Stability Evaluation of Approximate Riemann Solvers Using Direct Lyapunov Method

Core Concepts

Understanding the origins of numerical shock instabilities in approximate Riemann solvers.

Abstract

The content delves into the stability evaluation of approximate Riemann solvers using the direct Lyapunov method. It discusses the causes of numerical shock instabilities in various Riemann solvers, proposing a shock-stable HLLEM scheme. The analysis covers different schemes like HLLE, Roe, HLLEM, HLLC, HLLCM, HLLEC, HLLS, and HLLES, highlighting the impact of pressure perturbations on stability. Linear perturbation analysis and the application of the direct Lyapunov method are explored to assess the stability of these schemes. The study emphasizes the importance of reducing pressure perturbations to mitigate numerical shock instabilities.

Stats

The pressure perturbation feeding the density and transverse momentum perturbations is identified as the cause of numerical shock instabilities.
The magnitude of numerical shock instabilities is proportional to the magnitude of pressure perturbations.
The HLLEM scheme is proposed as shock-stable based on insights from the analysis.

Quotes

"The pressure perturbations feed the density perturbations in the complete approximate Riemann solvers."

Key Insights Distilled From

Stability evaluation of approximate Riemann solvers using the direct Lyapunov method

by Aishwarjya G... at arxiv.org 03-27-2024

https://arxiv.org/pdf/2403.17504.pdf

Deeper Inquiries

How can low Mach corrections reduce pressure perturbations in Riemann solvers

Low Mach corrections can reduce pressure perturbations in Riemann solvers by adjusting the pressure fluctuations to be consistent with the continuous Euler equations. In the context of the Roe scheme and other approximate Riemann solvers, the pressure perturbations are typically of the order of Mach number (O(M*)) at low Mach numbers. By applying low Mach corrections, the pressure perturbations can be reduced to the order of Mach number squared (O(M^2*)). This adjustment helps in minimizing the pressure perturbations that feed into density and momentum perturbations, ultimately reducing the magnitude of numerical shock instabilities. The corrections aim to align the pressure fluctuations with the expected behavior at low Mach numbers, leading to more stable and accurate numerical simulations.

What are the implications of pressure perturbations on the stability of numerical schemes beyond shock instabilities

Pressure perturbations in numerical schemes have implications beyond shock instabilities. While pressure perturbations are often associated with the onset of numerical shock instabilities, their impact extends to the overall stability and accuracy of the numerical scheme. In the context of approximate Riemann solvers like the Roe scheme, pressure perturbations can drive the scheme towards instability by feeding density and momentum perturbations. This can lead to phenomena like carbuncle formation, kinked Mach stems, and post-shock oscillations. Additionally, pressure perturbations can affect the convergence and robustness of the numerical scheme, influencing the overall solution quality. Therefore, controlling and minimizing pressure perturbations is crucial for ensuring the stability and reliability of numerical simulations in computational fluid dynamics.

How does the direct Lyapunov method enhance the understanding of stability in Riemann solvers

The direct Lyapunov method enhances the understanding of stability in Riemann solvers by providing a rigorous and systematic approach to assess the stability of non-linear systems. Unlike linear perturbation analysis or reduced Lyapunov methods, the direct Lyapunov method considers the non-linear nature of the system and evaluates the stability based on a Lyapunov function. By defining a suitable Lyapunov function that satisfies specific criteria, such as positivity and convergence properties, the method can determine the global asymptotic stability of the system. In the context of Riemann solvers like the HLL-family schemes, the direct Lyapunov method offers insights into the stability behavior beyond linear perturbations, highlighting the role of pressure perturbations and their impact on the overall stability of the numerical scheme. This method provides a more refined and comprehensive analysis of stability, aiding in the development of robust and accurate computational fluid dynamics solvers.