Abstract
A novel explicit three-sub-step time integration method is proposed. From linear analysis, it is designed to have at least second-order accuracy, tunable stability interval, tunable algorithmic dissipation and no overshooting behaviour. A distinctive feature is that the size of its stability interval can be adjusted to control the properties of the method. With the largest stability interval, the new method has better amplitude accuracy and smaller dispersion error for wave propagation problems, compared with some existing second-order explicit methods, and as the stability interval narrows, it shows improved period accuracy and stronger algorithmic dissipation. By selecting an appropriate stability interval, the proposed method can achieve properties better than or close to existing second-order methods, and by increasing or reducing the stability interval, it can be used with higher efficiency or stronger dissipation. The new method is applied to solve some illustrative wave propagation examples, and its numerical performance is compared with those of several widely used explicit methods.
Highlights
Direct time integration methods are frequently employed to compute numerical solutions of ordinary differential equations or differential-algebraic equations in multi-body dynamics, structural dynamics, wave propagation problems, and many other branches of science and engineering
While implicit, unconditionally stable methods have no limitations on the time step size other than those dictated by accuracy, explicit methods are more practical when their stability-critical time step size and that required by accuracy are of similar magnitude, such as wave propagation problems [7,12,28]
This paper focuses on explicit methods, so implicit ones are only briefly reviewed, only addressing those aspects that are essential for the present discussion
Summary
Direct time integration methods are frequently employed to compute numerical solutions of ordinary differential equations or differential-algebraic equations in multi-body dynamics, structural dynamics, wave propagation problems, and many other branches of science and engineering. The two-sub-step methods, represented by the Noh-Bathe method (NB) [28], the Kim-Lee method [17], the Soares method [32], the explicit method based on displacement–velocity relations [42], share equivalent spectral characteristics for undamped linear systems They are designed to have second-order accuracy, a large stability interval and tunable algorithmic dissipation. Since the explicit methods are commonly used to solve wave propagation problems, the dispersion analysis for these problems is performed, to support the selection of the parameters, including τb, ρb, and the corresponding CFL number For these problems, high dispersion accuracy is expected to provide accurate solutions, while strong algorithmic dissipation is important to filter out the inaccurate high-frequency dynamics, which can greatly spoil the overall accuracy as the errors accumulate.
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