An optimized Runge–Kutta–Chebyshev method for simulating compressible complex flows

The explicit time-marching method with a stability interval as large as possible is desirable for solving hyperbolic conservation equations in the simulation of compressible flows. The Runge–Kutta–Chebyshev (RKC) methods for the ordinary differential equation take advantage of the properties of the Chebyshev polynomials, and hence, their stability intervals are extended much larger than the traditional ones. In this paper, we find that, with a large time step, the excessive numerical dissipation and dispersion of the RKC schemes will generate some abnormal numerical phenomena or even make the computation interruption. Hence, we first propose a generalized stability polynomial for the second-order RKC methods, and then, an optimized RKC method is constructed by applying the low-dissipation low-dispersion optimization to determine the related parameters. Several numerical experiments involving complex flow fields including shock waves and vortex structures demonstrate that, for simulating the complex flow fields, the new method has improved performances, such as the capability of maintaining vortex shape and the robustness (the relaxed Courant–Friedrichs–Lewy condition for hyperbolic conservation equations).