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

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.