A well-balanced scheme for kinetic models of chemotaxis derived from one-dimensional local forward–backward problems

Numerical approximation of one-dimensional kinetic models for directed motion of bacterial populations in response to a chemical gradient, usually called chemotaxis, is considered in the framework of well-balanced (WB) schemes. The validity of one-dimensional models have been shown to be relevant for the simulation of more general situations with symmetry in all but one direction along which appears the chemical attractant gradient. Two main categories are considered depending on whether or not the kinetic equation with specular boundary conditions admits non-constant macroscopic densities for large times. The WB schemes are endowed with the property of having zero artificial viscosity at steady-state; in particular they furnish numerical solutions for which the macroscopic flux vanishes, a feature that more conventional discretizations can miss. A class of equations which admit constant asymptotic states can be treated by a slight variation of the method of Case’s elementary solutions originally developed for radiative transfer problems. More involved models which can display concentrations are handled through a different, but closely related, treatment of the tumbling term at the computational grid’s interfaces. Both types of WB algorithms can be implemented efficiently relying on the Sherman–Morrison formula for computing interface values. Transient and stationary numerical results are displayed for several test-cases.

► New scheme for kinetic models in the discrete ordinate approximation using Caseology.
► Well-balanced property allow to stabilize onto asymptotic patterns with strong gradients.
► Sherman–Morrison formula reduces CPU cost while residues in time decay more strongly.