CFL Condition

PDEs · Axiom Academy

Understanding the Courant-Friedrichs-Lewy stability condition for numerical PDE solutions 1. Physical Domain of Dependence For hyperbolic PDEs like the wave equation, information propagates along characteristics at a finite speed. The solution at point (x, t) depends only on values within the domain of dependence - the region bounded by characteristics passing through that point. 2. Numerical Domain of Dependence A numerical scheme also has a domain of dependence determined by its stencil. For an explicit scheme, the new value at grid point (j, n+1) depends on values at time level n within the numerical stencil. 3. CFL Condition for the Wave Equation For the wave equation u t + c u x = 0 with an explicit scheme using timestep t and spatial grid spacing x, the CFL condition requires that information cannot propagate faster than the numerical grid can resolve. The CFL number (or Courant number) quantifies how far information propagates in one timestep relative to the grid spacing. It's a dimensionless parameter that governs stability. 5. Necessary but Not Always Sufficient The CFL condition is necessary for stability of explicit schemes, but it's not always sufficient . Some schemes may be unstable even when the CFL condition is satisfied, requiring additional stability constraints. For the FTCS (Forward Time, Centered Space) scheme for the heat equation, CFL-like conditions exist but involve the diffusion number rather than wave speed.

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