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

Time integrators are widely used numerical methods for analyzing large-scale first-order dynamical systems. For the time integrator to be competitive, it must be self-starting, single-solve, identical second-order accuracy, and controllable numerical dampings. However, the existing self-starting single-solve time integrators without auxiliary techniques fail to simultaneously achieve both controllable numerical dampings and identical second-order accuracy. To overcome this limitation, this paper first constructs a family of self-starting single-solve time integrators, then derives algebraic conditions for the integrators to achieve identical second-order accuracy, and finally proposes both implicit and explicit time integrators with identical second-order accuracy. The new implicit integrators are unconditionally stable, with numerical high-frequency dampings controlled via two user-defined parameters that govern the eigenvalues of the numerical amplification matrix in the high-frequency limit. When these two parameters are equal, the implicit integrator reduces to a single-parameter scheme that optimally balances numerical dampings and minimizes the error constant. The new explicit integrators are conditionally stable and utilize a user-defined spectral radius at the bifurcation point to control numerical dampings. When the recommended parameter values are used, the explicit integrator achieves a reasonable balance between the stability region and accuracy. Performance comparisons and numerical examples validate the theoretical analysis and highlight the advantages of the new time integrators in terms of accuracy and damping capabilities.