Finite-Volume Schemes on Unstructured Meshes: Analyzing the Order of Accuracy

Core Concepts

The truncation error on (p+1)-th order polynomials has zero average over the mesh period, which is a necessary condition for finite-volume schemes to exhibit the (p+1)-th order convergence on unstructured meshes.

Abstract

The paper analyzes the convergence rate of finite-volume schemes for linear hyperbolic systems with constant coefficients on unstructured meshes. It focuses on schemes with polynomial reconstruction, the cell-centered multislope method, and edge-based schemes, including the flux correction method.
The key insights are:

If a finite-volume scheme is p-exact on non-uniform meshes and (p+1)-exact on uniform meshes, it may exhibit the (p+1)-th order convergence on non-uniform meshes, a phenomenon known as supra-convergence.

The authors propose a "zero mean error" condition, which states that the truncation error on (p+1)-th order polynomials should have zero average over the mesh period. This condition is necessary for supra-convergence and, under additional assumptions, sufficient for the (p+1)-th order convergence.

The authors verify the zero mean error condition heuristically and rigorously for the schemes considered. This explains the supra-convergence observed in previous studies and provides a unified framework to predict the convergence rate.

Numerical results are presented to demonstrate the accuracy of the multislope method for high-Reynolds number flows, which is attributed to the zero mean error property.

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Key Insights Distilled From

On the order of accuracy of finite-volume schemes on unstructured meshes

by Pavel Bakhva... at arxiv.org 04-08-2024

https://arxiv.org/pdf/2404.04157.pdf

Deeper Inquiries

What are the implications of the zero mean error condition beyond the finite-volume schemes considered in this paper

The zero mean error condition has significant implications beyond the finite-volume schemes discussed in the paper. This condition ensures that the truncation error on higher-order polynomials averages out to zero over the mesh period. This property is crucial for achieving higher-order convergence rates and improving the accuracy of numerical solutions.
The zero mean error condition can be extended to other numerical methods for hyperbolic problems on unstructured meshes. By verifying this condition, developers can enhance the convergence properties and accuracy of various numerical schemes, leading to more reliable simulations and predictions in computational fluid dynamics, structural mechanics, and other fields where hyperbolic problems are prevalent.

Can it be extended to other numerical methods for hyperbolic problems on unstructured meshes

Characterizing the constant CA in the error estimate, which is challenging to quantify, can be improved by considering the mesh scaling strategy. Mesh scaling offers advantages in this regard by providing a systematic approach to refine the mesh while maintaining the properties of the numerical scheme.
By utilizing mesh scaling, the constant CA can be better understood and quantified, as the scaling factor influences the error estimate and convergence properties of the scheme. This strategy allows for a more controlled analysis of the error behavior and helps in optimizing the numerical method for improved accuracy and efficiency.

How can the constant CA, which appears in the error estimate and is hard to quantify, be better characterized

Practical guidelines and heuristics can assist designers of finite-volume schemes in ensuring the zero mean error property during the scheme development process. Some key strategies include:

Periodic Mesh Analysis: Verify the zero mean error condition on periodic meshes to ensure convergence properties hold across the entire computational domain.

Error Analysis: Conduct thorough error analysis on different mesh configurations to identify patterns and trends in the truncation error behavior.

Mesh Refinement Strategies: Implement mesh refinement strategies that preserve the zero mean error property, such as scaling or adaptive mesh techniques.

Numerical Validation: Validate the scheme on benchmark problems with known solutions to confirm the accuracy and convergence properties.

By following these guidelines and incorporating heuristics based on the zero mean error condition, designers can develop finite-volume schemes that exhibit higher-order convergence and improved accuracy on unstructured meshes.