发表机构
Southern University of Science and Technology（南方科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出RK对齐的振荡消除DG框架，通过同步伪时间步与RK级系数恢复超收敛性，证明线性守恒律下达到(k+2)阶超收敛，并保持无振荡激波捕捉能力。
AI 中文摘要
非线性稳定化对于双曲守恒律的不连续伽辽金（DG）离散格式不可或缺，然而它通常会破坏超收敛分析所需的精细误差结构。因此，现有理论在很大程度上局限于缺乏振荡控制的线性或半离散格式。本文通过提出一种龙格-库塔（RK）对齐的振荡消除（OE）DG框架来弥合这一差距，该框架恢复了超收敛性质。通过将OE过程中的伪时间步与累积的RK级系数同步，我们解锁了标准OEDG格式无法实现的一种低阶界面误差的抵消机制。我们严格证明了该全离散格式在一维和二维线性守恒律中，对精确解的特定投影实现了$(k+2)$阶超收敛，同时保持了原始方法的无振荡激波捕捉能力。此外，我们建立了一个通用指导原则，用于设计一类展现此类超收敛性的OE型DG格式。关键的理论创新包括构造级对齐修正函数以补偿非线性OE源，以及发现一个保持出流边平均值的二维投影算子，这一性质对于闭合离散移位估计至关重要。数值实验证实了预测的超收敛阶数，并表明RK对齐保留了OEDG在强间断问题上的无参数鲁棒性。
英文摘要
Nonlinear stabilization is indispensable for discontinuous Galerkin (DG) discretizations of hyperbolic conservation laws, yet it typically disrupts the delicate error structure required for superconvergence analysis. Consequently, existing theory has largely been restricted to linear or semi-discrete schemes lacking oscillation control. This paper bridges this gap by proposing a Runge--Kutta (RK) aligned oscillation-eliminating (OE) DG framework that restores the superconvergence properties. By synchronizing the pseudo-time step in the OE procedure with the cumulative RK stage coefficients, we unlock a cancellation mechanism for low-order interface errors that is inaccessible to standard OEDG formulations. We rigorously prove that this fully discrete scheme achieves $(k+2)$-th order superconvergence to a tailored projection of the exact solution for linear conservation laws in both one and two dimensions, while maintaining the non-oscillatory shock-capturing capabilities of the original method. Moreover, we establish a general guiding principle for designing a class of OE-type DG schemes that exhibit such superconvergence. Key theoretical innovations include the construction of stage-aligned correction functions to compensate for nonlinear OE sources and the discovery of a two-dimensional projection operator that preserves outflow-edge averages, a property essential to close the discrete shift estimates. Numerical experiments confirm the predicted superconvergence rates and demonstrate that RK alignment preserves the parameter-free robustness of OEDG for problems with strong discontinuities.
