Asymptotic high order mass-preserving schemes for a hyperbolic model of chemotaxis

Roberto Natalini (1) , Magali Ribot (2, 3)

1 IAC - Istituto per le Applicazioni del Calcolo "Mauro Picone"
2 LJAD - Laboratoire Jean Alexandre Dieudonné
3 COFFEE - COmplex Flows For Energy and Environment

We introduce a new class of finite difference schemes for approximating the solutions to an initial-boundary value problem on a bounded interval for a one dimensional dissipative hyperbolic system with an external source term, which arises as a simple model of chemotaxis. Since the solutions to this problem may converge to non constant asymptotic states for large times, standard schemes usually fail to yield a good approximation. Therefore, we propose a new class of schemes, which use an asymptotic higher order correction, second and third order in our examples, to balance the effects of the source term and the influence of the asymptotic solutions. A special care is needed to deal with boundary conditions, to avoid harmful loss of mass. Convergence results are proven for these new schemes, and several numerical tests are presented and discussed to verify the effectiveness of their behavior.

Mots clés: hyperbolic systems with source; initial-boundary value problem; asymptotic behavior; finite difference schemes; chemotaxis

Domaines: Analyse numérique [math.NA]; Equations aux dérivées partielles [math.AP]

Roberto Natalini, Magali Ribot. Asymptotic high order mass-preserving schemes for a hyperbolic model of chemotaxis. SIAM Journal on Numerical Analysis, 2012, 50 (2), pp.883-905. ⟨10.1137/100803067⟩. ⟨hal-00765703⟩