In this talk, we present a general high-order fully explicit scheme based on projective integration methods for solving systems of degenerate parabolic equations in general dimensions [4]. The method relies on a discrete BGK approximation of the advection–diffusion equation [1], where projective integration is used as a time integrator to handle the stiff relaxation term. The discrete BGK formulation allows one to exploit the underlying kinetic structure, similarly to lattice Boltzmann methods, by representing the dynamics through multiple velocity scales. On the other hand, the projective integration approach exploits the clear spectral gap in the eigenvalue distribution of the kinetic equation, combining a sequence of small time steps to damp out stiff components with an extrapolation step over a larger time interval [2, 3]. The time step restriction for the projective step is analogous to the CFL condition for advection–diffusion equations. The scheme is validated on several test cases, providing accurate solutions in one and two spatial dimensions.

Bibliographie :

[1] Denise Aregba-Driollet, Roberto Natalini, and Shaoqiang Tang. Explicit diffusive kinetic schemes for nonlinear degenerate parabolic systems. Mathematics of computation, 73(245):63–94, 2004.
[2] C William Gear and Ioannis G Kevrekidis. Projective methods for stiff differential equations: problems with gaps in their eigenvalue spectrum. SIAM Journal on Scientific Computing, 24(4):1091–1106, 2003.
[3] Pauline Lafitte, Annelies Lejon, and Giovanni Samaey. A high-order asymptotic-preserving scheme for kinetic equations using projective integration. SIAM Journal on Numerical Analysis, 54(1):1–33, 2016.
[4] Tommaso Tenna. Projective integration schemes for nonlinear degenerate parabolic systems. arXiv preprint arXiv:2503.05017, 2025.