Courant–Friedrichs–Lewy condition

The Courant–Friedrichs–Lewy condition is a necessary convergence condition for certain numerical schemes solving hyperbolic partial differential equations. It requires the numerical domain of dependence to contain the physical domain of dependence.

Domain of dependence: The region of initial data that can influence a solution at a specified point and time. Convergence requires the scheme’s domain of dependence to cover the PDE’s physical domain of dependence.
Hyperbolic partial differential equation: A partial differential equation whose solutions typically describe finite-speed propagation of waves or signals. The classical CFL condition concerns numerical approximations to equations in this family.
Finite volume method: A numerical method that evolves cell averages using fluxes across cell boundaries. Explicit finite-volume solvers commonly limit time steps using local wave speeds and cell sizes.
A-stability: A stability property of time-integration methods that remain stable for every problem with a nonpositive linear eigenvalue. A-stable time integrators can avoid explicit stability limits, though accuracy and spatial propagation still matter.
Courant number: A dimensionless ratio comparing a signal’s travel distance during one time step with the spatial grid spacing. It expresses the CFL restriction for many schemes as a bound on wave travel per step.
Wave equation: A second-order partial differential equation describing propagation of waves at a finite speed. Its characteristic propagation speed makes the domain-of-dependence restriction concrete.
Computational fluid dynamics: The numerical study of fluid flow using discretized governing equations. CFL-based time-step control is routine in explicit flow simulations.
Lax equivalence theorem: A theorem stating that a consistent finite-difference scheme for a well-posed linear initial-value problem converges if and only if it is stable. It distinguishes stability from convergence; satisfying CFL alone does not establish both.
Finite difference method: A numerical method that approximates derivatives using values on a discrete grid. Its stencil determines which grid values can influence each computed solution value.
Method of lines: A numerical approach that discretizes spatial variables while leaving time continuous before time integration. Its spatial discretization and time integrator jointly set the allowable time step.

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