05 | 4.10e−09 | 3.80 |
1/256 | 1/256 | 7.76e−11 | 3.90 | 1.05e−10 | 4.08 | 1.68e−10 | 4.05 | 2.49e−10 | 4.04 |
1/512 | 1/512 | 5.62e−12 | 3.79 | 6.23e−12 | 4.08 | 1.01e−11 | 4.05 | 1.56e−11 | 4.00 |