This paper discusses a new class of optimized implicit-explicit Runge-Kutta methods for the numerical solution of the dispersive and non-dispersive hyperbolic systems. Optimized implicit-explicit methods are formulated for the better stability and dispersion properties. Moreover, for present methods inversion of the coefficient matrices is not necessary, which makes these methods very attractive in terms of computational cost and complexity. To validate the efficiency of the developed methods, we have solved the one- and two-dimensional dispersive rotating shallow water equations and benchmark problems from acoustics. Computed solutions are also compared with the exact and experimental results available in the literature. The present methods compete well with the existing multi-stage time-integration methods in terms of accurately resolving the physical characteristics for the chosen problems. Furthermore, the computational costs of the proposed methods are significantly lower as compared to the four-stage, fourth-order explicit Runge-Kutta (${\mathrm{RK}}_4$) method.

本文讨论了一类用于色散和非色散双曲系统数值解的优化隐式-显式 Runge-Kutta 方法。优化的隐式-显式方法是为获得更好的稳定性和分散特性而制定的。此外，对于目前的方法，不需要系数矩阵的反演，这使得这些方法在计算成本和复杂性方面非常有吸引力。为了验证所开发方法的效率，我们解决了一维和二维色散旋转浅水方程和声学基准问题。计算的解决方案也与文献中可用的精确和实验结果进行了比较。本方法在准确解决所选问题的物理特性方面与现有的多阶段时间积分方法竞争良好。此外，与四阶段、四阶显式 Runge-Kutta 相比，所提出方法的计算成本显着降低（${\mathrm{RK}}_4$） 方法。