In this paper, we present a novel class of high-order Runge--Kutta (RK) discontinuous Galerkin (DG) schemes for hyperbolic conservation laws. The new method extends beyond the traditional method of lines framework and utilizes stage-dependent polynomial spaces for the spatial discretization operators. To be more specific, two different DG operators, associated with $\mathcal{P}^k$ and $\mathcal{P}^{k-1}$ piecewise polynomial spaces, are used at different RK stages. The resulting method is referred to as the sdRKDG method. It features fewer floating-point operations and may achieve larger time step sizes. For problems without sonic points, we observe optimal convergence for all the sdRKDG schemes; and for problems with sonic points, we observe that a subset of the sdRKDG schemes remains optimal. We have also conducted von Neumann analysis for the stability and error of the sdRKDG schemes for the linear advection equation in one dimension. Numerical tests, for problems including two-dimensional Euler equations for gas dynamics, are provided to demonstrate the performance of the new method.

翻译：本文提出了一类用于双曲守恒律的新型高阶Runge—Kutta（RK）间断伽辽金（DG）格式。新方法超越了传统的Method of Lines框架，在空间离散算子中采用了阶段依赖的多项式空间。具体而言，在不同RK阶段使用了分别与$\mathcal{P}^k$和$\mathcal{P}^{k-1}$分段多项式空间相关联的两种DG算子，所得方法称为sdRKDG格式。该格式具有更少的浮点运算次数，并可能实现更大的时间步长。对于不含声点的问题，我们观察到所有sdRKDG格式均达到最优收敛阶；对于含声点的问题，我们观察到部分sdRKDG格式仍保持最优收敛阶。我们还通过von Neumann分析了sdRKDG格式在一维线性平流方程中的稳定性和误差。数值实验（包括针对气体动力学二维欧拉方程的问题）验证了新方法的性能。