Jacobian-free explicit multiderivative general linear methods for hyperbolic conservation laws

We study explicit strong stability preserving (SSP) multiderivative general linear methods (MDGLMs) for the numerical solution of hyperbolic conservation laws. Sufficient conditions for MDGLMs up to four derivatives to be SSP are determined. In this work, we describe the construction of two external stage explicit SSP MDGLMs based on Taylor series conditions, and present examples of constructed methods up to order nine and three internal stages along with their SSP coefficients. It is difficult to apply these methods directly to the discretization of partial differential equations, as higher-order flux derivatives must be calculated analytically. We hence use a Jacobian-free approach based on the recent development of explicit Jacobian-free multistage multiderivative solvers (Chouchoulis et al. J. Sci. Comput. 90, 96, 2022) that provides a practical application of MDGLMs. To show the capability of our novel methods in achieving the predicted order of convergence and preserving required stability properties, several numerical test cases for scalar and systems of equations are provided.