Bound-preserving and entropy stable enriched Galerkin methods for nonlinear hyperbolic equations

TL;DR

This work addresses nonlinear hyperbolic conservation laws by enforcing bound-preserving and entropy-stable properties within an enriched Galerkin discretization. The authors develop a monolithic convex limiting (MCL) framework that splits the high-order EG discretization into a low-order, bound-preserving, entropy-stable part and a high-order antidiffusive correction, using a flux limiter for cell averages and a clip-and-scale limiter for CG nodal values. They prove discrete maximum principles and entropy inequalities for the semi-discrete schemes and demonstrate through two-dimensional nonlinear tests that the limiters prevent bound violations while preserving optimal convergence on smooth solutions. This constitutes the first successful application of algebraic flux correction to EG discretizations of nonlinear hyperbolic problems and suggests extensions to Euler, shallow-water, and porous-medium models.

Abstract

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.

Figures (7)

• Figure 1: Visualization of the notation for vertices, cells, and faces of a uniform quadrilateral mesh.
• Figure 2: Degrees of freedom for finite element approximations on a patch of four cells.
• Figure 3: Example 1. The convergence behavior of EG errors for LO, HO, and BP schemes.
• Figure 4: Example 2. The diagrams show (a) error behavior and (b) solution profiles along the line $y=0.5$.
• Figure 5: Example 4. Numerical solutions produced by different schemes at $T = 1.0$. The colormap of the filled contour plots represents 25 discrete values in the range $[\frac{\pi}{4},\frac{7\pi}{2}]$.

Theorems & Definitions (3)

• Remark 1
• Remark 2
• Remark 3