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.

我们研究显式强保稳（SSP）多导数一般线性方法（MDGLM）来求解双曲守恒律的数值解。确定了 MDGLM 最多四个导数成为 SSP 的充分条件。在这项工作中，我们描述了基于泰勒级数条件的两个外部阶段显式 SSP MDGLM 的构造，并给出了高达九阶和三个内部阶段及其 SSP 系数的构造方法的示例。很难将这些方法直接应用于偏微分方程的离散化，因为必须解析计算高阶通量导数。因此，我们使用基于最近开发的显式无雅可比多级多导数求解器（Chouchoulis 等人 J. Sci. Comput. 90 , 96, 2022）的无雅可比方法，该方法提供了 MDGLM 的实际应用。为了展示我们的新方法在实现预测的收敛阶数和保持所需稳定性特性方面的能力，提供了几个标量和方程组的数值测试用例。