A new comparison framework is developed to fairly evaluate and properly compare single-solve and sub-step time integration algorithms with controllable numerical dissipation. Unlike existing and/or past practices that rely only upon using different time step sizes to ensure comparable computational cost, this work emphasizes the necessity of simultaneously accounting for the effect of the infinity spectral radius, i.e., ρ ∞ ∈ 0 , 1 , which governs the amount of numerical dissipation in practical applications. Neglecting this consideration, as is common in much of the current literature, may result in less accurate conclusions and potentially overlook some aspects of the performance of traditional single-step single-solve time integration algorithms. Numerical implementations are presented to demonstrate how to compute spectral properties in the newly proposed comparison framework to achieve a fair and proper comparison. The numerical illustrations demonstrate that the two-sub-step ρ ∞ -Bathe method with ρ ∞ ∈ 0 , 0.1 exhibits improved numerical properties compared to the single-step single-solve KDP- α method. Likewise, the two-sub-step ρ ∞ -Bathe method with ρ ∞ ∈ 0 , 0.2 provides improvements compared to the single-step single-solve TPO/G- α method. However, outside these ranges, the ρ ∞ -Bathe method is inferior in the sense of spectral accuracy to the KDP- α and TPO/G- α methods. Additional comparisons involving three- and four-sub-step algorithms are also included to further highlight the trade-offs and advantages/disadvantages relative to single-step single-solve methods. Besides the theoretical formulations, the numerical analysis is additionally verified by the stiff spring–mass, hardening spring, and elastic spring-pendulum problems. Furthermore, other results from the literature are revisited for the proper comparison.
Wang et al. (Mon,) studied this question.