A Novel Formulation to Predict the Accuracy of Implicit Time Integration Schemes

The article presents a novel approach to re-pose the temporal approximation problem as a recursive scheme in terms of displacement and velocity. The recursion matrix is used to obtain the numerical approximation induced amplitude magnification and frequency dilation. From the amplitude magnification term, expressed as a function of the time-step size, one can easily deduce the stability of the method. Similarly, the frequency dilation term can be used to check the accuracy of the numerical scheme. For each time-approximation scheme, a critical parameter (θ = ωΔt) can be identified to guarantee stability and accuracy of the numerical scheme. The performance of classical time-marching schemes, e.g., Newmark scheme, and other single-step schemes is analyzed. Further, a family of Hermite polynomials-based time-approximation methodology is proposed, that can guarantee any desired higher rate of temporal approximation. A residual norm based a-posteriori error estimator has been proposed to investigate the error in the solution. Sample problems have been solved to demonstrate the effectiveness of the current approach.