A first-order asymptotic preserving scheme for front propagation in a one-dimensional kinetic reaction–transport equation

Abstract

In this work, we propose an asymptotic preserving scheme for a nonlinear kinetic reaction–transport equation, in the regime of sharp interface. With a nonlinear reaction term of KPP-type, a phenomenon of front propagation was proven in [9]. This behaviour can be highlighted by considering a suitable hyperbolic limit of the kinetic equation, using a Hopf–Cole transform. It was proven in [6,8,11] that the logarithm of the distribution function then converges to the viscosity solution of a constrained Hamilton–Jacobi equation.The hyperbolic scaling and the Hopf–Cole transform make the kinetic equation stiff. Thus, the numerical resolution of the problem is challenging, since the standard numerical methods usually lead to high computational costs in these regimes. Asymptotic Preserving (AP) schemes have typically been introduced to deal with this difficulty, since they are designed to be stable along the transition to the macroscopic regime. The scheme we propose is adapted to the non-linearity of the problem, enjoys a discrete maximum principle, and solves the limit equation in the sense of viscosity. It is based on a dedicated micro–macro decomposition attached to the Hopf–Cole transform. As it is well adapted to the singular limit, our scheme is able to cope with singular behaviours in space (sharp interface), and possibly in velocity (concentration in the velocity distribution). Various numerical tests are proposed to illustrate the properties and the efficiency of our scheme.