Theory and Modern Applications
Table 3 The errors of numerical solutions and rate of convergence for the BBM–Burgers equation (54) at \(T=20\) using \(\tau =h^{2}\)
From: Novel advances in high-order numerical algorithm for evaluation of the shallow water wave equations
h | ∥⋅∥ | Rate | \(\|\cdot \|_{\infty }\) | Rate | CPU time (s) | |
|---|---|---|---|---|---|---|
Scheme I | 0.5 | 9.4642 × 10−4 | – | 4.0490 × 10−4 | – | 0.1944 |
0.25 | 5.8886 × 10−5 | 4.0065 | 2.5164 × 10−5 | 4.0081 | 1.1024 | |
0.125 | 3.6789 × 10−6 | 4.0006 | 1.5740 × 10−6 | 3.9989 | 8.2897 | |
0.0625 | 2.2991 × 10−7 | 4.0001 | 9.8361 × 10−8 | 4.0002 | 103.6436 | |
Scheme II [30] | 0.5 | 2.3023 × 10−3 | – | 8.1853 × 10−4 | – | 0.1838 |
0.25 | 1.4378 × 10−4 | 4.0012 | 5.1110 × 10−5 | 4.0013 | 2.0441 | |
0.125 | 8.9858 × 10−6 | 4.0000 | 3.1958 × 10−6 | 3.9994 | 41.0603 | |
0.0625 | 5.6162 × 10−7 | 4.0000 | 1.9970 × 10−7 | 4.0003 | 837.0102 | |
Scheme III | 0.5 | 5.8714 × 10−3 | – | 2.0247 × 10−3 | – | 0.1222 |
0.25 | 1.0506 × 10−3 | 2.4825 | 3.6048 × 10−4 | 2.4897 | 1.4217 | |
0.125 | 2.3683 × 10−4 | 2.1493 | 8.1093 × 10−5 | 2.1523 | 20.3022 | |
0.0625 | 5.7601 × 10−5 | 2.0397 | 1.9710 × 10−5 | 2.0407 | 360.4731 |