New optimized implicit-explicit Runge-Kutta methods with applications to the hyperbolic conservation laws

Highlights: • Formulation of a new class of optimized implicit-explicit Runge-Kutta methods. • Developed methods do not require inversion of coefficient matrices-reduced computational cost. • Spectral analysis is used to assess the numerical properties in the spectral plane. • Constrained optimization is performed to freeze the values of free parameters in spectral plane. • Methods are gauged for non-dispersive benchmark problems from acoustics and dispersive shallow water equations. 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.