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.
- W. Melis and G. Samaey, Telescopic projective integration for kinetic equations with multiple relaxation times, 2016. Submitted.
- 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.
- 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.
- 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.