Enforcing the Courant-Friedrichs-Lewy Condition in Explicitly Conservative Local Time Stepping Schemes

TL;DR

This paper demonstrates how to enforce the CFL condition in local time stepping schemes for fluid dynamics, ensuring correct wave propagation without significant performance loss.

Contribution

It introduces a method to enforce the CFL condition locally by constraining neighboring patches' time steps, improving accuracy in explicit schemes.

Findings

Enforcing CFL condition reduces numerical errors.

Strict CFL enforcement maintains stability and accuracy.

Performance impact of CFL enforcement is minimal.

Abstract

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.