THE SELF-STARTING SINGLE-SOLVE TIME INTEGRATORS WITH IDENTICAL SECOND-ORDER ACCURACY

Time integrators are essential numerical methods frequently employed for analyzing large-scale first-order dynamical systems. To ensure the competitiveness of a time integrator, it must meet several key requirements, including being self-starting, single-solve, possessing identical second-order accuracy, and offering controllable numerical dampings. However, current self-starting single-solve time integrators that do not employ auxiliary techniques face a significant limitation: they are unable to simultaneously achieve both controllable numerical dampings and identical second-order accuracy. To address this shortcoming, this paper takes several critical steps. First, a new family of self-starting single-solve time integrators is introduced. These integrators are designed with the goal of achieving both the required numerical dampings and second-order accuracy. Following this, algebraic conditions are derived to ensure that the time integrators meet the criterion of identical second-order accuracy. In the final part of the study, self-starting single-solve implicit and explicit time integrators are proposed, both of which fulfill the condition of identical second-order accuracy. The newly developed implicit integrators are particularly notable for their unconditional stability. One of the distinguishing features of these implicit integrators is their ability to control numerical high-frequency dampings, which is achieved through two user-defined parameters. These parameters influence the eigenvalues of the numerical amplification matrix in the high-frequency limit, allowing for greater flexibility in damping control. When these two parameters are set equal, the implicit integrator simplifies into a single-parameter scheme that strikes an optimal balance between numerical dampings and minimizes the associated error constant. The new explicit integrators, on the other hand, are conditionally stable and incorporate a user-defined spectral radius at the bifurcation point, which serves to regulate the numerical dampings. By choosing the appropriate parameter values, the explicit integrator achieves a balanced trade-off between stability and solution accuracy. Theoretical analysis, along with performance comparisons and numerical examples, confirms the effectiveness of these new self-starting single-solve time integrators, demonstrating their improved accuracy and superior damping capabilities.