Theory and Modern Applications
Table 4 Error comparison at different time levels with parameters \((h, \Delta x)=(0.01,1/40)\) and \((\alpha _{1},\alpha _{2},\beta _{1},\beta _{2},\nu )=(2.0,2.0,0.2,0.8,1.0)\) for u and v on \(\Omega =[-20,20]\)
From: Algorithms for coupled Burgers’ equations by sharing characteristic curves within BSLM
t | \(\kappa _{0} = -1.0\) | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
CBSLM3 | SC-SL3 | SC-SL2 | CBSLM2 [2] | |||||||||
\(\mathrm{E}_{\infty}(u)\) | \(\mathrm{E}_{\infty}(v)\) | CPU | \(\mathrm{E}_{\infty}(u)\) | \(\mathrm{E}_{\infty}(v)\) | CPU | \(\mathrm{E}_{\infty}(u)\) | \(\mathrm{E}_{\infty}(v)\) | CPU | \(\mathrm{E}_{\infty}(u)\) | \(\mathrm{E}_{\infty}(v)\) | CPU | |
1.0 | 5.18e–9 | 1.41e–9 | 0.778 | 5.17e–9 | 2.01e–9 | 0.429 | 4.17e–8 | 1.12e–8 | 0.430 | 1.05e–6 | 2.94e–7 | 0.461 |
2.0 | 7.79e–9 | 2.01e–9 | 1.599 | 7.74e–9 | 3.31e–9 | 0.869 | 5.88e–8 | 1.77e–8 | 0.891 | 1.43e–6 | 4.27e–7 | 0.821 |
3.0 | 9.51e–9 | 2.37e–9 | 2.378 | 9.41e–9 | 4.32e–9 | 1.319 | 6.95e–8 | 2.27e–8 | 1.350 | 1.61e–6 | 5.17e–7 | 1.167 |
5.0 | 1.17e–8 | 2.84e–9 | 3.898 | 1.14e–8 | 5.86e–9 | 2.153 | 8.37e–8 | 3.07e–8 | 2.212 | 1.77e–6 | 6.61e–7 | 1.934 |
7.0 | 1.31e–8 | 3.18e–9 | 5.513 | 1.27e–8 | 7.09e–9 | 3.093 | 9.49e–8 | 3.78e–8 | 3.105 | 1.88e–6 | 7.98e–7 | 2.717 |
9.0 | 1.43e–8 | 3.49e–9 | 7.022 | 1.37e–8 | 8.18e–9 | 3.922 | 1.05e–7 | 4.45e–8 | 3.933 | 1.99e–6 | 9.36e–7 | 3.453 |
t | \(\kappa _{0} = -2.0\) | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
CBSLM3 | SC-SL3 | SC-SL2 | CBSLM2 [2] | |||||||||
\(\mathrm{E}_{\infty}(u)\) | \(\mathrm{E}_{\infty}(v)\) | CPU | \(\mathrm{E}_{\infty}(u)\) | \(\mathrm{E}_{\infty}(v)\) | CPU | \(\mathrm{E}_{\infty}(u)\) | \(\mathrm{E}_{\infty}(v)\) | CPU | \(\mathrm{E}_{\infty}(u)\) | \(\mathrm{E}_{\infty}(v)\) | CPU | |
1.0 | 1.36e–6 | 3.33e–7 | 0.768 | 1.34e–6 | 6.43e–7 | 0.430 | 9.78e–6 | 3.45e–6 | 0.446 | 5.36e–5 | 1.86e–5 | 0.401 |
2.0 | 1.74e–6 | 4.23e–7 | 1.590 | 1.67e–6 | 9.70e–7 | 0.873 | 1.28e–5 | 5.29e–6 | 0.885 | 6.02e–5 | 2.72e–5 | 0.815 |
3.0 | 2.02e–6 | 5.04e–7 | 2.335 | 1.89e–6 | 1.23e–6 | 1.295 | 1.56e–5 | 6.99e–6 | 1.325 | 6.67e–5 | 3.61e–5 | 1.160 |
5.0 | 2.64e–6 | 6.85e–7 | 4.011 | 2.34e–6 | 1.69e–6 | 2.190 | 2.25e–5 | 1.04e–5 | 2.236 | 8.18e–5 | 5.44e–5 | 1.955 |
7.0 | 3.41e–6 | 9.15e–7 | 5.469 | 2.91e–6 | 2.10e–6 | 3.052 | 3.16e–5 | 1.38e–5 | 3.069 | 9.83e–5 | 7.21e–5 | 2.695 |
9.0 | 4.23e–6 | 1.19e–6 | 7.081 | 3.55e–6 | 2.50e–6 | 3.915 | 4.09e–5 | 1.72e–5 | 3.918 | 1.17e–4 | 8.90e–5 | 3.485 |
t | \(\kappa _{0} = 1.0\) | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|
CBSLM3 | SC-SL3 | SC-SL2 | CBSLM2 [2] | |||||||||
\(\mathrm{E}_{\infty}(u)\) | \(\mathrm{E}_{\infty}(v)\) | CPU | \(\mathrm{E}_{\infty}(u)\) | \(\mathrm{E}_{\infty}(v)\) | CPU | \(\mathrm{E}_{\infty}(u)\) | \(\mathrm{E}_{\infty}(v)\) | CPU | \(\mathrm{E}_{\infty}(u)\) | \(\mathrm{E}_{\infty}(v)\) | CPU | |
1.0 | 5.18e–9 | 1.41e–9 | 0.797 | 5.17e–9 | 2.01e–9 | 0.442 | 4.17e–8 | 1.12e–8 | 0.443 | 1.05e–6 | 2.94e–7 | 0.442 |
2.0 | 7.79e–9 | 2.01e–9 | 1.577 | 7.74e–9 | 3.31e–9 | 0.864 | 5.88e–8 | 1.77e–8 | 0.869 | 1.43e–6 | 4.27e–7 | 0.790 |
3.0 | 9.51e–9 | 2.37e–9 | 2.328 | 9.41e–9 | 4.32e–9 | 1.324 | 6.95e–8 | 2.27e–8 | 1.327 | 1.61e–6 | 5.17e–7 | 1.236 |
5.0 | 1.17e–8 | 2.84e–9 | 3.891 | 1.14e–8 | 5.86e–9 | 2.203 | 8.37e–8 | 3.07e–8 | 2.207 | 1.77e–6 | 6.61e–7 | 1.977 |
7.0 | 1.31e–8 | 3.18e–9 | 5.433 | 1.27e–8 | 7.09e–9 | 3.056 | 9.49e–8 | 3.78e–8 | 3.086 | 1.88e–6 | 7.98e–7 | 2.713 |
9.0 | 1.43e–8 | 3.49e–9 | 7.127 | 1.37e–8 | 8.18e–9 | 3.947 | 1.05e–7 | 4.45e–8 | 3.961 | 1.99e–6 | 9.36e–7 | 3.464 |