Hierarchical higher-order dissipative methods for transient analysis

This work focuses on devising an efficient hierarchy of higher-order methods for linear transient analysis, equipped with an efficient dissipative action on the spurious high modes of the response. The proposed strategy stems from the Norsett idea and is based on a multi-stage algorithm, designed to hierarchically improve accuracy while retaining the desired dissipative behaviour. Computational efficiency is pursued by requiring that each stage should involve just one set of implicit equations of the size of the problem to be solved (as standard time integration methods) and, in addition, all the stages should share the same coefficient matrix. This target is achieved by rationally formulating the methods based on the discontinuous collocation approach. The resultant procedure is shown to be well suited for adaptive solution strategies. In particular, it embeds two natural tools to effectively control the error propagation. One estimates the local error through the next-stage solution, which is one-order more accurate, the other through the solution discontinuity at the beginning of the current time step, which is permitted by present formulation. The performance of the procedure and the quality of the two error estimators are experimentally verified on the different classes of problems. Some typical numerical tests in transient heat conduction and elasto-dynamics are presented.