Comparison of Maximum Principle Preserving and Approximately Maximum Principle Preserving Slope Limiters for High-Order Finite Volume Schemes

Core Concepts

There is a fundamental tradeoff between maximum principle violations, long time-scale numerical solution quality, and computational cost when using a priori and a posteriori slope limiters for high-order finite volume schemes.

Abstract

The content presents a comparison of a priori and a posteriori slope limiters for high-order finite volume schemes.
Key highlights:

Developed an experimental methodology to compare a priori and a posteriori slope limiters for arbitrarily high-order finite volume schemes.
Conducted numerical tests for the linear advection equation in one and two dimensions.
Demonstrated a tradeoff between maximum principle violations, long time-scale numerical solution quality, and computational cost.
The a priori limited high-order solutions can preserve a strict maximum principle, but suffer from numerical artifacts and diffusion at high orders and long time-scales.
The a posteriori limited high-order solutions exhibit great quality at long time-scales, but introduce consistent maximum principle violations.
The computational cost of the a priori limited scheme is significantly higher than the a posteriori limited scheme on CPU architecture, due to the reduced CFL condition and increased number of nodal reconstructions required. This cost difference can be almost entirely mitigated using a GPU implementation.

Stats

The maximum CFL factor CMPP and the corresponding L Gauss-Lobatto quadrature points used in the Zhang & Shu slope limiter for polynomial degree p = 0, ..., 7.

Quotes

"The high-order a posteriori limited solutions boast great quality at long time-scales, taking full advantage of the sharp gradients of the high-order finite volume method. However, they introduce consistent maximum principle violations."
"The high-order a priori limited solutions can preserve a strict maximum principle. Interestingly, this is still true when the classic fourth-order Runge-Kutta method is used, despite it not being classified as strong-stability-preserving."
"The a priori limited scheme is forced to use the more costly flux computation, making it significantly more expensive at higher-order than the a posteriori limited scheme on CPU architecture. This cost difference can be almost entirely mitigated using a GPU implementation of the same schemes, highlighting that GPUs are well-suited for high-order finite volume stencil operations."

Key Insights Distilled From

Comparison between a priori and a posteriori slope limiters for high-order finite volume schemes

by Jonathan Pal... at arxiv.org 04-30-2024

https://arxiv.org/pdf/2404.18037.pdf

Deeper Inquiries

How do the a priori and a posteriori limited schemes perform when applied to nonlinear conservation laws, where maximum principle violations can be catastrophic

In the context of nonlinear conservation laws, where maximum principle violations can have catastrophic consequences, the performance of a priori and a posteriori limited schemes differs significantly.

A Priori Limited Schemes: These schemes revise the high-order solution based only on data at the current time, tn. While a priori limited schemes can maintain a strict maximum principle, they may struggle with preserving solution quality over long time scales. This is particularly evident in cases where the solution contains discontinuities, leading to unphysical oscillations and violations of the maximum principle. In such scenarios, a priori limited schemes may exhibit numerical artifacts and diffusion, impacting the overall solution quality.

A Posteriori Limited Schemes: On the other hand, a posteriori limited schemes involve computing a candidate solution at tn+1 and iteratively revising it until certain conditions are met. These schemes are more adaptive and responsive to the evolving nature of the solution, making them better suited for handling discontinuities and maintaining solution quality over extended time scales. However, a common tradeoff with a posteriori schemes is the potential for maximum principle violations, which can be a concern in critical applications of nonlinear conservation laws.
In summary, while a priori limited schemes offer the advantage of strict maximum principle preservation, they may struggle with long-term solution quality in the presence of discontinuities. A posteriori limited schemes, although more adaptive and capable of maintaining solution quality over time, may face challenges with maximum principle violations in certain scenarios.

What modifications or alternative approaches could be explored to mitigate the tradeoffs between maximum principle preservation, solution quality, and computational cost observed in this study

To mitigate the tradeoffs observed in the comparison of slope limiters for finite volume schemes, several modifications and alternative approaches could be explored:

Adaptive Slope Limiting: Implementing adaptive slope limiting techniques that dynamically adjust the degree of slope limitation based on the local characteristics of the solution could help balance maximum principle preservation, solution quality, and computational cost. By selectively activating slope limiting in regions with oscillations or discontinuities, the scheme can maintain accuracy in smooth regions while addressing issues in problematic areas.

Hybrid Approaches: Combining the strengths of a priori and a posteriori limited schemes through hybrid methods could offer a comprehensive solution. By integrating the adaptability of a posteriori schemes with the strict maximum principle preservation of a priori schemes, a hybrid approach could provide a balanced tradeoff between solution quality, maximum principle adherence, and computational efficiency.

Advanced Numerical Techniques: Exploring advanced numerical techniques such as artificial viscosity methods or higher-order flux reconstruction methods could enhance the performance of slope limiters in handling discontinuities and preserving solution quality. These techniques could help mitigate numerical artifacts and diffusion while maintaining the accuracy of the solution.

Machine Learning-Based Approaches: Leveraging machine learning algorithms to adaptively adjust slope limiting parameters based on the evolving solution characteristics could offer a data-driven approach to optimizing the tradeoffs between maximum principle preservation, solution quality, and computational cost. By training models on a diverse set of scenarios, these approaches could enhance the adaptability and efficiency of slope limiters.

What insights from this comparison of slope limiters for finite volume schemes could be extended to the context of high-order finite element methods, where slope limiting is also a critical challenge

Insights from the comparison of slope limiters for finite volume schemes can be extended to the context of high-order finite element methods, where slope limiting is also a critical challenge:

Adaptive Slope Limiting: Similar to finite volume schemes, high-order finite element methods can benefit from adaptive slope limiting techniques that adjust the degree of limitation based on the local solution characteristics. By incorporating adaptive strategies, finite element methods can effectively handle discontinuities while maintaining solution quality and adherence to physical constraints.

A Priori and A Posteriori Limiters: The distinction between a priori and a posteriori slope limiters in finite volume schemes can be translated to finite element methods. A priori limiters can offer strict maximum principle preservation, while a posteriori limiters provide adaptability and responsiveness to evolving solution features. Understanding the tradeoffs between these two approaches is crucial for optimizing the performance of slope limiters in high-order finite element methods.

Smooth Extrema Detection: The implementation of smooth extrema detection, as discussed in the context of finite volume schemes, can also be applied to high-order finite element methods. By identifying smooth extrema and selectively deactivating slope limiting in such regions, finite element methods can enhance solution accuracy and reduce unnecessary limitations in smooth areas.

Hybrid Approaches: Exploring hybrid approaches that combine the strengths of different slope limiting strategies, such as a priori and a posteriori methods, can provide a comprehensive solution for high-order finite element methods. By integrating adaptive, maximum principle-preserving, and solution quality-focused techniques, hybrid approaches can optimize the performance of slope limiters in complex simulations.