Convergence of a numerical method (${\max }_{t_n\le T}\Vert y_n-y(t_n)\Vert \to 0$)

For an initial value problem, convergence means that the largest global error on each fixed finite time interval tends to zero as the step size tends to zero and the starting errors vanish. For linear multistep methods, the Dahlquist equivalence theorem separates this requirement into consistency of a numerical method and zero-stability.