Convergence and Error Estimates of Penalization Finite Volume Method for Compressible Navier-Stokes System

Core Concepts

Penalized finite volume method converges to strong solution.

Abstract

The content discusses the convergence and error estimates of a penalization finite volume method for the compressible Navier-Stokes system. It introduces the penalty approach to control domain-related discretization errors, showing that numerical solutions converge to a dissipative weak solution. Extensive numerical experiments confirm theoretical results.

Introduction to compressible fluid flow modeled by Navier-Stokes equations.
Application of penalty method for domain approximation in numerical simulations.
Theoretical analysis on convergence and error estimates of finite volume method.
Comparison between numerical and strong solutions with relative energy tool.
Detailed organization of the paper with sections dedicated to different aspects.

Stats

In numerical simulations a smooth domain occupied by a fluid has to be approximated by a computational domain that typically does not coincide with a physical domain.
We show that numerical solutions of the penalized problem converge to a generalized, the so-called dissipative weak, solution of the original problem.

Quotes

Key Insights Distilled From

Convergence and error estimates of a penalization finite volume method for the compressible Navier-Stokes system

by Mári... at arxiv.org 03-22-2024

https://arxiv.org/pdf/2209.02344.pdf

Deeper Inquiries

How does the penalty approach compare to other methods in controlling discretization errors

The penalty approach in controlling discretization errors offers a unique advantage compared to other methods. By embedding the physical domain into a larger computational domain and imposing penalty terms, this method effectively handles domain-related discretization errors. The use of penalties helps maintain stability and accuracy in numerical simulations by penalizing violations of boundary conditions or constraints within the computational domain. This approach is particularly useful when dealing with complex geometries or boundary conditions that are challenging to represent accurately in numerical models.

What implications do variational crimes have on numerical simulations in fluid dynamics

Variational crimes can significantly impact numerical simulations in fluid dynamics by introducing errors related to the approximation of smooth domains with computational domains. These errors arise due to discrepancies between the physical domain occupied by the fluid and the polygonal computational domain used for simulation. Variational crimes necessitate careful consideration during analysis as they can affect convergence rates, error estimates, and overall accuracy of numerical solutions. By applying methods like penalization techniques, researchers aim to mitigate variational crimes and improve the reliability of simulation results.

How can these findings be applied to real-world engineering problems involving fluid flow

The findings from this study on convergence and error estimates in fluid dynamics simulations using a penalization finite volume method have practical implications for real-world engineering problems involving fluid flow. Engineers working on projects such as aerodynamics, hydrodynamics, or heat transfer can benefit from these insights when developing numerical models for complex systems with irregular boundaries or intricate geometries. By understanding how penalization methods control discretization errors and handle variational crimes, engineers can enhance the accuracy and efficiency of their simulations, leading to better design decisions and optimized performance outcomes in various engineering applications related to fluid dynamics.