Hydrodynamic singular regimes in 1 + 1 kinetic models and spectral numerical methods

• Applies spectral theory of stationary kinetic equations to contemporary models of confined matter dynamics.
• Derives original well-balanced schemes involving S-matrices (following early Glimm–Sharp's ideas) for kinetic models like chemotaxis or Vlasov–Fokker–Planck coupled to attractive Poisson equation.
• Hybridizes this framework in order to support strongly singular hydrodynamic limits, where kinetic distributions concentrate into Dirac atoms, for which rigorous results are proved.

Classical results from spectral theory of stationary linear kinetic equations are applied to efficiently approximate two physically relevant weakly nonlinear kinetic models: a model of chemotaxis involving a biased velocity-redistribution integral term, and a Vlasov–Fokker–Planck (VFP) system. Both are coupled to an attractive elliptic equation producing corresponding mean-field potentials. Spectral decompositions of stationary kinetic distributions are recalled, based on a variation of Case's elementary solutions (for the first model) and on a Sturm–Liouville eigenvalue problem (for the second one). Well-balanced Godunov schemes with strong stability properties are deduced. Moreover, in the stiff hydrodynamical scaling, an hybridized algorithm is set up, for which asymptotic-preserving properties can be established under mild restrictions on the computational grid. Several numerical validations are displayed, including the consistency of the VFP model with Burgers–Hopf dynamics on the velocity field after blowup of the macroscopic density into Dirac masses.