Enforcing the Courant–Friedrichs–Lewy condition in explicitly conservative local time stepping schemes

Journal Article · Tue Jan 30 00:00:00 EST 2018 · Journal of Computational Physics

DOI:https://doi.org/10.1016/j.jcp.2018.01.008· OSTI ID:1420907

Gnedin, Nickolay Y. ^[1]; Semenov, Vadim A. ^[2]; Kravtsov, Andrey V. ^[3]

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1. Fermi National Accelerator Lab. (FNAL), Batavia, IL (United States); The Univ. of Chicago, Chicago, IL (United States)
2. The Univ. of Chicago, Chicago, IL (United States)
3. The Univ. of Chicago, Chicago, IL (United States); Fermi National Accelerator Lab. (FNAL), Batavia, IL (United States)

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 Courant-Friedrichs-Lewy (CFL) condition, which is manifestly non-local. 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.

• Research Organization: Fermi National Accelerator Laboratory (FNAL), Batavia, IL (United States)
• Sponsoring Organization: USDOE Office of Science (SC), High Energy Physics (HEP)
• Grant/Contract Number: AC02-07CH11359
• OSTI ID: 1420907
• Alternate ID(s): OSTI ID: 1548848
• Report Number(s): arXiv:1801.03108; FERMILAB-PUB-18-025-A; 1647346; TRN: US1801505
• Journal Information: Journal of Computational Physics, Vol. 359, Issue C; ISSN 0021-9991
• Publisher: Elsevier
• Country of Publication: United States
• Language: English

Related Subjects

79 ASTRONOMY AND ASTROPHYSICS
Numerical methods
Computational fluid dynamics
Partial differential equations