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

TL;DR

This paper develops kinetic schemes for linear hyperbolic systems of conservation laws, demonstrating their effectiveness through numerical convergence tests and highlighting the importance of discrete convex extensions for accuracy.

Contribution

It introduces a methodology for constructing kinetic schemes that satisfy a discrete convex extension, improving numerical solutions of linear symmetric-hyperbolic systems.

Findings

Kinetic schemes with discrete convex extensions improve convergence.

Parameter choices significantly affect error magnitude.

Higher CFL numbers reduce discretization errors.

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.