Asymptotic-preserving projective integration methods for kinetic equations

Kinetic equations describe a particle system in a position-velocity phase space. In the limit of small mean free path (whether in a hydrodynamic or a diffusive scaling), one obtains a macroscopic limiting equation of hyperbolic of diffusive type.  Improving this approximation by taking into account kinetic effects often results in a very stiff partial differential equation to solve.

Asymptotic-preserving methods aim at solving such equations by constructing schemes that fall onto a scheme for the limiting macroscopic equation in the appropriate limit, resulting in a discretization of which the computational costs is essentially independent of the mean free path.  Projective integration is such a technique that is general, easy to implement, fully explicit and of arbitrary order.

Recent publications

  1. W. Melis, T. Rey, and G. Samaey, Projective and telescopic projective integration for the nonlinear BGK and Boltzmann equations, 2017. Submitted.
  2. W. Melis and G. Samaey, Telescopic projective integration for kinetic equations with multiple relaxation times, Journal of Scientific Computing 2017. In press.
  3. P. Lafitte, W. Melis and G. Samaey, A relaxation method with projective integration for solving nonlinear systems of hyperbolic conservation laws, Journal of Computational Physics 340:1-25, 2017.
  4. P. Lafitte, A. Lejon and G. Samaey, A high-order asymptotic-preserving scheme for kinetic equations using projective integration, SIAM Journal on Numerical Analysis 54:1-33, 2015.
  5. P. Lafitte and G. Samaey, Asymptotic-preserving projective integration schemes for kinetic equations in the diffusion limit, SIAM Journal on Scientific Computing 34:A579-A602, 2012.

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