Table 4
Summary of the tests for the red giant.
| Method | CFLhydro | CFLadv | CFLrad |
|
|
|
|
|
|
| Explicit schemes | |||||||||
| MRAI | 1.50 | 3.8(−2) | 0.28 | – | – | – | – | – | 371 |
| Adam-Bashforth | 0.10 | 2.5(−3) | 1.5(−2) | – | – | – | – | – | 201 |
|
|
|||||||||
| Implicit schemes | |||||||||
| CFLadv,max = 0.5 | |||||||||
| Broyden(10-2) - rILU(1) | 19.1 | 0.50 | 5.0 | 7.3 | 4.2 | 56.7% | 13.9% | 11.4% | 404 |
| Broyden(10-1) - rILU(1) | 19.1 | 0.50 | 5.0 | 7.7 | 2.8 | 57.7% | 13.5% | 10.1% | 390 |
| Broyden(10-4) - rILU(1) | 19.1 | 0.50 | 5.0 | 7.3 | 7.2 | 53.3% | 12.5% | 17.5% | 364 |
| Broyden(10-1) - rILU(2) | 19.1 | 0.50 | 5.0 | 7.7 | 2.5 | 47.5% | 26.9% | 9.7% | 336 |
| Broyden(10-2) - rILU(2) | 19.1 | 0.50 | 5.0 | 7.3 | 3.8 | 46.3% | 26.4% | 12.3% | 330 |
| Broyden(10-4) - rILU(2) | 19.1 | 0.50 | 5.0 | 7.3 | 6.8 | 42.6% | 23.1% | 20.8% | 287 |
| Broyden(10-2) - rILU(3) | 19.1 | 0.50 | 5.0 | 7.3 | 3.7 | 33.2% | 41.3% | 14.5% | 237 |
| Broyden(10-1) - rILU(3) | 19.1 | 0.50 | 5.0 | 7.7 | 2.5 | 34.6% | 41.2% | 12.4% | 235 |
| Broyden(10-4) - rILU(3) | 19.1 | 0.50 | 5.0 | 7.3 | 6.7 | 31.1% | 38.5% | 20.2% | 221 |
| Newton(10-2) - rILU(1) | 19.1 | 0.50 | 5.0 | 5.2 | 4.5 | 87.5% | 4.9% | 3.0% | 140 |
|
|
|||||||||
| CFLadv,max = 1 | |||||||||
| Broyden(10-1) - rILU(1) | 40.2 | 0.99 | 8.7 | 9.1 | 4.5 | 53.4% | 12.7% | 13.8% | 779 |
| Broyden(10-2) - rILU(1) | 40.2 | 0.99 | 8.7 | 8.7 | 6.5 | 50.7% | 11.8% | 19.2% | 719 |
| Broyden(10-1) - rILU(2) | 40.2 | 0.98 | 8.6 | 9.3 | 3.5 | 44.4% | 24.4% | 13.9% | 659 |
| Broyden(10-2) - rILU(2) | 40.4 | 0.99 | 8.8 | 8.7 | 5.4 | 42.8% | 23.8% | 17.6% | 632 |
| Broyden(10-4) - rILU(1) | 40.4 | 0.99 | 8.8 | 8.7 | 11.7 | 44.1% | 10.3% | 29.7% | 632 |
| Broyden(10-4) - rILU(2) | 40.4 | 0.99 | 8.8 | 8.7 | 10.0 | 37.8% | 21.1% | 27.2% | 559 |
| Broyden(10-1) - rILU(3) | 40.4 | 0.99 | 8.7 | 9.1 | 3.4 | 31.9% | 39.8% | 15.8% | 473 |
| Broyden(10-2) - rILU(3) | 40.2 | 0.99 | 8.7 | 8.8 | 5.1 | 30.9% | 38.7% | 18.8% | 454 |
| Broyden(10-4) - rILU(3) | 40.4 | 0.99 | 8.8 | 8.7 | 9.7 | 26.6% | 33.4% | 30.1% | 393 |
| Newton(10-2) - rILU(1) | 40.5 | 0.99 | 8.8 | 5.9 | 7.3 | 86.8% | 4.3% | 4.5% | 262 |
|
|
|||||||||
| CFLadv,max = 1.5 | |||||||||
| Broyden(10-1) - rILU(1) | 57.3 | 1.49 | 11.7 | 11.0 | 5.5 | 49.1% | 11.3% | 18.0% | 1008 |
| Broyden(10-2) - rILU(1) | 57.3 | 1.49 | 11.7 | 10.5 | 8.3 | 45.5% | 10.1% | 25.4% | 896 |
| Broyden(10-2) - rILU(2) | 57.3 | 1.49 | 11.7 | 10.6 | 6.5 | 39.3% | 21.1% | 22.7% | 812 |
