Improved second-order unconditionally stable schemes of linear multi-step and equivalent single-step integration methods
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ORIGINAL PAPER
Improved second-order unconditionally stable schemes of linear multi-step and equivalent single-step integration methods Huimin Zhang1,2 · Runsen Zhang1,2 · Pierangelo Masarati2 Received: 11 June 2020 / Accepted: 1 October 2020 © The Author(s) 2020
Abstract Second-order unconditionally stable schemes of linear multi-step methods, and their equivalent single-step methods, are developed in this paper. The parameters of the linear two-, three-, and four-step methods are determined for optimal accuracy, unconditional stability and tunable algorithmic dissipation. The linear three- and four-step schemes are presented for the first time. As an alternative, corresponding single-step methods, spectrally equivalent to the multi-step ones, are developed by introducing the required intermediate variables. Their formulations are equivalent to that of the corresponding multistep methods; their use is more convenient, owing to being self-starting. Compared with existing second-order methods, the proposed ones, especially the linear four-step method and its alternative single-step one, show higher accuracy for a given degree of algorithmic dissipation. The accuracy advantage and other properties of the newly developed schemes are demonstrated by several illustrative examples. Keywords Linear multi-step method · Accuracy · Unconditional stability · Alternative single-step method
1 Introduction Direct time integration methods are powerful numerical tools for solving the ordinary differential and differential-algebraic equations arising in structural dynamics, multi-body dynamics and many other fields. A lot of excellent integration methods have been developed and are available in the literature; they are often classified as explicit and implicit methods. Explicit methods are easy to implement, but their intrinsic conditional stability limits the step size. Implicit methods can be designed to be unconditionally stable, so they are suitable when the accuracy requirements are not very strict. In addition, time integration methods also can be categorized as multi-step and multi-stage techniques. The multistep methods employ the states of several previous steps to express the current one, whereas the multi-stage utilize the states of selected collocation points. The single-step methods, which only use the information of the last step, may be
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Huimin Zhang [email protected]
1
School of Aeronautic Science and Engineering, Beihang University, Beijing 100083, China
2
Dipartimento di Scienze e Tecnologie Aerospaziali, Politecnico di Milano, 20156 Milan, Italy
included in both classes and represent the connecting link between them. The most representative methods in the multi-stage class belong to the Runge-Kutta family [6], including the explicit [7], the diagonally implicit [19], and the fully implicit Runge–Kutta methods [16]. Based on the Gauss quadrature, the fully implicit s-stage Runge–Kutta methods can reach up to 2sth-order accuracy. However, the improvement in accuracy is realized at a dramatic increas
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