Von Neumann stability and CFL conditions

For a scalar linear one-step scheme on a periodic uniform grid, each discrete Fourier mode is multiplied by a number G(theta) per step. The discrete Fourier transform is unitary after normalization in the h-weighted L2 norm. Therefore a bound|G|≤1 for every represented frequency implies one-step and repeated-step contraction in that norm. A slightly weaker bound|G|≤1+C time step can still yield a finite-time stability bound independent of the number of steps, provided C is uniform across refining grids.

Testing a scalar eigenvalue is less decisive for coupled nonnormal update matrices: uniformly bounded powers and a uniformly controlled diagonalization may matter. A periodic Fourier calculation also does not see an unstable inflow closure at a physical boundary. Consequently the analysis should specify periodicity or an infinite grid, constant coefficients, the scalar one-step structure and the norm. Treating it as a universal proof for variable-coefficient systems suppresses precisely the assumptions that make the transform argument work.

For centered-space Forward Euler heat flow, let r=kappa times time step divided by h squared. Its multiplier is one minus 4r sin squared(theta/2). Requiring its magnitude at most one for all frequencies in the standard infinite-grid or all-grid-size test gives 0≤r≤1/2. The same interval makes the physical-space coefficients nonnegative and summing to one. A particular finite grid lacking the highest frequency may permit a slightly larger L2 threshold; that does not change the uniform sufficient condition.

For positive-speed upwind advection, the update mixes the current and left-neighbor values with weights 1−nu and nu, where nu=a time step/h. Its squared amplification magnitude is one minus 4nu(1−nu)sin squared(theta/2), giving contraction for 0≤nu≤1. Here the restriction also ensures a monotone convex-combination update. These are scheme-specific sufficient results. The centered-space Forward Euler advection counterexample meets the geometric condition for small nu yet still amplifies every nontrivial mode.

For an explicit local advection stencil with a fixed finite spatial reach per step, a convergent refinement must allow its numerical domain of dependence to contain the PDE’s characteristic domain. This is the geometric CFL principle. It is a necessary condition in that class of methods, not a sufficient theorem for all formulas with the same stencil reach. Implicit methods have a different numerical dependence structure, so the same finite-reach argument should not be imposed on them as an explicit-step necessity.

Stability must also be combined with consistency to obtain convergence in an appropriate well-posed linear problem; nonlinear conservation laws need additional entropy and compactness arguments. Report whether a proposed time step is justified by a symbol bound, monotonicity, an energy estimate or only a characteristic-speed heuristic. This makes it possible to adapt the calculation when the spatial operator, boundary treatment or time integrator changes, instead of applying one remembered CFL constant to unrelated algorithms.