Algorithms without accuracy saturation and exponentially convergent algorithms 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 depends automatically on 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^-fA and of the operator cosine family with a constant operator A from earlier papers possessing an exponential accuracy and based on the Sinc-quadrature approximations of the corresponding Dunford-Cauchy integral representations of solutions or the solution operators.