Controlling spurious oscillations is crucial for designing reliable numerical schemes for hyperbolic conservation laws. This paper proposes a novel, robust, and efficient oscillation-eliminating discontinuous Galerkin (OEDG) method on general meshes, motivated by the damping technique in [Lu, Liu, and Shu, SIAM J. Numer. Anal., 59:1299-1324, 2021]. The OEDG method incorporates an OE procedure after each Runge-Kutta stage, devised by alternately evolving conventional semidiscrete DG scheme and a damping equation. A novel damping operator is carefully designed to possess scale-invariant and evolution-invariant properties. We rigorously prove optimal error estimates of the fully discrete OEDG method for linear scalar conservation laws. This might be the first generic fully-discrete error estimates for nonlinear DG schemes with automatic oscillation control mechanism. The OEDG method exhibits many notable advantages. It effectively eliminates spurious oscillations for challenging problems across various scales and wave speeds, without problem-specific parameters. It obviates the need for characteristic decomposition in hyperbolic systems. It retains key properties of conventional DG method, such as conservation, optimal convergence rates, and superconvergence. Moreover, it remains stable under normal CFL condition. The OE procedure is non-intrusive, facilitating integration into existing DG codes as an independent module. Its implementation is easy and efficient, involving only simple multiplications of modal coefficients by scalars. The OEDG approach provides new insights into the damping mechanism for oscillation control. It reveals the role of damping operator as a modal filter and establishes close relations between the damping and spectral viscosity techniques. Extensive numerical results confirm the theoretical analysis and validate the effectiveness and advantages of the OEDG method.

翻译：抑制伪振荡是设计双曲守恒律可靠数值格式的关键。受[Lu, Liu, and Shu, SIAM J. Numer. Anal., 59:1299-1324, 2021]中阻尼技术的启发，本文提出了一种新颖、鲁棒且高效的消振荡间断伽辽金（OEDG）方法，适用于一般网格。OEDG方法在每个龙格-库塔阶段后引入一个消振荡（OE）步骤，该步骤通过交替演化经典半离散DG格式与一个阻尼方程实现。我们精心设计了一种新的阻尼算子，使其具有尺度不变性和演化不变性。针对线性标量守恒律，我们严格证明了全离散OEDG方法的最优误差估计。这可能是首个针对具有自动振荡控制机制的非线性DG格式的通用全离散误差估计。OEDG方法展现出诸多显著优势：它能有效消除跨尺度和波速的难题中的伪振荡，且无需问题相关参数；避免了双曲系统中所需的特征分解；保留了经典DG格式的关键性质，如守恒性、最优收敛率和超收敛性；此外，在标准CFL条件下保持稳定。OE步骤是非侵入式的，便于作为独立模块集成到现有DG代码中。其实现简单高效，仅需将模态系数与标量进行简单乘法。OEDG方法为振荡控制的阻尼机制提供了新见解，揭示了阻尼算子在模态滤波中的作用，并建立了阻尼与谱粘性技术之间的紧密联系。大量数值结果证实了理论分析，验证了OEDG方法的有效性与优势。