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This paper presents a new computational method for solving hyperbolic conservation laws using discontinuous Galerkin (DG) schemes that simultaneously eliminates spurious oscillations near discontinuities and maintains high-order accuracy in smooth regions. By aligning the oscillation-elimination procedure with Runge-Kutta time-stepping coefficients, the authors prove that their fully discrete scheme achieves superconvergence with order k+2 for polynomial degree k, even with nonlinear stabilization active. The theoretical framework includes novel correction functions and projection operators that enable rigorous mathematical proofs in both one and two spatial dimensions.
Why it matters
This advancement resolves a longstanding theoretical gap in numerical methods for shock-dominated flows, such as those in aerospace and fluid dynamics simulations. The method maintains both mathematical rigor and practical robustness without requiring user-tunable parameters, potentially improving the reliability and accuracy of computational predictions for problems involving discontinuities like shock waves.
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⚠️ Preprint – Noch nicht peer-reviewed
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Abstract: 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.