Improved Runge–Kutta–Chebyshev methods

Abstract

This study proposes a class of improved Runge–Kutta–Chebyshev (RKC) methods for the stiff systems arising from the spatial discretization of partial differential equations. We can obtain the improved first-order and second-order RKC methods by introducing an appropriate combination technique. The main advantage of our improved RKC methods is that the width of the stability domain along the imaginary axis is significantly increased while the length along the negative real axis has almost no reduction. This implies that our improved RKC methods can extend the application scope of the classical RKC methods. The results of five numerical examples (including the advection–diffusion–reaction equations with dominating advection) show that our improved RKC methods can perform very well.