On the optimization of n-sub-step composite time integration methods

Abstract

A family of n-sub-step composite time integration methods, which employs the trapezoidal rule in the first $$n-1$$ n - 1 sub-steps and a general formula in the last one, is discussed in this paper. A universal approach to optimize the parameters is provided for any cases of $$n\ge 2$$ n ≥ 2 , and two optimal sub-families of the method are given for different purposes. From linear analysis, the first sub-family can achieve nth-order accuracy and unconditional stability with controllable algorithmic dissipation, so it is recommended for high-accuracy purposes. The second sub-family has second-order accuracy, unconditional stability with controllable algorithmic dissipation, and it is designed for heuristic energy-conserving purposes, by preserving as much low-frequency content as possible. Finally, some illustrative examples are solved to check the performance in linear and nonlinear systems.

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

“…For solving DAEs, the stiffly-accurate Runge-Kutta methods without the quadrature step are more practical, because the constraints are satisfied at the final stage, but may not be satisfied after implementing the quadrature formula. Multi-stage methods can be designed to have higher-order accuracy and A-stability simultaneously, as in [1,6,21,35]. Considering the computational cost, the singly diagonally-implicit Runge-Kutta methods [27], which perform the computation of each stage in sequence, are more convenient and recommended.…”

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

“…This paper focuses on explicit methods, so implicit ones are only briefly reviewed, only addressing those aspects that are essential for the present discussion. 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.…”

“…It was shown that the methods using solutions at more previous steps have higher low-frequency accuracy under the same amount of algorithmic dissipation. The composite multi-sub-step methods, such as the two-sub-step ones [4,5,16,18,29], the three-sub-step ones [13,23], and the general multisub-step ones [9,44], have received a lot of attention during the past decade. Although these methods can be designed to have higher-order accuracy and unconditional stability by using more sub-steps, their second-order formulations are of most concern, since they can simultaneously offer strong high-frequency dissipation and desirable low-frequency accuracy.…”

“…Although these methods can be designed to have higher-order accuracy and unconditional stability by using more sub-steps, their second-order formulations are of most concern, since they can simultaneously offer strong high-frequency dissipation and desirable low-frequency accuracy. Two sets of optimal schemes of the general n-sub-step method considering different conditions can be found in [44].…”

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

“…[20][21] Using similar idea, asymptotically stable three-sub-step [22][23] and four-sub-step [24][25] methods with higher accuracy were later proposed. By replacing the backward difference formula with interpolation formulae, some multi-sub-step methods [26][27][28][29][30] with controllable dissipation were constructed, but developing such methods with more sub-steps encounters difficulty. Moreover, compared with higher-order methods, the improvement of accuracy by second-order linear multi-step methods or multi-sub-step methods is not significant.…”