Classical numerical analysis rests on a chain: consistency plus stability implies convergence. If your numerical scheme accurately represents the differential equation (consistency) and doesn't amplify errors (stability), then the discrete solution approaches the continuous one as the mesh refines (convergence). Stability is the load-bearing assumption. Without it, errors compound and the scheme diverges.
Entropy-stable split-form discretizations converge for nonlinear hyperbolic PDEs even when they are locally linearly unstable.
The result, established for polynomial degrees exceeding 1 + d/2, shows that error bounds remain finite and vanish with mesh refinement despite the fact that a classical stability analysis — linearize the scheme around the numerical solution and check the eigenvalues — predicts growth. The eigenvalues are wrong. Not wrong in the sense of computed incorrectly, but wrong in the sense of irrelevant. The linear analysis describes perturbation growth in a neighborhood of the solution, but the entropy structure provides a nonlinear bound that the linear analysis cannot see.
The mechanism: entropy-stable schemes satisfy a discrete analog of the continuous entropy inequality. This inequality bounds the total “energy” of the solution regardless of what local perturbations do. A local mode can grow — can amplify errors in its neighborhood — but the global entropy constraint prevents the total error from escaping. The instability is real but caged.
This is a concrete mathematical instance of a general principle. Local fragility and global robustness can coexist when the system satisfies the right conservation constraint. The conservation law does not prevent local failures. It prevents local failures from cascading into global ones. The system absorbs perturbations not by suppressing them locally but by redistributing them globally within an invariant budget.
Classical stability theory, by linearizing, asks whether the system can resist a specific perturbation. The entropy structure asks a different question: whether the system can contain the consequences. The answers need not agree. A system that fails the first test and passes the second is locally unstable and globally convergent — fragile in detail, robust in aggregate.