A three-time-level a posteriori error estimator for GS4-2 framework: Adaptive time stepping for second-order transient systems

Abstract

The most commonly used adaptive time stepping technique is mainly based on the simple notion of a local error estimation. Accordingly, a variety of error estimators have mostly been designed for particular algorithms on an algorithm by algorithm selection basis. In this paper, a novel design of a three-time-level general purpose a posteriori error estimator is demonstrated for second-order transient systems, providing features of: (a) being universal for the most popular and entire class of second-order accurate unconditionally stable implicit LMS methods with/without numerical dissipation developed over the past few decades and on newer and optimal designs (the focus is on unconditionally stable implicit LMS methods and not explicit methods); (b) being simple for numerical implementation in the general purpose GS4-2 framework encompassing the entire class of LMS methods that are second order time accurate such that only the acceleration at three time levels is required; (c) preserving the third-order convergence in time for the local error; (d) predicting accurate local error such that the effectivity index is close to unity. In contrast to the existing a posteriori error estimators which are designed for certain algorithms and on algorithm by algorithm basis, the newly proposed approach can be directly applied to all the known and existing and new second-order accurate time integration algorithms in the entire class of the unconditionally stable implicit LMS family under the umbrella of the GS4-2 framework; thereby, the cumbersome work of designing different error estimators for different algorithms is avoided. In particular, the proposed error estimator can be directly applied also to the novel V0 family of algorithms without any limitation. Besides the limited selection of estimators existing within the U0 family of algorithms where most of the traditional methods belong to, the remainder of the algorithms within the U0 family and several others such as the V0 family do not have estimators that can be readily designed as the V0 family of algorithms only exist in the GS4-2 framework, which are entirely novel and new and are highly competitive. The numerical examples of stiff nonlinear spring pendulum, van der Pol equation, wave propagation, and phase-field modeling are described here to show the benefits to the computational cost whilst preserving generality, accuracy, and the like when the adaptive time stepping approach coupled with the proposed error estimator is applied to the transient simulations.