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:
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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.
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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.