Spectral Deferred Correction Methods for Second-Order Problems#

Python code for the plots of the paper Spectral Deferred Correction Methods for Second-Order Problems by Akramov, Götschel, Minion, Ruprecht and Speck (SIAM Journal on Scientific Computing, 2024).

You are welcome to use and adapt this code under the terms of the BSD license. If you use it, in whole or in part, for a publication, please cite the paper.

Reproducing Figures from the Publication#

  • Fig. 1: Execute harmonic_oscillator_run_stability.py while setting kappa_max=18 and mu_max=18.

  • Fig. 2: Run harmonic_oscillator_run_stability.py with the following configurations:

    • Set kappa_max=30 and mu_max=30.

    • Adjust maxiter to 1, 2, 3, or 4 and execute each individually.

  • Table 1: Execute harmonic_oscillator_run_stab_interval.py:

    • For the Picard iteration set: Picard=True

    • To save the results set: save_interval_file=True

  • Use the script harmonic_oscillator_run_points.py to create a table based on given $(kappa, mu)$ points. This table assists in determining suitable values for M, K, and quadrature nodes to ensure stability in the SDC method.

    • To save the results set: save_points_file=True

  • Fig. 3: Run penningtrap_run_error.py (Run local convergence: conv.run_local_error()) with dt=0.015625/4 and axes=(0,).

  • Fig. 4: Run penningtrap_run_error.py (Run local convergence: conv.run_local_error()) using dt=0.015625*4 and axes=(2,).

  • Fig. 5: Run penningtrap_run_error.py (Run global convergence: conv.run_global_error()) with dt=0.015625*2:

    • Note: Perform each run individually: first with axes=(0,), then with axes=(2,).

    • Manually set y-axis limits in penningtrap_run_error.py, specifically in lines 147-148.

  • Table 2: Execute penningtrap_run_error.py (Run global convergence: conv.run_global_error()) with the following settings:

    • Expected order and approximate order are saved in the file: data/global_order_vs_approx_order.csv

    • Set: K_iter=(2, 3, 4, 10)

    • For M=2:

      • Set dt=0.015625 / 2 and num_nodes=2 (Both Horizontal and Vertical axes)

    • For M=3:

      • Set dt=0.015625 and num_nodes=3 (Both Horizontal and Vertical axes)

    • For M=4:

      • Set dt=0.015625 * 4 and num_nodes=4 (Both Horizontal and Vertical axes)

  • Fig. 6: Run penningtrap_run_Hamiltonian_error.py:

    • Note: Execute each computation individually in case of crashes.

    • Data is saved in the data folder.

  • Fig. 7: Execute penningtrap_run_work_precision.py with the following configurations:

    • Set dt=0.015625*2 and K_iter=(1, 2, 3) for axis=(2,).

    • Set dt=0.015625*4, and K_iter=(2, 4, 6), and dt_cont=2 for axis=(0,).

Contact#

This code is written by Ikrom Akramov

Papers#

The results of this project are published in:

  • Ikrom Akramov, Sebastian Götschel, Michael Minion, Daniel Ruprecht, Robert Speck, Spectral Deferred Correction Methods for Second-Order Problems, SIAM Journal on Scientific Computing 46(3), A1690–A1713, 2024, https://doi.org/10.1137/23M1592596

    BibTeX
    @article{akramov2024spectral,
    author = {Akramov, Ikrom and Götschel, Sebastian and Minion, Michael and Ruprecht, Daniel and Speck, Robert},
    title = {Spectral Deferred Correction Methods for Second-Order Problems},
    journal = {SIAM Journal on Scientific Computing},
    volume = {46},
    number = {3},
    pages = {A1690--A1713},
    year = {2024},
    doi = {10.1137/23M1592596}
}