High-order implicit time integration scheme based on Padé expansions

Abstract

A single-step high-order implicit time integration scheme for the solution of transient as well as wave propagation problems is presented. It is constructed from the Padé expansion of the matrix exponential solution of a system of first-order ordinary differential equations formulated in the state–space. By exploiting the techniques of polynomial factorization and partial fractions of rational functions, and by decoupling the solution for the displacement and velocity vectors a computationally efficient scheme is developed. An important feature of the novel algorithm is that no direct inversion of the mass matrix is required. Based on the diagonal Padé expansion of order M a time-stepping scheme of order 2M is constructed. When M is odd, one system of equation with a real matrix and (M−1)/2 systems of equations with complex matrices are solved recursively. When M is even, there are M/2 systems of equations with complex matrices. These systems are sparse and comparable in complexity to the standard Newmark method, i.e., the effective system matrix is a linear combination of the static stiffness, damping, and mass matrices. The derived second-order scheme is analytically equivalent to Newmark constant average acceleration method. The proposed time integrator has been implemented in MATLAB and FORTRAN using direct linear equation solvers. In this article, numerical examples featuring nearly one million degrees of freedom are presented to highlight the exceptional accuracy and efficiency of the novel time-stepping scheme in comparison with established second-order methods. The MATLAB- and FORTRAN-implementations are available from the authors upon request.