An accurate and simple universal a posteriori error estimator for GS4-1 framework: Adaptive time stepping in first-order transient systems

The “Generalized Single Step Single Solve” framework for first-order transient systems (GS4-1) is considered to develop adaptive time stepping via a newly proposed a posteriori error estimator in this paper. Traditional approaches design and employ particular error estimators for particular algorithms on algorithm by algorithm selection basis. In sharp contrast, in this paper, an accurate and simple universal a posteriori error estimator that is agnostic is newly designed to foster features that enable the adaptive time stepping for the entire subset of second order time accurate linear multi-step (LMS) algorithms in the GS4-1 family. The core contributions are: (a) The estimated local error has the same order of convergence and error constant as that of the exact local error; thereby the underlying issues that under-/over-estimate the error for optimizing the time step are addressed with rigour. The validation tests for linear/nonlinear problems show the third-order convergence rate in time for the local error, and the overlapped profiles of estimated local error and exact local error demonstrate the accuracy of the proposed error estimator. (b) Unlike several previously reported error estimators designed for some specific existing time integration algorithms in the literature, the proposed error estimator is universal, and is independent of the algorithmic parameters; thereby it can be applied to all the numerically dissipative/non-dissipative schemes without any limitations in the GS4-1 family. (c) The adaptive time stepping computational strategy emanating from the proposed error estimator achieves an excellent balance between the solution accuracy and CPU cost. In addition, several linear/nonlinear heat transfer problems with multi-degree of freedom are solved via the proposed method to further demonstrate the effectiveness of the numerical implementation.