Strong Stability Preserving Second Derivative General Linear Methods

In this paper, we find sufficient strong stability preserving (SSP) conditions for second derivative general linear methods (SGLMs). Then we construct some optimal SSP SGLMs of order p up to eight and stage order 1 ≤ q ≤ p with two external stages and 2 ≤ s ≤ 10 internal stages, which have larger effective Courant–Friedrichs–Lewy coefficients than the other existing class of the methods such as two derivative Runge–Kutta methods investigated by Christlieb et al., the class of two-step Runge–Kutta methods introduced by Ketcheson et al., the class of multistep multistage methods studied by Constantinescu and Sandu, and general linear methods investigated by Izzo and Jackiewicz. Some numerical experiments for scalar and systems of equations are provided which demonstrate that the constructed SSP SGLMs achieve the predicted order of convergence and preserve monotonicity.