Algorithms without accuracy saturation for evolution equations in Hilbert and Banach spaces

We consider the Cauchy problem for the first and the second order differential equations in Banach and Hilbert spaces with an operator coefficient $A(t)$ depending on the parameter $t$. We develop discretization methods with high parallelism level and without accuracy saturation; i.e., the accuracy adapts automatically to the smoothness of the solution. For analytical solutions the rate of convergence is exponential. These results can be viewed as a development of parallel approximations of the operator exponential $e^{-tA}$ and of the operator cosine family $\cos{\sqrt{A} t}$ with a constant operator $A$ possessing exponential accuracy and based on the Sinc-quadrature approximations of the corresponding Dunford-Cauchy integral representations of solutions or the solution operators.