On second-order s-sub-step explicit algorithms with controllable dissipation and adjustable bifurcation point for second-order hyperbolic problems

This paper proposes a self-starting, second-order accurate, composite s-sub-step explicit method, within which the first five explicit members are developed, analyzed, and compared. Each member attains maximal stability bound, reaching 2×s, where s denotes the number of sub-steps. Identical diagonal elements in the amplification matrix are required in the undamped case, and algorithmic dissipation is controlled and measured at the bifurcation point. Further, if the bifurcation point is regarded as a free parameter during the dissipation analysis, the methods can achieve simultaneously controllable algorithmic dissipation and adjustable bifurcation point. Apart from these, the optimization of low-frequency dissipation is performed in the four- and five-sub-step members. As the number of sub-steps increases, the proposed explicit members can embed better numerical characteristics. This paper provides two approaches to constructing the updating velocity schemes in all sub-steps. The first approach enlarges the stability in the damped case, while the second one enhances the solution accuracy for damping problems. Numerical examples are solved to validate the proposed explicit members’ numerical performance. The novel algorithms are often more accurate and efficient than the existing methods.