Criterion for the Stability of Difference Schemes for Nonlinear Differential Equations

For abstract nonlinear difference schemes with operators acting in finite-dimensional Banach spaces, a stability criterion is stated and proved; namely, for a consistent finite-difference approximation to a well-posed differential problem, the solution of the difference scheme converges if and only if the scheme is unconditionally stable. In a sense, this criterion generalizes Lax's equivalence theorem to nonlinear differential problems. The results obtained are used to study the stability of difference schemes that approximate quasilinear parabolic equations with nonlinearities of unbounded growth.