Continuously Bounds-Preserving Discontinuous Galerkin Methods with Exact Nonlinear Limiting

Core Concepts

A modified limiting approach for discontinuous Galerkin methods that achieves exact enforcement of arbitrary nonlinear quasiconcave constraints, reducing unnecessary numerical dissipation.

Abstract

This work proposes an improvement to the continuously bounds-preserving limiting approach for discontinuous Galerkin (DG) schemes introduced in prior work. The key contribution is a modification to the limiting functional that allows for exact enforcement of arbitrary nonlinear quasiconcave constraint functionals, as opposed to just ensuring a sufficient (but not minimum necessary) amount of limiting.
The original approach relied on a linearization of the constraint functionals, which was exact for linear constraints but only sufficient for nonlinear constraints. The proposed modification replaces this linearization with a nonlinear formulation that solves an intersection/root-finding problem to determine the minimum necessary limiting factor. This can be done analytically for some constraints (e.g., quadratic pressure) and numerically for others (e.g., nonlinear entropy).
The benefits of this approach are demonstrated through examples involving the compressible Euler equations, where the new nonlinear limiting approach is shown to significantly reduce numerical dissipation compared to the original linear limiting, while still ensuring the solution satisfies the constraints continuously across the entire element. This is particularly important for applications where the DG solution needs to be evaluated at arbitrary locations, such as in coupled solvers or adaptive mesh refinement.

Stats

The minimum density and pressure values were set as 𝜌min = 𝑃min = 10−11.
For the near-vacuum isentropic Euler vortex problem:

The minimum density at the vortex peak was 𝜌= 8⋅10−9.
The minimum pressure at the vortex peak was 𝑃= 2⋅10−11 = 2𝑃min.

Quotes

"The proposed modification significantly reduces the overall error in the pressure field, with both approximation orders showing decreases between 20−70% across the varying levels of mesh resolution."
"These results highlight the reduced numerical dissipation that stems from the modified constraint functional proposed in this note."

Key Insights Distilled From

A note on higher-order and nonlinear limiting approaches for continuously bounds-preserving discontinuous Galerkin methods

by Tarik Dzanic at arxiv.org 04-22-2024

https://arxiv.org/pdf/2404.12965.pdf

Deeper Inquiries

How could this nonlinear limiting approach be extended to handle more complex constraint functionals, such as those involving coupled systems of equations or non-quasiconcave constraints

To extend the nonlinear limiting approach to handle more complex constraint functionals, such as those involving coupled systems of equations or non-quasiconcave constraints, several modifications and enhancements can be considered:

Coupled Systems of Equations: For systems of equations, the nonlinear limiting approach can be adapted to consider the interdependencies between different variables. This would involve developing a framework to identify the critical constraints across the coupled equations and ensuring that the limiting procedure enforces these constraints accurately. By incorporating cross-variable interactions into the limiting process, the method can be extended to handle complex coupled systems effectively.

Non-Quasiconcave Constraints: Non-quasiconcave constraints pose a challenge due to their nonlinear and potentially non-convex nature. To address this, advanced optimization techniques can be employed to handle the non-quasiconcave nature of the constraints. This may involve utilizing optimization algorithms that can handle non-convex functions to accurately determine the limiting factors for such constraints. By incorporating sophisticated numerical methods, the nonlinear limiting approach can be extended to accommodate a wider range of constraint types.

Adaptive Limiting Strategies: Implementing adaptive strategies that adjust the limiting procedure based on the characteristics of the constraint functionals can enhance the versatility of the approach. By dynamically modifying the limiting factors according to the specific constraints encountered, the method can effectively handle diverse constraint types, including those that are more complex or non-quasiconcave. Adaptive algorithms can optimize the limiting process based on the nature of the constraints, ensuring accurate and efficient enforcement.

By incorporating these enhancements, the nonlinear limiting approach can be tailored to address the intricacies of complex constraint functionals, including those arising from coupled systems of equations or non-quasiconcave constraints.

What are the potential drawbacks or limitations of the proposed nonlinear limiting method, and how could they be addressed

While the proposed nonlinear limiting method offers significant advantages in terms of reducing numerical dissipation and improving accuracy, there are potential drawbacks and limitations that need to be considered:

Computational Complexity: The nonlinear nature of the proposed method, especially when dealing with non-quasiconcave constraints, can lead to increased computational complexity. Solving intersection problems or employing iterative root-finding algorithms may require additional computational resources, potentially impacting the overall efficiency of the simulation.

Convergence and Stability: The iterative nature of solving nonlinear constraints could introduce challenges related to convergence and stability. Convergence issues may arise when iteratively determining the limiting factors for complex constraints, affecting the overall robustness of the method. Ensuring convergence and stability under varying conditions is crucial for the method's reliability.

Generalizability: The applicability of the nonlinear limiting method to a wide range of problems and constraint types needs to be validated. While the approach shows promise in reducing numerical dissipation for specific cases, its generalizability to diverse fluid dynamics problems with varying constraints requires thorough testing and validation.

To address these limitations, further research could focus on optimizing the computational efficiency of the method, enhancing convergence properties, and conducting extensive validation studies across a broader spectrum of fluid dynamics problems.

Given the reduced numerical dissipation, how might this approach impact the long-term stability and accuracy of DG simulations for challenging fluid dynamics problems

The reduced numerical dissipation resulting from the proposed nonlinear limiting approach can have significant implications for the long-term stability and accuracy of Discontinuous Galerkin (DG) simulations, particularly for challenging fluid dynamics problems. Here are some ways in which this approach might impact the stability and accuracy of DG simulations:

Improved Accuracy: By minimizing unnecessary numerical dissipation, the nonlinear limiting method can lead to more accurate solutions, especially in regions where constraints are critical. This enhanced accuracy can result in better representation of physical phenomena and improved predictive capabilities of the DG simulations.

Enhanced Stability: Reduced numerical dissipation can contribute to the overall stability of the simulations, particularly in scenarios where constraints play a crucial role in maintaining the stability of the solution. The method's ability to enforce constraints accurately and continuously can prevent unphysical oscillations or instabilities, enhancing the overall stability of the DG scheme.

Long-Term Behavior: Over extended simulation periods, the decreased numerical dissipation can help maintain the fidelity of the solution by preserving important physical properties and preventing the accumulation of errors. This long-term behavior is essential for simulations of complex fluid dynamics problems where accuracy and stability are paramount.

Overall, the nonlinear limiting approach's impact on DG simulations is expected to result in more stable, accurate, and reliable solutions, particularly in challenging fluid dynamics scenarios where maintaining constraints is crucial for the simulation's fidelity.