The Runge--Kutta (RK) discontinuous Galerkin (DG) method is a mainstream numerical algorithm for solving hyperbolic equations. In this paper, we use the linear advection equation in one and two dimensions as a model problem to prove the following results: For an arbitrarily high-order RKDG scheme in Butcher form, as long as we use the $P^k$ approximation in the final stage, even if we drop the $k$th-order polynomial modes and use the $P^{k-1}$ approximation for the DG operators at all inner RK stages, the resulting numerical method still maintains the same type of stability and convergence rate as those of the original RKDG method. Numerical examples are provided to validate the analysis. The numerical method analyzed in this paper is a special case of the Class A RKDG method with stage-dependent polynomial spaces proposed in arXiv:2402.15150. Our analysis provides theoretical justifications for employing cost-effective and low-order spatial discretization at specific RK stages for developing more efficient DG schemes without affecting stability and accuracy of the original method.

翻译：龙格-库塔（RK）间断伽辽金（DG）方法是求解双曲型方程的主流数值算法。本文以一维和二维线性平流方程为模型问题，证明以下结论：对于任意高阶布彻形式的RKDG格式，只要在最终阶段采用$P^k$近似，即便将所有内RK阶段的DG算子中的$k$次多项式模态降阶为$P^{k-1}$近似，所得数值方法仍保持与原RKDG方法相同类型的稳定性和收敛阶。本文通过数值算例验证了理论分析结果。本文分析的数值方法是arXiv:2402.15150提出的具有阶段依赖多项式空间的A类RKDG方法的特例。我们的分析为在不影响原方法稳定性与精度的前提下，通过在特定RK阶段采用低阶空间离散构造更高效DG格式提供了理论依据。