Theory and Modern Applications
Table 1 \(L^{\infty }\)-norm errors and convergence orders of Ũ obtained by the fourth-order scheme for Example 1
τ | h | α = 1.2 | α = 1.5 | α = 1.8 | α = 2.0 | ||||
|---|---|---|---|---|---|---|---|---|---|
\( \Vert u^{n}-\widetilde{U}^{n} \Vert _{\infty }\) | Order | \(\Vert u^{n}-\widetilde{U}^{n} \Vert _{\infty }\) | Order | \(\Vert u^{n}-\widetilde{U}^{n} \Vert _{\infty }\) | Order | \(\Vert u^{n}-\widetilde{U}^{n} \Vert _{\infty }\) | Order | ||
1/32 | 1/32 | 1.61e−08 | – | 2.83e−08 | – | 1.05e−07 | – | 1.68e−07 | – |
1/64 | 1/64 | 8.78e−10 | 4.20 | 1.81e−09 | 3.97 | 2.66e−09 | 5.31 | 3.15e−09 | 5.74 |
1/128 | 1/128 | 5.75e−11 | 3.93 | 1.05e−10 | 4.11 | 1.48e−10 | 4.16 | 2.24e−10 | 3.81 |
1/256 | 1/256 | 3.93e−12 | 3.87 | 6.14e−12 | 4.09 | 8.99e−12 | 4.05 | 1.30e−11 | 4.11 |
1/512 | 1/512 | 2.67e−13 | 3.88 | 3.63e−13 | 4.08 | 5.40e−13 | 4.06 | 8.07e−13 | 4.01 |