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Suppressing spurious oscillations is crucial for designing reliable high-order numerical schemes for hyperbolic conservation laws, yet it has been a challenge actively investigated over the past several decades. This paper proposes a novel, robust, and efficient oscillation-eliminating discontinuous Galerkin (OEDG) method on general meshes, motivated by the damping technique (see J. Lu, Y. Liu, and C. W. Shu [SIAM J. Numer. Anal. 59 (2021), pp. 1299–1324]). The OEDG method incorporates an oscillation-eliminating (OE) procedure after each Runge–Kutta stage, and it is devised by alternately evolving the conventional semidiscrete discontinuous Galerkin (DG) scheme and a damping equation. A novel damping operator is carefully designed to possess bothscale-invariantandevolution-invariantproperties. We rigorously prove the optimal error estimates of the fully discrete OEDG method for smooth solutions of linear scalar conservation laws. This might be the first generic fully discrete error estimate fornonlinearDG schemes with an automatic oscillation control mechanism. The OEDG method exhibits many notable advantages. It effectively eliminates spurious oscillations for challenging problems spanning various scales and wave speeds, without necessitating problem-specific parameters for all the tested cases. It also obviates the need for characteristic decomposition in hyperbolic systems. Furthermore, it retains the key properties of the conventional DG method, such as local conservation, optimal convergence rates, and superconvergence. Moreover, the OEDG method maintains stability under the normal Courant–Friedrichs–Lewy (CFL) condition, even in the presence of strong shocks associated with highly stiff damping terms. The OE procedure isnonintrusive, facilitating seamless integration into existing DG codes as an independent module. Its implementation is straightforward 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 the damping operator as a modal filter, establishing close relations between the damping technique and spectral viscosity techniques.Extensive numerical results validate the theoretical analysis and confirm the effectiveness and advantages of the OEDG method. 

We develop a new second-order unstaggered semidiscrete path-conservative central- upwind (PCCU) scheme for ideal and shallow water magnetohydrodynamics (MHD) equations. The new scheme possesses several important properties: it locally preserves the divergence-free constraint, it does not rely on any (approximate) Riemann problem solver, and it robustly produces high- resolution and nonoscillatory results. The derivation of the scheme is based on the Godunov-Powell nonconservative modifications of the studied MHD systems. The local divergence-free property is enforced by augmenting the modified systems with the evolution equations for the corresponding derivatives of the magnetic field components. These derivatives are then used to design a special piecewise linear reconstruction of the magnetic field, which guarantees a nonoscillatory nature of the resulting scheme. In addition, the proposed PCCU discretization accounts for the jump of the nonconservative product terms across cell interfaces, thereby ensuring stability. We test the proposed PCCU scheme on several benchmarks for both ideal and shallow water MHD systems. The obtained numerical results illustrate the performance of the new scheme, its robustness, and its ability not only to achieve high resolution, but also to preserve the positivity of computed quantities such as density, pressure, and water depth.