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Construction and Performance of Kinetic Schemes for Linear Systems of Conservation Laws

Emmanuel Audusse, Sébastien Boyaval, Virgile Dubos, Minh-Hoang Le

TL;DR

The paper develops vectorial kinetic schemes for linear symmetric-hyperbolic conservation laws, using linear acoustics/elastodynamics as a benchmark. It constructs Maxwellians that enforce a discrete convex entropy extension and leverages isotropy-preserving lattices (D2Q5/D2Q9) to obtain stable, high-fidelity discretizations. Through a parameter-focused design (minimizing diffusion and optimizing CFL), it demonstrates that large-CFL kinetic schemes can outperform standard FD/FV methods in accuracy for smooth solutions. Numerical experiments on Cartesian grids show first-order convergence for ω=1 and second-order convergence for ω=2, with stability and error closely tied to the discrete entropy conditions and diffusion minimization criteria. The results motivate further rigorous analysis and extension of kinetic schemes to broader symmetric-hyperbolic systems.

Abstract

We describe a methodology to build vectorial kinetic schemes, targetting the numerical solution of linear symmetric-hyperbolic systems of conservation laws -a minimal application case for those schemes. Precisely, we fully detail the construction of kinetic schemes that satisfy a discrete equivalent to a convex extension (an additional non-trivial conservation law) of the target system -the (linear) acoustic and elastodynamics systems, specifically -. Then, we evaluate numerically the convergence of various possible kinetic schemes toward smooth solutions, in comparison with standard finite-difference and finite-volume discretizations on Cartesian meshes. Our numerical results confirm the interest of ensuring a discrete equivalent to a convex extension, and show the influence of remaining parameter variations in terms of error magnitude, both for ''first-order'' and ''second-order'' kinetic schemes\,: the parameter choice with largest CFL number (equiv., smallest spurious diffusion in the equivalent equation analysis) has the smallest discretization error.

Construction and Performance of Kinetic Schemes for Linear Systems of Conservation Laws

TL;DR

The paper develops vectorial kinetic schemes for linear symmetric-hyperbolic conservation laws, using linear acoustics/elastodynamics as a benchmark. It constructs Maxwellians that enforce a discrete convex entropy extension and leverages isotropy-preserving lattices (D2Q5/D2Q9) to obtain stable, high-fidelity discretizations. Through a parameter-focused design (minimizing diffusion and optimizing CFL), it demonstrates that large-CFL kinetic schemes can outperform standard FD/FV methods in accuracy for smooth solutions. Numerical experiments on Cartesian grids show first-order convergence for ω=1 and second-order convergence for ω=2, with stability and error closely tied to the discrete entropy conditions and diffusion minimization criteria. The results motivate further rigorous analysis and extension of kinetic schemes to broader symmetric-hyperbolic systems.

Abstract

We describe a methodology to build vectorial kinetic schemes, targetting the numerical solution of linear symmetric-hyperbolic systems of conservation laws -a minimal application case for those schemes. Precisely, we fully detail the construction of kinetic schemes that satisfy a discrete equivalent to a convex extension (an additional non-trivial conservation law) of the target system -the (linear) acoustic and elastodynamics systems, specifically -. Then, we evaluate numerically the convergence of various possible kinetic schemes toward smooth solutions, in comparison with standard finite-difference and finite-volume discretizations on Cartesian meshes. Our numerical results confirm the interest of ensuring a discrete equivalent to a convex extension, and show the influence of remaining parameter variations in terms of error magnitude, both for ''first-order'' and ''second-order'' kinetic schemes\,: the parameter choice with largest CFL number (equiv., smallest spurious diffusion in the equivalent equation analysis) has the smallest discretization error.

Paper Structure

This paper contains 9 sections, 5 theorems, 100 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Vectorial kinetic schemes
  3. Construction of kinetic schemes for 2D elastodynamics
  4. D2Q5 scheme
  5. D2Q9 scheme
  6. Numerical evaluation
  7. Finite Difference and Finite Volume schemes
  8. Finite-Difference scheme
  9. Finite-Volume methods

Key Result

Proposition 1

Given $\{c_\zeta, \zeta=0\dots Q-1\}$ consider Maxwellians $\bar{f}_{\zeta}:\mathbb{R}^L\rightarrow\mathbb{R}^L$linear, that satisfy req1--req2 and Then, for any smooth solution to linear conservations laws eq:scl with initial condition $q(t=0)=q^0 \in [H^2(\mathbb{R}^d)]^L$, there exists $C_T>0$ such that it holds $\forall k=T/N$, $N\in\mathbb{N}^*$: when, for each lattice $\{\boldsymbol{a}_{i

Figures (2)

  • Figure 1: Solution \ref{['eq:anasol']} for $\kappa = 1, \mu =2$ computed at $t\in\{0;0.5;1;2\}$ as a function of $r$ using gsl.
  • Figure 2: Errors in $L_2(\{\|\boldsymbol{a}\|\le 2\})$ of approximations of \ref{['eq:anasol']} at $t=1$ when $\kappa=1$, $\mu=2$, $c=\frac{1}{\sqrt{2}}$, by (i) various D2Q5, D2Q9 kinetic schemes, with various values of parameters $\alpha,\gamma,\lambda,\omega$, and (ii) the FD, FV schemes with CFL = $1/2$ -- on $\mathcal{D}_T=(-4,4)^2$ --.

Theorems & Definitions (7)

  • Proposition 1
  • proof
  • Proposition 2
  • Proposition 3
  • proof
  • Proposition 4
  • Proposition 5