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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 multi-step 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.
Highlights
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
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
This paper focuses on second-order accurate and unconditionally stable schemes of the linear multi-step methods
Summary
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. The improvement in accuracy is realized at a dramatic increase in computational cost, so these higher-order schemes are not very useful in practice In this family, the explicit and diagonally implicit Runge– Kutta methods are more widely-used. This paper focuses on second-order accurate and unconditionally stable schemes of the linear multi-step methods To acquire these accuracy and stability properties, the linear two-step method has been originally studied in [31,32,44]. The linear three-, and four-step methods are discussed in this paper From linear analysis, their optimal parameters are determined by minimizing the global error, under the constraints of second-order accuracy, unconditional stability and tunable algorithmic dissipation.
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