Provably Stable Weighted State Redistribution Algorithm for Finite Volume Methods on Cut Cells

Conceitos essenciais

The authors propose a practical finite volume method on cut cells using a provably monotone, total variation diminishing, and GKS stable weighted state redistribution algorithm. The algorithm shuts off continuously as the cut cell size approaches a target value and maintains conservation.

Resumo

The authors introduce a new weighted state redistribution (SRD) algorithm for finite volume methods on cut cells. The key highlights and insights are:

The original SRD algorithm is provably monotone, total variation diminishing, and GKS stable in many situations, but can be subject to a slightly smaller CFL condition.

The new weighted algorithm is designed to be monotone and TVD, while retaining the full CFL condition in most cases. This is achieved by carefully choosing the weights used in the state redistribution step.

The authors analyze the new weighted algorithm and prove its monotonicity and TVD properties for a model linear advection problem in 1D and 2D. They show that pre-merging the initial conditions is crucial for ensuring monotonicity.

The new weights approach zero continuously as the cut cell volume fraction approaches a target threshold, providing a smooth transition between the regular finite volume update and the stabilized cut cell update.

Computational experiments in 2D and 3D demonstrate the improved accuracy and robustness of the new weighted SRD algorithm compared to the original SRD approach, especially in the presence of shocks and complex geometries.

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Principais Insights Extraídos De

A new provably stable weighted state redistribution algorithm

by Marsha Berge... às arxiv.org 04-03-2024

https://arxiv.org/pdf/2308.16332.pdf

Perguntas Mais Profundas

How can the weighted SRD algorithm be extended to handle more complex flow physics, such as viscous effects or chemical reactions, while maintaining the proven stability and monotonicity properties

To extend the weighted SRD algorithm to handle more complex flow physics, such as viscous effects or chemical reactions, while maintaining stability and monotonicity properties, several considerations need to be taken into account.

Viscous Effects:

Introducing viscosity into the algorithm requires modifying the flux terms to include viscous contributions. This can be achieved by adding terms related to the velocity gradients and the dynamic viscosity of the fluid.
Implementing a slope limiter that accounts for viscosity effects can help maintain stability and prevent oscillations near shocks or discontinuities.
Adjusting the weights in the SRD algorithm to consider the viscosity of the fluid and its impact on the flow dynamics can help ensure accurate and stable solutions.

Chemical Reactions:

For systems involving chemical reactions, additional terms related to reaction rates and species concentrations need to be incorporated into the governing equations.
The weighted SRD algorithm can be extended to handle chemical reactions by including reaction terms in the flux calculations and updating the solution based on the reaction kinetics.
Careful consideration of the weights in the algorithm to preserve mass conservation and ensure monotonicity in the presence of chemical reactions is essential.

Complex Flow Physics:

To address more complex flow physics, the algorithm can be adapted to handle multi-phase flows, turbulence models, or other advanced fluid dynamics phenomena.
Incorporating higher-order numerical schemes, such as second-order accurate gradients or TVD Runge-Kutta methods, can improve the accuracy and stability of the algorithm in complex flow scenarios.
Validating the extended algorithm through numerical experiments and benchmark tests for a wide range of flow conditions is crucial to ensure its reliability and effectiveness in handling diverse flow physics.

What are the potential drawbacks or limitations of the weighted SRD approach compared to other cut cell methods, such as cell merging or flux redistribution, and how can these be addressed

The weighted SRD approach offers several advantages in terms of stability, monotonicity, and conservation properties compared to other cut cell methods like cell merging or flux redistribution. However, there are potential drawbacks and limitations that need to be addressed:

Drawbacks of Weighted SRD:

Computational Cost: The weighted SRD algorithm may involve more complex calculations due to the weight adjustments and neighborhood considerations, leading to increased computational overhead.
Accuracy: While the algorithm is designed to be stable and monotone, there may be limitations in accurately capturing sharp gradients or resolving small-scale features in the flow field.
Extension to 3D: Generalizing the algorithm to three-dimensional curved geometries can pose challenges in maintaining stability and monotonicity properties across complex surfaces.

Addressing Limitations:

Improved Weighting Schemes: Developing more sophisticated weighting schemes that adapt to the flow physics and geometry can enhance the accuracy and stability of the algorithm.
Integration of Advanced Numerical Techniques: Incorporating higher-order numerical methods, adaptive mesh refinement, or hybrid approaches can mitigate the limitations of the weighted SRD approach.
Validation and Benchmarking: Conducting thorough validation studies and comparing the algorithm against benchmark problems can help identify and address limitations effectively.

By addressing these drawbacks and limitations through algorithmic enhancements and rigorous testing, the weighted SRD approach can be further optimized for a wider range of flow physics scenarios.

The analysis in this work focuses on linear advection and planar embedded boundaries. How can the theoretical results be generalized to handle nonlinear hyperbolic systems and more complex curved geometries

Generalizing the theoretical results of the analysis to handle nonlinear hyperbolic systems and complex curved geometries involves several key considerations:

Nonlinear Hyperbolic Systems:

Extending the analysis to nonlinear hyperbolic systems requires incorporating nonlinear terms in the governing equations and adapting the stability and monotonicity criteria accordingly.
Implementing appropriate numerical flux functions, slope limiters, and weight adjustments to handle the nonlinearities in the system is essential for maintaining stability and accuracy.

Complex Curved Geometries:

Adapting the theoretical framework to handle complex curved geometries involves developing geometrically flexible algorithms that can accurately represent curved surfaces and boundaries.
Utilizing advanced mesh generation techniques, such as unstructured grids or adaptive mesh refinement, can improve the resolution of curved geometries and enhance the algorithm's performance.
Considering surface reconstruction methods and curvature-aware numerical schemes can help ensure the algorithm's stability and monotonicity properties on non-planar surfaces.

Validation and Verification:

Validating the generalized algorithm through a series of test cases involving nonlinear hyperbolic systems and complex geometries is crucial to verify its stability, accuracy, and robustness.
Conducting convergence studies, benchmark tests, and comparison with analytical solutions can provide insights into the algorithm's performance across a diverse range of flow scenarios.

By systematically extending the theoretical results to nonlinear systems and curved geometries, and rigorously validating the algorithm through numerical experiments, the algorithm's applicability to a broader class of flow physics problems can be established.