| Broyden(10-1) - rILU(2) | 55.7 | 1.42 | 10.8 | 12.4 | 4.0 | 41.0% | 21.6% | 18.1% | 767 |
| Broyden(10-4) - rILU(1) | 57.3 | 1.49 | 11.7 | 10.5 | 19.6 | 33.4% | 7.5% | 45.2% | 664 |
| Broyden(10-2) - rILU(3) | 57.3 | 1.49 | 11.7 | 10.6 | 6.2 | 28.9% | 34.0% | 24.5% | 599 |
| Broyden(10-4) - rILU(2) | 57.3 | 1.49 | 11.7 | 10.5 | 17.6 | 28.5% | 15.3% | 44.0% | 590 |
| Broyden(10-1) - rILU(3) | 55.8 | 1.43 | 10.9 | 12.3 | 3.9 | 29.7% | 36.0% | 20.4% | 541 |
| Broyden(10-4) - rILU(3) | 57.3 | 1.49 | 11.7 | 10.5 | 17.2 | 18.6% | 21.3% | 52.2% | 374 |
| Newton(10-2) - rILU(1) | 56.2 | 1.45 | 11.1 | 7.5 | 8.8 | 86.8% | 3.4% | 5.5% | 286 |
|
|
|||||||||
| CFLadv,max = 2 | |||||||||
| Broyden(10-1) - rILU(1) | 70.4 | 1.95 | 13.9 | 13.5 | 6.9 | 43.4% | 9.3% | 25.3% | 1015 |
| Broyden(10-2) - rILU(1) | 70.2 | 1.94 | 14.0 | 13.1 | 10.5 | 40.5% | 9.0% | 30.7% | 960 |
| Broyden(10-2) - rILU(2) | 69.7 | 1.92 | 13.9 | 13.6 | 7.5 | 35.7% | 18.4% | 28.0% | 813 |
| Broyden(10-1) - rILU(2) | 67.9 | 1.85 | 13.8 | 13.8 | 4.6 | 38.5% | 18.6% | 24.1% | 785 |
| Broyden(10-1) - rILU(3) | 70.4 | 1.95 | 13.9 | 13.8 | 4.5 | 29.0% | 32.5% | 23.3% | 696 |
| Broyden(10-2) - rILU(3) | 70.2 | 1.94 | 14.1 | 13.1 | 6.9 | 25.4% | 27.7% | 34.4% | 576 |
| Broyden(10-4) - rILU(1) | 69.2 | 1.90 | 14.0 | 13.4 | 50.3 | 16.3% | 3.5% | 72.4% | 353 |
| Broyden(10-4) - rILU(2) | 70.2 | 1.94 | 14.1 | 13.4 | 41.4 | 14.2% | 7.2% | 71.5% | 329 |
| Newton(10-1) - rILU(1) | 69.5 | 1.91 | 13.9 | 9.9 | 7.7 | 88.3% | 2.6% | 4.9% | 269 |
| Broyden(10-4) - rILU(3) | 68.6 | 1.89 | 13.8 | 13.4 | 60.5 | 7.4% | 8.0% | 81.1% | 163 |
Notes. Nomenclature for the methods. “Broyden(10-2)”: Broyden method with a tolerance η = 10-2 on the linear solver. “rILU(k)”: recycled ILU preconditioner with fill-in factor k. The columns: the mean values of the three CFL numbers defined in Sect. 2, the average number of Newton-Raphson iterations per time step, the average number of GMRES iterations per Newton-Raphson iteration, the percentage of the wall time spent in: i) computing the Jacobian matrix, ii) building the ILU preconditioner, iii) performing the GMRES solve, and finally the ratio of the simulated time by the wall time, which measures the computational efficiency of the method. Methods are shown in each subsection in order of decreasing efficiency. In each case, only the most efficient Newton method is shown.
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