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

Abstract

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 (RK4) method.