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Asymptotic-preserving finite difference method for partially dissipative hyperbolic systems

Timothée Crin-Barat, Dragoş Manea

TL;DR

The paper addresses preserving the asymptotic behavior and diffusion (parabolic) limits of partially dissipative hyperbolic systems under discretization. It shows the central finite difference scheme is asymptotic-preserving and relax- preserving, using time-weighted hypocoercive energies and a novel discrete Littlewood–Paley framework with numerically adapted Besov spaces. The authors prove uniform-in-grid decay rates and a strong relaxation limit: the discrete Euler system with damping converges to the discrete heat equation at rate O(ε^2), independently of the mesh width, supported by a frequency-based analysis and Besov estimates. Numerical simulations validate the decay and relaxation results and illustrate uniform convergence with respect to the grid. Overall, the work provides a rigorous, scalable framework for preserving long-time and diffusion-limit behavior in discrete approximations of partially dissipative hyperbolic systems.

Abstract

In this paper, we analyze the preservation of asymptotic properties of partially dissipative hyperbolic systems when switching to a discrete setting. We prove that one of the simplest consistent and unconditionally stable numerical methods - the central finite difference scheme - preserves both the asymptotic behaviour and the parabolic relaxation limit of one-dimensional partially dissipative hyperbolic systems which satisfy the Kalman rank condition. The large time asymptotic-preserving property is achieved by conceiving time-weighted perturbed energy functionals in the spirit of the hypocoercivity theory. For the relaxation-preserving property, drawing inspiration from the observation that solutions in the continuous case exhibit distinct behaviours in low and high frequencies, we introduce a novel discrete Littlewood-Paley theory tailored to the central finite difference scheme. This allows us to prove Bernstein-type estimates for discrete differential operators and leads to a new relaxation result: the strong convergence of the discrete linearized compressible Euler system with damping towards the discrete heat equation, uniformly with respect to the mesh parameter.

Asymptotic-preserving finite difference method for partially dissipative hyperbolic systems

TL;DR

The paper addresses preserving the asymptotic behavior and diffusion (parabolic) limits of partially dissipative hyperbolic systems under discretization. It shows the central finite difference scheme is asymptotic-preserving and relax- preserving, using time-weighted hypocoercive energies and a novel discrete Littlewood–Paley framework with numerically adapted Besov spaces. The authors prove uniform-in-grid decay rates and a strong relaxation limit: the discrete Euler system with damping converges to the discrete heat equation at rate O(ε^2), independently of the mesh width, supported by a frequency-based analysis and Besov estimates. Numerical simulations validate the decay and relaxation results and illustrate uniform convergence with respect to the grid. Overall, the work provides a rigorous, scalable framework for preserving long-time and diffusion-limit behavior in discrete approximations of partially dissipative hyperbolic systems.

Abstract

In this paper, we analyze the preservation of asymptotic properties of partially dissipative hyperbolic systems when switching to a discrete setting. We prove that one of the simplest consistent and unconditionally stable numerical methods - the central finite difference scheme - preserves both the asymptotic behaviour and the parabolic relaxation limit of one-dimensional partially dissipative hyperbolic systems which satisfy the Kalman rank condition. The large time asymptotic-preserving property is achieved by conceiving time-weighted perturbed energy functionals in the spirit of the hypocoercivity theory. For the relaxation-preserving property, drawing inspiration from the observation that solutions in the continuous case exhibit distinct behaviours in low and high frequencies, we introduce a novel discrete Littlewood-Paley theory tailored to the central finite difference scheme. This allows us to prove Bernstein-type estimates for discrete differential operators and leads to a new relaxation result: the strong convergence of the discrete linearized compressible Euler system with damping towards the discrete heat equation, uniformly with respect to the mesh parameter.
Paper Structure (31 sections, 15 theorems, 122 equations, 6 figures)

This paper contains 31 sections, 15 theorems, 122 equations, 6 figures.

Table of Contents

  1. Introduction and informal results
  2. Partially dissipative systems -- propagation of damping through hyperbolic dynamics
  3. Asymptotic-preserving schemes for partially dissipative systems
  4. Relaxation-preserving scheme
  5. Main results
  6. Asymptotic-preserving property of the central finite difference scheme
  7. Strong relaxation limit in the semi-discrete setting
  8. A new frequency-based discrete framework
  9. Discrete strong relaxation of the incompressible linearized Euler system
  10. Presentation of the discrete framework
  11. Discrete Fourier transform
  12. Standard Finite difference schemes
  13. The Fourier transform of finite difference schemes and stability results
  14. Properties of the central finite difference operator
  15. Discrete Besov spaces
  16. ...and 16 more sections

Key Result

Theorem 1.2

Let $U_{0}\in H^1(\mathbb{R})$, $A$ and $B$ be symmetric $N\times N$ matrices, with $B$ as in BD, satisfying the Kalman rank condition. Then, for all $t>0$, the solution $U$ of SystGen with the initial datum $U_{0}$ satisfies where $C>0$ is a constant independent of time and $U_0$.

Figures (6)

  • Figure 1: In blue: Plot of the function $\zeta\rightarrow \frac{\sin(\zeta)}{\zeta}$. $M_c$ is the value of the function at the high-frequency thresholds $\pm(\pi-c)$, where $c$ is a constant in $\left(0,\frac{\pi}{2}\right)$. Below this value, i.e below the red line, the analysis needs special treatment compared to the continuous setting.
  • Figure 2: The decomposition of the frequency space in the continuous case \ref{['eq:intro-continuous-freq-band']} (left) and in the discrete setting \ref{['eq:intro-discrete-freq-band']} (right), for $h=2^{-4}$.
  • Figure 3: The semi-log plot of the large time behaviour of the solution of \ref{['mainR:inEuler']} with parameters $\varepsilon=1$ and $h=2^{-4}$.
  • Figure 4: The first component $\rho^{\varepsilon}$ of the solution of \ref{['mainr:Systn']} (blue) approximates the solution $\rho$ of the heat equation \ref{['mainR:heat']} (red) as ${\varepsilon}\rightarrow 0$. The plots were generated for $h=2^{-4}$ and $T=5$.
  • Figure 5: The log-log plot of the approximation error and Darcy law in $l^\infty_h$, obtained in Corollary \ref{['cor:lInftyRelax']}, as a function of $\varepsilon$, for fixed $h=2^{-4}$ and $T=5$.
  • ...and 1 more figures

Theorems & Definitions (31)

  • Definition 1.1
  • Theorem 1.2: CBSZ
  • Remark 1.3
  • Theorem 2.1: Numerical hypocoercivity for hyperbolic systems
  • Remark 2.2
  • Theorem 2.3: Uniform Besov estimates with respect to the grid width
  • Proposition 2.4
  • Definition 2.5: $(s',h)$-truncation
  • Theorem 2.6: Numerical relaxation limit
  • Corollary 2.7
  • ...and 21 more