Preserving Asymptotic and Relaxation Properties in Numerical Discretization of Partially Dissipative Hyperbolic Systems

Concepts de base

The central finite-difference scheme preserves both the asymptotic behavior and the parabolic relaxation limit of one-dimensional partially dissipative hyperbolic systems satisfying the Kalman rank condition.

Résumé

The paper analyzes the preservation of asymptotic properties and relaxation limits when transitioning from the continuous setting of partially dissipative hyperbolic systems to a discrete numerical framework. The key findings are:

1. Asymptotic-preserving property:

   • The central finite-difference scheme preserves the large-time asymptotics of partially dissipative hyperbolic systems under the Kalman rank condition.
   • Time-decay estimates similar to the continuous case are derived for the semi-discretized version of the system, uniformly with respect to the mesh size.

2. Relaxation-preserving property:

   • A novel discrete Littlewood-Paley theory tailored to the central finite-difference scheme is introduced.
   • This allows proving Bernstein-type estimates for discrete differential operators and establishing a new relaxation result.
   • The strong convergence of the discrete linearized compressible Euler equations with damping towards the discrete heat equation is shown, uniformly with respect to the mesh parameter.

The authors develop a refined frequency-based functional framework to handle the stiff relaxation procedures for hyperbolic systems in the discrete setting. The results demonstrate the ability of the central finite-difference scheme to effectively capture the hypocoercive and relaxation properties inherent to partially dissipative hyperbolic systems.

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Idées clés tirées de

Asymptotic-preserving finite difference method for partially dissipative hyperbolic systems

by Timo... à arxiv.org 04-10-2024

https://arxiv.org/pdf/2404.06380.pdf

Questions plus approfondies

How can the proposed numerical framework be extended to handle partially dissipative hyperbolic systems with time- and space-dependent coefficients

To extend the proposed numerical framework to handle partially dissipative hyperbolic systems with time- and space-dependent coefficients, we would need to adapt the discrete framework to accommodate the varying coefficients. This adaptation would involve modifying the discrete operators and the construction of the discrete Besov spaces to account for the spatial and temporal variations in the coefficients. Specifically, the discrete Fourier transform and the localization operators would need to be adjusted to handle the changing coefficients in the hyperbolic systems. Additionally, the stability analysis and the integration by parts formula may need to be revised to account for the time- and space-dependent nature of the coefficients. By incorporating these modifications, the numerical framework can be extended to effectively handle partially dissipative hyperbolic systems with time- and space-dependent coefficients.

What are the potential challenges in applying the asymptotic-preserving and relaxation-preserving properties to nonlinear partially dissipative hyperbolic systems

Applying the asymptotic-preserving and relaxation-preserving properties to nonlinear partially dissipative hyperbolic systems may pose several challenges. One significant challenge is the complexity introduced by the nonlinearity of the systems, which can affect the stability and convergence properties of the numerical methods. Nonlinear systems often exhibit more intricate behavior, requiring sophisticated numerical techniques to accurately capture the asymptotic and relaxation limits. Additionally, the interaction between the dissipative and hyperbolic components in nonlinear systems can lead to more intricate dynamics, making it challenging to preserve the asymptotic and relaxation properties accurately. Ensuring the stability and convergence of the numerical methods in the presence of nonlinearity is a key challenge that needs to be addressed when dealing with nonlinear partially dissipative hyperbolic systems.

Can the discrete Littlewood-Paley theory developed in this work be adapted to study other types of discrete differential operators beyond the central finite-difference scheme

The discrete Littlewood-Paley theory developed in this work can potentially be adapted to study other types of discrete differential operators beyond the central finite-difference scheme. By modifying the frequency localization operators and the construction of the discrete Besov spaces, the theory can be extended to analyze the properties of different discrete differential operators. The adaptation may involve adjusting the localization functions and the frequency decomposition to suit the characteristics of the specific differential operators under consideration. Additionally, the Bernstein-type estimates and the embedding results of the discrete Besov spaces can be generalized to apply to a broader range of discrete differential operators, providing a comprehensive framework for studying various types of discrete differential equations.