Enforcing the Courant–Friedrichs–Lewy condition in explicitly conservative local time stepping schemes
In this study, an optimally efficient explicit numerical scheme for solving fluid dynamics equations, or any other parabolic or hyperbolic system of partial differential equations, should allow local regions to advance in time with their own, locally constrained time steps. However, such a scheme can result in violation of the CourantFriedrichsLewy (CFL) condition, which is manifestly nonlocal. Although the violations can be considered to be "weak" in a certain sense and the corresponding numerical solution may be stable, such calculation does not guarantee the correct propagation speed for arbitrary waves. We use an experimental fluid dynamics code that allows cubic "patches" of grid cells to step with independent, locally constrained time steps to demonstrate how the CFL condition can be enforced by imposing a condition on the time steps of neighboring patches. We perform several numerical tests that illustrate errors introduced in the numerical solutions by weak CFL condition violations and show how strict enforcement of the CFL condition eliminates these errors. In all our tests the strict enforcement of the CFL condition does not impose a significant performance penalty.
 Authors:

^{[1]};
^{[2]};
^{[3]}
 Fermi National Accelerator Lab. (FNAL), Batavia, IL (United States); The Univ. of Chicago, Chicago, IL (United States)
 The Univ. of Chicago, Chicago, IL (United States)
 The Univ. of Chicago, Chicago, IL (United States); Fermi National Accelerator Lab. (FNAL), Batavia, IL (United States)
 Publication Date:
 Report Number(s):
 arXiv:1801.03108; FERMILABPUB18025A
Journal ID: ISSN 00219991; 1647346; TRN: US1801505
 Grant/Contract Number:
 AC0207CH11359
 Type:
 Accepted Manuscript
 Journal Name:
 Journal of Computational Physics
 Additional Journal Information:
 Journal Volume: 359; Journal Issue: C; Journal ID: ISSN 00219991
 Publisher:
 Elsevier
 Research Org:
 Fermi National Accelerator Lab. (FNAL), Batavia, IL (United States)
 Sponsoring Org:
 USDOE Office of Science (SC), High Energy Physics (HEP) (SC25)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 79 ASTRONOMY AND ASTROPHYSICS; Numerical methods; Computational fluid dynamics; Partial differential equations
 OSTI Identifier:
 1420907
Gnedin, Nickolay Y., Semenov, Vadim A., and Kravtsov, Andrey V.. Enforcing the Courant–Friedrichs–Lewy condition in explicitly conservative local time stepping schemes. United States: N. p.,
Web. doi:10.1016/j.jcp.2018.01.008.
Gnedin, Nickolay Y., Semenov, Vadim A., & Kravtsov, Andrey V.. Enforcing the Courant–Friedrichs–Lewy condition in explicitly conservative local time stepping schemes. United States. doi:10.1016/j.jcp.2018.01.008.
Gnedin, Nickolay Y., Semenov, Vadim A., and Kravtsov, Andrey V.. 2018.
"Enforcing the Courant–Friedrichs–Lewy condition in explicitly conservative local time stepping schemes". United States.
doi:10.1016/j.jcp.2018.01.008.
@article{osti_1420907,
title = {Enforcing the Courant–Friedrichs–Lewy condition in explicitly conservative local time stepping schemes},
author = {Gnedin, Nickolay Y. and Semenov, Vadim A. and Kravtsov, Andrey V.},
abstractNote = {In this study, an optimally efficient explicit numerical scheme for solving fluid dynamics equations, or any other parabolic or hyperbolic system of partial differential equations, should allow local regions to advance in time with their own, locally constrained time steps. However, such a scheme can result in violation of the CourantFriedrichsLewy (CFL) condition, which is manifestly nonlocal. Although the violations can be considered to be "weak" in a certain sense and the corresponding numerical solution may be stable, such calculation does not guarantee the correct propagation speed for arbitrary waves. We use an experimental fluid dynamics code that allows cubic "patches" of grid cells to step with independent, locally constrained time steps to demonstrate how the CFL condition can be enforced by imposing a condition on the time steps of neighboring patches. We perform several numerical tests that illustrate errors introduced in the numerical solutions by weak CFL condition violations and show how strict enforcement of the CFL condition eliminates these errors. In all our tests the strict enforcement of the CFL condition does not impose a significant performance penalty.},
doi = {10.1016/j.jcp.2018.01.008},
journal = {Journal of Computational Physics},
number = C,
volume = 359,
place = {United States},
year = {2018},
month = {1}
}