【Abstract】 We consider the relations between the long-time stability, compatility and convergence of the numerical methods for evolution equations. The key to establish the long-time convergence lies in introducing the idea of the γ-compatility, stability and strict stability on the infinity time interval. For finite difference methods of homogeneous linear initial-value problem, we extend the Lax equivalence theorem to infinite time interval by using the γ-compatility. For nonlinear initial-value problem, we give two kinds of long-time convergence theorems. In roughly speaking, longtime stability (or strict stability) and γ-compatility (or compatility) implicate longtime convergence. At last, we indicate several applications on the FEM and FDM for evolution equations.更多还原