Improved second-order unconditionally stable schemes of linear multi-step and equivalent single-step integration methods

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.

Performance of implicit A-stable time integration methods for multibody system dynamics

“…They were initially designed to solve second-order ODEs in structural dynamics, and some of their improved formulations can also be used to solve DAEs and general first-order differential equations [2,8,20]. Some comparisons between the linear two-step method and single-step methods have already been presented in [34,36] and are not reproduced here. Therefore, these single-step methods are not considered further in this work.…”

“…In the class of multi-step methods, the linear two-step method [24], and several backward difference formulas (BDFs) [19,28], have been efficiently used in multibody system dynamics. The optimal parameters of the linear three-and four-step methods with secondorder accuracy, A-stability and tunable algorithmic dissipation were given in [36]. According to Dahlquist's second barrier [12], the linear multi-step methods cannot exceed secondorder accuracy to possess A-stability.…”

A novel explicit three-sub-step time integration method for wave propagation problems

“…Most implicit methods proposed since the 1970s are designed to have second-order accuracy, unconditional stability and tunable algorithmic dissipation in the linear regime, including the single-step single-solve methods [6,11,36,45], the linear multi-step methods [26,27,43], and the composite multi-sub-step methods [5,9,13,16,18,23,29,40,44]. The single-step singlesolve methods, such as the HHT-α method [11] proposed by Hilber, Hughes and Taylor, the generalized-α method [6,31,41], as well as the linear multi-step methods (see for example [43]) are limited by Dahlquist's barrier [8], which states that methods of higher than second-order accuracy cannot achieve unconditional stability, so higher-order formulations of those schemes are not so attractive in practice. Optimal schemes of second-order linear two-, three-, and four-step methods, and their equivalent single-step formulations, were recently proposed in [43].…”

“…The single-step singlesolve methods, such as the HHT-α method [11] proposed by Hilber, Hughes and Taylor, the generalized-α method [6,31,41], as well as the linear multi-step methods (see for example [43]) are limited by Dahlquist's barrier [8], which states that methods of higher than second-order accuracy cannot achieve unconditional stability, so higher-order formulations of those schemes are not so attractive in practice. Optimal schemes of second-order linear two-, three-, and four-step methods, and their equivalent single-step formulations, were recently proposed in [43]. It was shown that the methods using solutions at more previous steps have higher low-frequency accuracy under the same amount of algorithmic dissipation.…”

Highly precise and efficient solution strategy for linear heat conduction and structural dynamics

“…Linear multi-step methods [14][15][16][17] and multi-sub-step composite methods [18][19][20][21][22][23][24][25][26][27][28][29][30] can improve accuracy without efficiency loss. Compared with most second-order single-step methods, they are more accurate, and their accuracy can be enhanced by employing the states of more past steps or more sub-steps.…”