Use of post-processing to increase the order of accuracy of the trapezoidal rule at time integration of linear elastodynamics problems

In this paper, we suggest a very simple and effective post-processing procedure to increase the order of accuracy in time for numerical results obtained by the trapezoidal rule. We first derive a new exact, closed-form, a-priori error estimator for time integration of linear elastodynamics equations by the trapezoidal rule with non-uniform time increments. Based on this error estimator, we suggest a new post-processing procedure (containing additional time integration of elastodynamics equations by the trapezoidal rule with few time increments) that systematically improves the order of accuracy of numerical results, with the increase in the number of additional time increments used for post-processing. For example, the use of just one additional time increment for post-processing after time integration with any number of uniform time increments, renders the order of accuracy of numerical results equal to 10/3. Numerical examples of the application of the new techniques to a system with a single degree of freedom and to a multi-degree system confirm the corresponding increase in the order of convergence of numerical results after post-processing. Because the same trapezoidal rule is used for basic computations and post-processing, the new technique retains all of the properties of the trapezoidal rule, requires no writing of a new computer program for its implementation, and can be easily used with any existing commercial and research codes for elastodynamics.