insight - Scientific Computing - # Computational Fluid Dynamics

Monolithic Convex Limiting and Implicit Pseudo-Time Stepping for Calculating Steady-State Solutions of the Euler Equations: A Proof of Concept and Practical Implementation

Q: How does the computational cost of the proposed implicit MCL scheme compare to that of explicit predictor-corrector approaches for achieving the same level of accuracy in steady-state solutions?

While the paper doesn't provide a direct comparison of computational cost against explicit predictor-corrector approaches, we can infer some key points: Implicit MCL Advantages: Large time steps: Implicit schemes are not bound by the restrictive CFL stability conditions of explicit methods. This allows for significantly larger pseudo-time steps, potentially leading to faster convergence for steady-state solutions. Convergence at high CFL: The paper mentions achieving convergence with CFL numbers as high as 105. This strongly suggests a considerable speed-up compared to explicit methods, especially for problems with stiff source terms or where the steady-state solution is the primary interest. Explicit Predictor-Corrector Advantages: Simpler implementation: Explicit methods are generally easier to implement, as they don't require solving large linear systems. Lower cost per time step: Each time step in an explicit method involves simpler computations compared to an implicit method, which requires solving a linear system. Overall Comparison: For steady-state solutions: The implicit MCL scheme likely holds a significant advantage due to its ability to take much larger time steps. This advantage would be particularly pronounced in cases where high accuracy is desired, as fewer time steps would be needed. For transient solutions: Explicit predictor-corrector methods might be more efficient if high temporal accuracy is required, as the computational overhead of solving the implicit system at each time step could become prohibitive. Further Considerations: The specific implementation details of both methods (e.g., linear solver used for the implicit MCL scheme) would significantly influence the actual computational cost. The problem size and desired accuracy level would also play a role in determining the most efficient approach.

Q: Could the MCL approach be adapted to handle discontinuous solutions, such as shocks, which are common in high-speed compressible flows?

While the MCL approach presented focuses on positivity preservation, which is crucial for stability and physical realizability, handling shocks effectively requires additional considerations: Challenges with Shocks: Oscillations: High-order methods, even with limiters, can introduce spurious oscillations near discontinuities like shocks. Accuracy: Simply limiting the solution to preserve positivity might not be sufficient to accurately capture the sharp gradients present in shocks. Potential Adaptations: Shock capturing: Incorporating nonlinear shock-capturing schemes, such as artificial viscosity or slope limiters specifically designed for shock discontinuities, could improve the solution quality near shocks. Adaptive limiting: Implementing adaptive limiting strategies that adjust the amount of limiting based on local solution smoothness could help maintain high accuracy in smooth regions while providing sufficient dissipation near shocks. Discontinuous Galerkin (DG) methods: Exploring the MCL approach within a DG framework, which naturally handles discontinuities, could be a promising direction. Existing Work: The paper mentions the use of the Rusanov flux, which is a robust Riemann solver often employed in shock-capturing schemes. This suggests that the MCL framework could potentially be extended to incorporate more sophisticated shock-capturing techniques. Overall: Adapting the MCL approach to handle shocks effectively would require further research and development, particularly in integrating appropriate shock-capturing mechanisms while maintaining the method's positivity-preserving properties.

Core Concepts

This paper presents a novel approach for calculating steady-state solutions of the Euler equations using a combination of monolithic convex limiting (MCL) and implicit pseudo-time stepping, proving its invariant domain preserving (IDP) property and demonstrating its effectiveness through numerical experiments.

Abstract

Bibliographic Information: Moujaes, P., & Kuzmin, D. (2024). Monolithic convex limiting and implicit pseudo-time stepping for calculating steady-state solutions of the Euler equations. Journal of Computational Physics. arXiv:2407.03746v2 [math.NA]
Research Objective: To develop an accurate and efficient numerical method for calculating steady-state solutions of the compressible Euler equations while ensuring the preservation of physically admissible states (positivity of density and pressure).
Methodology: The authors employ a continuous Galerkin (CG) finite element method with monolithic convex limiting (MCL) for spatial discretization and a backward Euler scheme for implicit pseudo-time stepping. They prove the IDP property of the fully discrete scheme using a Krasnoselskii-type fixed-point theorem. A low-order Jacobian approximation based on the homogeneity of the flux function is used to construct an efficient iterative solver. Adaptive explicit underrelaxation based on minimizing nodal entropy residuals is employed to accelerate convergence to steady-state solutions.
Key Findings: The proposed MCL scheme with implicit pseudo-time stepping is proven to be IDP, guaranteeing the preservation of physically admissible states. The iterative solver with a low-order Jacobian approximation and adaptive underrelaxation demonstrates robust convergence behavior for a range of test cases, including those with high CFL numbers and Mach numbers.
Main Conclusions: The combination of MCL and implicit pseudo-time stepping provides an effective and robust approach for calculating steady-state solutions of the Euler equations while preserving physical admissibility. The proposed methodology offers advantages over existing explicit predictor-corrector approaches, which are limited by small time steps and do not guarantee positivity preservation for converged solutions.
Significance: This work contributes to the advancement of bound-preserving numerical methods for computational fluid dynamics, particularly for steady-state simulations of compressible flows. The proposed approach addresses limitations of existing methods and offers a promising avenue for developing more robust and accurate solvers for challenging problems in aerodynamics and other fields involving hyperbolic conservation laws.
Limitations and Future Research: The paper primarily focuses on the Euler equations with an ideal gas equation of state. Further research is needed to extend the methodology to more complex equations of state and other hyperbolic systems. Investigating the performance of the proposed approach for three-dimensional problems and exploring alternative limiting strategies are also promising directions for future work.

