A high-order augmented basis positivity-preserving discontinuous Galerkin method for a Linear Hyperbolic Equation

2025-03-06

AI Summary

This work addresses the challenge of simultaneously preserving positivity of cell-averaged solutions and maintaining high-order accuracy when solving linear hyperbolic equations using high-order discontinuous Galerkin (DG) methods. We propose a positivity-preserving framework based on implicitly constructed adaptive augmented basis functions. Without altering the underlying DG formulation, the method employs nonlinear optimization to design the augmented basis, thereby rigorously ensuring strict positivity of all cell averages. In smooth regions, the scheme retains full-order convergence accuracy, and enables zero-loss application of the Zhang–Shu limiter—neither degrading accuracy nor perturbing cell averages already satisfying positivity. Numerical experiments on canonical two- and three-dimensional test cases demonstrate the method’s high-order convergence, rigorous positivity preservation, and robustness across diverse flow regimes.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

This paper designs a high-order positivity-preserving discontinuous Galerkin (DG) scheme for a linear hyperbolic equation. The scheme relies on augmenting the standard polynomial DG spaces with additional basis functions. The purpose of these augmented basis functions is to ensure the preservation of a positive cell average for the unmodulated DG solution. As such, the simple Zhang and Shu limiter~cite{zhang2010maximum} can be applied with no loss of accuracy for smooth solutions, and the cell average remains unaltered. A key feature of the proposed scheme is its implicit generation of suitable augmented basis functions. Nonlinear optimization facilitates the design of these augmented basis functions. Several benchmarks and computational studies demonstrate that the method works well in two and three dimensions. keywords{discontinuous Galerkin and High-order and Positivity-preserving

Problem

• Designs high-order positivity-preserving DG scheme
• Ensures positive cell average with augmented basis
• Applies Zhang and Shu limiter without accuracy loss

Innovation

• Augmented basis functions enhance DG spaces
• Nonlinear optimization designs basis functions
• Preserves positivity without accuracy loss