WIAS Preprint No. 607, (2000)

Kinetic schemes and initial boundary value problems for the Euler system


  • Dreyer, Wolfgang
  • Herrmann, Michael
  • Kunik, Matthias

2010 Mathematics Subject Classification

  • 82C40 76P05 65P05 76L05


  • Kinetic theory of gases, extended thermodynamics, Maximum Entropy Principle, shock waves




We study kinetic solutions, including shocks, of initial and boundary value problems for the Euler equations of gases. In particular we consider moving adiabatic boundaries, which may be driven either by a given path or because they are subjected to forces.

In the latter case we consider a gas in a cylinder, and the boundary may represent a piston that suffers forces by the incoming and outgoing gas particles. Moreover, we will study periodic boundary conditions.

A kinetic scheme consists of three ingredients: (i) There are periods of free flight of duration τM, where the gas particles move according to the free transport equation. (ii) It is assumed that the distribution of the gas particles at the beginning of each of these periods is given by a MAXWELLian. (iii) The interaction of gas particles with a boundary is described by a so called extension law, that determines the phase density at the boundary, and provides additionally continuity conditions for the the fields at the boundary in order to achieve convergence.

The EULER equations result in the limit τM → 0.

We prove rigorous results for these kinetic schemes concerning (i) regularity, (ii) weak conservation laws, (iii) entropy inequality and (iv) continuity conditions for the fields at the boundaries. The study is supplemented by some numerical examples.

This approach is by no mean restricted to EULER equations or to adiabatic boundaries, but it holds also for other hyperbolic systems, namely those that rely on a kinetic formulation.

Appeared in

  • Transport Theory Statist. Phys., 31 (2002), pp. 1-33, with new title: Kinetic schemes and initial boundary value problems

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