Table 1
Comparative analysis of the explicit and PIROCK methods with varying tolerances.
| Steps (rej. steps) | ![$\[\overline{\Delta t}\]$](/articles/aa/full_html/2025/03/aa52351-24/aa52351-24-eq39.png){kind=small} |
FD evals | FA evals | smax | L2 norm | CPU time/step (s) | Tot. time (s) | |
|---|---|---|---|---|---|---|---|---|
| explicit | 2203 (0) | 8.8 × 10−5 | 6609 | 6609 | 0 | 2 × 10−6 | 6 × 10−3 | 13.3 |
| PIR. (10−2) | 169 (0) | 1.3 × 10−3 | 2602 | 507 | 12 | 7.6 × 10−5 | 2.25 × 10−2 | 3.8 |
| PIR. (10−3) | 169 (0) | 1.3 × 10−3 | 2605 | 507 | 12 | 7 × 10−5 | 2.31 × 10−2 | 3.9 |
| PIR. (10−4) | 260 (1) | 8.1 × 10−4 | 3497 | 780 | 10 | 3.4 × 10−5 | 1.96 × 10−2 | 5.1 |
| PIR. (10−5) | 800 (1) | 2.7 × 10−4 | 7234 | 2400 | 4 | 1 × 10−5 | 1.4 × 10−2 | 11.3 |
| PIR. (10−8) | 7318 (379) | 3.3 × 10−5 | 23 472 | 22 047 | 1 | 3.5 × 10−8 | 6 × 10−3 | 42.2 |
Notes. Comparison of the number of steps (including rejected steps), average timestep value (Δt), function evaluations for the FD and FA terms, the maximum number of stages considered by the PIROCK method for the hyperdiffusive term, the L2 norm between the solution and the reference solution on the whole domain for the density variable ρ, the average CPU time per step for each case and the total computational time. Moreover, the computations were performed using a 2.4 GHz 8-Core Intel Core i9 processor.
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