TVD and TVB preservation without TVD time discretization for discontinuous Galerkin methods

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Original abstract

Total variation diminishing (TVD) and total variation bounded (TVB) properties are crucial for controlling spurious oscillations in numerical solutions of conservation laws. In the classical Runge--Kutta (RK) discontinuous Galerkin (DG) framework, enforcing these properties is intrinsically tied to TVD time integrators, more commonly known today as strong-stability-preserving (SSP) methods. This reliance imposes severe structural restrictions, including order barriers and incompatibility with various fully discrete DG formulations, ranging from the recent RKDG method with compact stencils (cRKDG) to the widely established Arbitrary DERivative (ADER) DG method. To bypass these constraints, we propose a novel trace-limited corrector framework that preserves the TVD/TVB-in-the-means properties using generic, non-SSP time stepping. Based on Harten's lemma, our key insight is that total variation stability is dictated solely by the cell-average update in the final corrector stage. Consequently, we modify the traces in the numerical fluxes exclusively in the final stage, leaving the intermediate predictor stages unconstrained. This strategy decouples oscillation control from the SSP restriction, accommodates standard RKDG, cRKDG, and ADER-DG predictors, and retains the compactness of the cRKDG framework. We also prove that the limiter does not activate in smooth regions, thereby preserving the underlying accuracy. Finally, numerical experiments are presented to demonstrate the capabilities and robustness of the method.

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