Table 1
Comparison between the methods tsRK4(4,4,4) and ARS3(4,4,3) for the problem (22), with α = 2/3. Time steps Δt = 2π/m and integration intervals [0, 2πN].
| m | N | error tsRK4(4,4,4) | error ARS3(4,4,3) |
|---|---|---|---|
| 5 | 5 | 8.7501e-02 | 6.6770e-01 |
| 10 | 5 | 6.4467e-03 | 1.2622e-01 |
| 20 | 5 | 4.2897e-04 | 1.6895e-02 |
| 40 | 5 | 2.7854e-05 | 2.1340e-03 |
| 5 | 10 | 1.8045e-01 | 9.1760e-01 |
| 10 | 10 | 1.3314e-02 | 2.4161e-01 |
| 20 | 10 | 8.7283e-04 | 3.4335e-02 |
| 40 | 10 | 5.5842e-05 | 4.3733e-03 |
| 5 | 20 | 3.5877e-01 | 1.0068e+00 |
| 10 | 20 | 2.7080e-02 | 4.2989e-01 |
| 20 | 20 | 1.7635e-03 | 6.8352e-02 |
| 40 | 20 | 1.1197e-04 | 8.8442e-03 |
Table 2
Comparison between the methods tsRK4(4,4,4) and ARS3(4,4,3) for the problem (24), with ω = 100, ε = 0.05, time steps Δt =2π/m, integration intervals [0, 2πN] and error = |un – u(2πN)|, with n = mN.
| m | N | error tsRK4(4,4,4) | error ARS3(4,4,3) |
|---|---|---|---|
| 10 | 10 | 2.2533e-01 | 6.7569e-01 |
| 20 | 10 | 1.5140e-02 | 1.1932e-01 |
| 40 | 10 | 1.0841e-03 | 1.5515e-02 |
| 80 | 10 | 4.7040e-04 | 2.2383e-03 |
| 160 | 10 | 3.3149e-04 | 8.3100e-04 |
| 320 | 10 | 5.6479e-04 | 8.8426e-04 |
| 10 | 20 | 4.1622e-01 | 9.3054e-01 |
| 20 | 20 | 3.0132e-02 | 2.2622e-01 |
| 40 | 20 | 2.0105e-03 | 3.1081e-02 |
| 80 | 20 | 4.7033e-04 | 4.1364e-03 |
| 160 | 20 | 3.3283e-04 | 1.0762e-03 |
| 320 | 20 | 5.6482e-04 | 9.1561e-04 |