A simple and effective single‐step time marching technique based on adaptive time integrators

Abstract

Summary In this work, a novel time marching procedure is presented in which adaptive time integration parameters are locally computed, allowing different spatial and temporal distributions. As a consequence, a more versatile technique arises, enabling an enhanced performance. The main features of the proposed technique are: (1) it is simple; (2) it has guaranteed stability; (3) it is an efficient non‐iterative adaptive single‐step procedure; (4) it provides enhanced accuracy; (5) it enables advanced controllable algorithmic dissipation in the higher modes; (6) it is truly self‐starting; and (7) it is entirely automatic, requiring no input parameter from the user. The proposed technique is very complete, providing the main positive attributes that are requested from an effective time marching procedure. Numerical results are presented along the paper, illustrating the excellent performance of the method. Copyright © 2016 John Wiley & Sons, Ltd.

A model/solution‐adaptive explicit‐implicit time‐marching technique for wave propagation analysis

“…For these time steps, considering the new method, the left half of the rod is automatically discretized by implicit elements, whereas the remaining half is discretized by explicit elements. In Figure , relative error results for the computed displacements at the middle of the rod are depicted, taking into account the proposed explicit‐implicit technique, the TR, and the techniques described in the works of Soares . As one can observe, once again, very good accuracy is provided by the new approach, and this enhanced precision is obtained employing reduced computational efforts.…”

“…The basic framework that is employed here was introduced by Soares . Later on, this basic framework was reconfigured, and local adaptive time integrators were presented, taking into account implicit analyses . In this work, explicit analyses are discussed, and a hybrid explicit‐implicit formulation is proposed.…”

“…This is the case because explicit subdomains may contribute with just diagonal entries to the effective matrix of the model, which can be very easily cast out of the final system of equations, reducing its dimension and, as a consequence, the computational complexity of the analysis. In addition, in this work, the expressions of the adaptive time integrators are expanded, allowing them to also account for the local physical damping features of the model, which is a further advance regarding the formulation that is presented in the work of Soares …”

“…44 Later on, this basic framework was reconfigured, and local adaptive time integrators were presented, taking into account implicit analyses. 45 In this work, explicit analyses are discussed, and a hybrid explicit-implicit formulation is proposed. Following this mixed formulation, contracted systems of equations can be obtained, enabling reduced computational efforts and a much more effective approach.…”

“…In addition, in this work, the expressions of the adaptive time integrators are expanded, allowing them to also account for the local physical damping features of the model, which is a further advance regarding the formulation that is presented in the work of Soares. 45 This manuscript is organized as follows. First, the governing equations of the model and the time integration strategy are presented, describing how the adaptive computation of the time integration parameters is carried out.…”