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Controlling spurious oscillations is crucial for designing reliable numericalschemes for hyperbolic conservation laws. This paper proposes a novel, robust,and efficient oscillation-eliminating discontinuous Galerkin (OEDG) method ongeneral meshes, motivated by the damping technique in [Lu, Liu, and Shu, SIAMJ. Numer. Anal., 59:1299-1324, 2021]. The OEDG method incorporates an OEprocedure after each Runge-Kutta stage, devised by alternately evolvingconventional semidiscrete DG scheme and a damping equation. A novel dampingoperator is carefully designed to possess scale-invariant andevolution-invariant properties. We rigorously prove optimal error estimates ofthe fully discrete OEDG method for linear scalar conservation laws. This mightbe the first generic fully-discrete error estimates for nonlinear DG schemeswith automatic oscillation control mechanism. The OEDG method exhibits manynotable advantages. It effectively eliminates spurious oscillations forchallenging problems across various scales and wave speeds, withoutproblem-specific parameters. It obviates the need for characteristicdecomposition in hyperbolic systems. It retains key properties of conventionalDG method, such as conservation, optimal convergence rates, andsuperconvergence. Moreover, it remains stable under normal CFL condition. TheOE procedure is non-intrusive, facilitating integration into existing DG codesas an independent module. Its implementation is easy and efficient, involvingonly simple multiplications of modal coefficients by scalars. The OEDG approachprovides new insights into the damping mechanism for oscillation control. Itreveals the role of damping operator as a modal filter and establishes closerelations between the damping and spectral viscosity techniques. Extensivenumerical results confirm the theoretical analysis and validate theeffectiveness and advantages of the OEDG method.

OEDG: Oscillation-eliminating discontinuous Galerkin method for hyperbolic conservation laws