Theory and Modern Applications
Table 1 The errors of numerical solutions and rate of convergence at \(T=20\) using \(h=\tau \)
From: Novel advances in high-order numerical algorithm for evaluation of the shallow water wave equations
τ = h | ∥⋅∥ | Rate | \(\|\cdot \|_{\infty }\) | Rate | CPU time (s) | |
|---|---|---|---|---|---|---|
Scheme I | 0.5 | 5.4875 × 10−6 | – | 2.2094 × 10−6 | – | 0.0857 |
0.25 | 3.3976 × 10−7 | 4.0136 | 1.3672 × 10−7 | 4.0144 | 0.3099 | |
0.125 | 2.1207 × 10−8 | 4.0019 | 8.5434 × 10−9 | 4.0003 | 1.0363 | |
0.0625 | 1.3250 × 10−9 | 4.0004 | 5.3515 × 10−10 | 3.9968 | 8.8850 | |
Scheme II [30] | 0.5 | 1.8618 × 10−2 | – | 7.0602 × 10−3 | – | 0.0813 |
0.25 | 4.7664 × 10−3 | 1.9657 | 1.8134 × 10−3 | 1.9610 | 0.4885 | |
0.125 | 1.1996 × 10−3 | 1.9903 | 4.5739 × 10−4 | 1.9872 | 5.5283 | |
0.0625 | 3.0053 × 10−4 | 1.9970 | 1.1465 × 10−4 | 1.9962 | 47.0629 | |
0.5 | 1.4946 × 10−3 | – | 5.2670 × 10−4 | – | 0.0591 | |
0.25 | 1.3373 × 10−4 | 1.9808 | 5.1894 × 10−4 | 1.9777 | 0.3434 | |
0.125 | 9.5127 × 10−5 | 1.9930 | 3.3644 × 10−5 | 1.9909 | 2.4583 | |
0.0625 | 2.3828 × 10−5 | 1.9972 | 8.4277 × 10−6 | 1.9971 | 22.0431 |