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CFL numbers as high as 105.
Mach numbers as high as 20.

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Key Insights Distilled From

Monolithic convex limiting and implicit pseudo-time stepping for calculating steady-state solutions of the Euler equations

by Paul Moujaes... at arxiv.org 11-12-2024

https://arxiv.org/pdf/2407.03746.pdf

Monolithic convex limiting and implicit pseudo-time stepping for calculating steady-state solutions of the Euler equations

Deeper Inquiries

How does the computational cost of the proposed implicit MCL scheme compare to that of explicit predictor-corrector approaches for achieving the same level of accuracy in steady-state solutions?

While the paper doesn't provide a direct comparison of computational cost against explicit predictor-corrector approaches, we can infer some key points:
Implicit MCL Advantages:

Large time steps: Implicit schemes are not bound by the restrictive CFL stability conditions of explicit methods. This allows for significantly larger pseudo-time steps, potentially leading to faster convergence for steady-state solutions.
Convergence at high CFL: The paper mentions achieving convergence with CFL numbers as high as 105. This strongly suggests a considerable speed-up compared to explicit methods, especially for problems with stiff source terms or where the steady-state solution is the primary interest.
Explicit Predictor-Corrector Advantages:

Simpler implementation: Explicit methods are generally easier to implement, as they don't require solving large linear systems.
Lower cost per time step: Each time step in an explicit method involves simpler computations compared to an implicit method, which requires solving a linear system.
Overall Comparison:

For steady-state solutions: The implicit MCL scheme likely holds a significant advantage due to its ability to take much larger time steps. This advantage would be particularly pronounced in cases where high accuracy is desired, as fewer time steps would be needed.
For transient solutions: Explicit predictor-corrector methods might be more efficient if high temporal accuracy is required, as the computational overhead of solving the implicit system at each time step could become prohibitive.
Further Considerations:

The specific implementation details of both methods (e.g., linear solver used for the implicit MCL scheme) would significantly influence the actual computational cost.
The problem size and desired accuracy level would also play a role in determining the most efficient approach.

Could the MCL approach be adapted to handle discontinuous solutions, such as shocks, which are common in high-speed compressible flows?

While the MCL approach presented focuses on positivity preservation, which is crucial for stability and physical realizability, handling shocks effectively requires additional considerations:
Challenges with Shocks:

Oscillations:  High-order methods, even with limiters, can introduce spurious oscillations near discontinuities like shocks.
Accuracy:  Simply limiting the solution to preserve positivity might not be sufficient to accurately capture the sharp gradients present in shocks.
Potential Adaptations:

Shock capturing: Incorporating nonlinear shock-capturing schemes, such as artificial viscosity or slope limiters specifically designed for shock discontinuities, could improve the solution quality near shocks.
Adaptive limiting:  Implementing adaptive limiting strategies that adjust the amount of limiting based on local solution smoothness could help maintain high accuracy in smooth regions while providing sufficient dissipation near shocks.
Discontinuous Galerkin (DG) methods:  Exploring the MCL approach within a DG framework, which naturally handles discontinuities, could be a promising direction.
Existing Work:

The paper mentions the use of the Rusanov flux, which is a robust Riemann solver often employed in shock-capturing schemes. This suggests that the MCL framework could potentially be extended to incorporate more sophisticated shock-capturing techniques.
Overall:
Adapting the MCL approach to handle shocks effectively would require further research and development, particularly in integrating appropriate shock-capturing mechanisms while maintaining the method's positivity-preserving properties.

What are the potential applications of this research in fields beyond aerodynamics, such as weather forecasting or astrophysical simulations?

The development of accurate and robust numerical methods for compressible Euler equations has far-reaching implications beyond aerodynamics. Here are some potential applications in other fields:
Weather Forecasting:

Atmospheric modeling: The Euler equations form the basis of many atmospheric models used in weather prediction. The proposed implicit MCL scheme, with its ability to handle high CFL numbers and maintain positivity, could improve the accuracy and efficiency of these simulations, leading to more reliable forecasts.
Cloud resolution: Accurately resolving cloud formation and dynamics is crucial for accurate weather prediction. The positivity-preserving property of the MCL scheme could be particularly beneficial in simulating the condensation and evaporation processes involved.
Astrophysical Simulations:

Star formation: The collapse of interstellar gas clouds leading to star formation involves highly compressible flows and shocks. The MCL approach, potentially enhanced with shock-capturing techniques, could provide valuable insights into these complex phenomena.
Supernova explosions: Simulating supernova explosions requires handling extreme conditions and strong shocks. Robust and accurate numerical methods like the proposed MCL scheme are essential for studying these energetic events.
Accretion disks: Accretion disks around compact objects like black holes involve compressible flows and intricate shock structures. The MCL approach could contribute to a better understanding of these systems.
Other Fields:

Combustion modeling:  Simulating combustion processes requires solving the compressible Euler equations coupled with chemical reaction models. The positivity-preserving property of the MCL scheme is crucial for maintaining the physical feasibility of these simulations.
Multiphase flows:  The MCL approach could be extended to handle multiphase flows, such as bubbly flows or droplet dynamics, where maintaining positivity of each phase's volume fraction is essential.
Overall Impact:
The research presented has the potential to advance numerical simulations in various fields involving compressible flows, ultimately leading to a better understanding of complex physical phenomena and improved predictive capabilities.