Maximum-principle-preserving high-order implicit discontinuous Galerkin methods with local Lagrange multipliers for scalar conservation laws

Document Type

Article

Publication Date

12-1-2026

Department

Department of Mathematical Sciences

Abstract

In this paper, we propose a high-order implicit maximum-principle-preserving discontinuous Galerkin method with Lagrange multipliers for scalar conservation laws, which enforces physical bounds while maintaining weakly local mass conservation. Different from the classical local conservation property, where mass conservation is usually imposed on each individual cell, our method preserves mass within each merged region consisting of a small number of cells. We observe that most previous related works adopted the global correction mechanism and may introduce spurious global oscillations for problems with large gradients, as out-of-bound values are corrected through global adjustments. To overcome this limitation, we develop a localized correction strategy based on a bound-preserving cell-merging procedure. Specifically, any cell whose average value violates the admissible bounds is adaptively merged with neighboring cells until the cell average over the merged region satisfies the prescribed bounds. A localized bound-preserving correction is then applied within each merged region. In practice, each merged region contains only a small number of cells and remains significantly smaller than the global domain. This localized mechanism enhances robustness for problems involving sharp gradients, and we theoretically prove that it preserves the original convergence order of the DG approximation under reasonable assumptions. In addition, we extend the proposed approach to the two-dimensional incompressible Euler equations. Since the system contains coupled nonlinear terms, a fully implicit discretization is computationally expensive. To address this difficulty, we employ a tailored implicit–explicit Runge–Kutta scheme that decouples the system into a sequence of linear subproblems. This treatment significantly improves computational efficiency while preserving high-order temporal accuracy. Finally, the performance of the proposed method is examined through numerical tests in terms of accuracy, robustness, and efficiency.

Publication Title

Journal of Computational Physics

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