In this paper, we develop monolithic limiting techniques for enforcing nonlinear stability constraints in enriched Galerkin (EG) discretizations of nonlinear scalar hyperbolic equations. To achieve local mass conservation and gain control over the cell averages, the space of continuous (multi-)linear finite element approximations is enriched with piecewise-constant functions. The resulting spatial semi-discretization has the structure of a variational multiscale method. For linear advection equations, it is inherently stable but generally not bound preserving. To satisfy discrete maximum principles and ensure entropy stability in the nonlinear case, we use limiters adapted to the structure of our locally conservative EG method. The cell averages are constrained using a flux limiter, while the nodal values of the continuous component are constrained using a clip-and-scale limiting strategy for antidiffusive element contributions. The design and analysis of our new algorithms build on recent advances in the fields of convex limiting and algebraic entropy fixes for finite element methods. In addition to proving the claimed properties of the proposed approach, we conduct numerical studies for two-dimensional nonlinear hyperbolic problems. The numerical results demonstrate the ability of our limiters to prevent violations of the imposed constraints, while preserving the optimal order of accuracy in experiments with smooth solutions.

翻译：本文针对非线性标量双曲方程的增广Galerkin（EG）离散格式，发展了用于强化非线性稳定性约束的整体限制技术。为实现局部质量守恒并获得对单元平均值的控制，我们在连续（多重）线性有限元逼近空间中引入了分片常数函数进行增广。所得空间半离散格式具有变分多尺度方法的结构。对于线性平流方程，该方法具有固有稳定性但通常不满足保界性。为满足离散极值原理并确保非线性情形下的熵稳定性，我们采用了适应于局部守恒EG方法结构的限制器：单元平均值通过通量限制器进行约束，而连续分量的节点值则通过针对反扩散单元贡献的裁剪-缩放限制策略进行约束。新算法的设计与分析建立在有限元法中凸限制技术和代数熵修正的最新进展基础上。除证明所提方法具备的理论特性外，我们还对二维非线性双曲问题开展了数值研究。数值结果表明，我们的限制器能够有效防止对设定约束的违反，同时在光滑解实验中保持最优收敛阶。