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In this contribution, we study a stability notion for a fundamental linearone-dimensional lattice Boltzmann scheme, this notion being related to themaximum principle. We seek to characterize the parameters of the scheme thatguarantee the preservation of the non-negativity of the particle distributionfunctions. In the context of the relative velocity schemes, we derive necessaryand sufficient conditions for the non-negativity preserving property. Theseconditions are then expressed in a simple way when the relative velocity isreduced to zero. For the general case, we propose some simple necessaryconditions on the relaxation parameters and we put in evidence numerically thenon-negativity preserving regions. Numerical experiments show finally that nooscillations occur for the propagation of a non-smooth profile if thenon-negativity preserving property is satisfied.

A stability property for a mono-dimensional three velocities scheme with relative velocity