Table 4.
Numerical errors in terms of the maximum norm and CPU time for the various methods on the example in Section 5.1 at T = 1 with diffusion coefficient d = 2. Here N is the number of grid points in the spatial discretization (Δx = π/2N), and the time step Δt = 0.1Δx. “CPU time 1” is the CPU time for initializing the matrices (Appendix C), “CPU time 2” is the CPU time for the iterations, and “CPU time” is the sum of the two.
| N | L∞ error | Order | CPU time (s) | CPU time 1 (s) | CPU time 2 (s) | |
|---|---|---|---|---|---|---|
| IIF2 | 8 | 0.00228 | - | 0.09 | 0.05 | 0.04 |
| 16 | 0.000591 | 1.95 | 0.04 | 0.02 | 0.02 | |
| 32 | 0.000198 | 1.58 | 0.07 | 0.03 | 0.04 | |
| 64 | 7.81e-05 | 1.34 | 0.13 | 0.04 | 0.09 | |
| 128 | 0.000108 | −0.46 | 0.54 | 0.07 | 0.47 | |
| 256 | 5.18e−05 | 1.06 | 1.26 | 0.23 | 1.03 | |
| 512 | 1.83e−05 | 1.50 | 4.00 | 1.39 | 2.61 | |
| 1024 | 2.07e−05 | −0.18 | 28.30 | 7.75 | 20.55 | |
| 2048 | 1.07e−05 | 0.96 | 168.12 | 42.10 | 126.02 | |
| 4096 | 5.35e−06 | 1.00 | 1148.42 | 265.35 | 883.07 | |
| N | L∞ error | Order | CPU time (s) | CPU time 1 (s) | CPU time 2 (s) | |
| iETD2 | 8 | 0.00216 | - | 0.07 | 0.04 | 0.03 |
| 16 | 0.000539 | 2.00 | 0.07 | 0.04 | 0.03 | |
| 32 | 0.000135 | 2.00 | 0.12 | 0.06 | 0.06 | |
| 64 | 3.37e−05 | 2.00 | 0.80 | 0.07 | 0.73 | |
| 128 | 8.41e−06 | 2.00 | 3.78 | 0.16 | 3.62 | |
| 256 | 2.1e−06 | 2.00 | 22.99 | 0.54 | 22.45 | |
| 512 | 5.26e−07 | 2.00 | 289.66 | 2.70 | 286.96 | |
| 1024 | 1.32e−07 | 2.00 | 2841.66 | 14.65 | 2827.01 | |
| 2048 | 3.31e−08 | 1.99 | 35348.32 | 91.84 | 35256.48 | |
| 4096 | - | - | too long | - | - | |
| N | L∞ error | Order | CPU time (s) | CPU time 1 (s) | CPU time 2 (s) | |
| hIFE2 | 8 | 0.00217 | - | 0.12 | 0.09 | 0.03 |
| 16 | 0.000544 | 1.99 | 0.06 | 0.04 | 0.02 | |
| 32 | 0.000137 | 1.99 | 0.08 | 0.05 | 0.03 | |
| 64 | 3.42e−05 | 2.00 | 0.16 | 0.08 | 0.08 | |
| 128 | 8.75e−06 | 1.97 | 0.76 | 0.17 | 0.59 | |
| 256 | 2.21e−06 | 1.99 | 1.85 | 0.54 | 1.31 | |
| 512 | 5.53e−07 | 2.00 | 9.17 | 2.61 | 6.56 | |
| 1024 | 1.49e−07 | 1.89 | 61.82 | 14.20 | 47.62 | |
| 2048 | 3.93e−08 | 1.93 | 419.24 | 89.49 | 329.75 | |
| 4096 | 1.12e−08 | 1.81 | 3096.23 | 603.04 | 2493.19 | |
| N | L∞ error | Order | CPU time (s) | CPU time 1 (s) | CPU time 2 (s) | |
| fEIF2 | 8 | 0.00216 | - | 0.37 | 0.37 | 0.00 |
| 16 | 0.00054 | 2.00 | 0.04 | 0.04 | 0.00 | |
| 32 | 0.000135 | 2.00 | 0.07 | 0.07 | 0.00 | |
| 64 | 3.38e−05 | 2.00 | 0.09 | 0.08 | 0.01 | |
| 128 | 8.44e−06 | 2.00 | 0.54 | 0.18 | 0.36 | |
| 256 | 2.11e−06 | 2.00 | 1.41 | 0.69 | 0.72 | |
| 512 | 5.28e−07 | 2.00 | 11.62 | 3.01 | 8.61 | |
| 1024 | 1.32e−07 | 2.00 | 84.11 | 16.11 | 68.00 | |
| 2048 | 3.31e−08 | 1.99 | 613.91 | 101.12 | 512.79 | |
| 4096 | 8.89e−09 | 1.90 | 4700.11 | 707.64 | 3992.47 |