On the precise integration methods based on Padé approximations

The precise time step integration method proposed for linear time-invariant homogeneous dynamic systems can give precise numerical results approaching exact solutions at the integration points, but it is conditionally stable. By combining Padé approximation and the generalized Padé approximation of the matrix exponential function in precise integration, and by using three different types of quadrature formulae, a new generalized family of precise time step integration methods is developed to achieve unconditional stability and arbitrary order of accuracy. Numerical studies indicate that they are unconditionally stable algorithms with controllable numerical dissipation. They also demonstrate the validity and efficiency of these algorithms.