In this contribution, we study a stability notion for a fundamental linear one-dimensional lattice Boltzmann scheme, this notion being related to the maximum principle. We seek to characterize the parameters of the scheme that guarantee the preservation of the non-negativity of the particle distribution functions. In the context of the relative velocity schemes, we derive necessary and sufficient conditions for the non-negativity preserving property. These conditions are then expressed in a simple way when the relative velocity is reduced to zero. For the general case, we propose some simple necessary conditions on the relaxation parameters and we put in evidence numerically the non-negativity preserving regions. Numerical experiments show finally that no oscillations occur for the propagation of a non-smooth profile if the non-negativity preserving property is satisfied.

在这一贡献中，我们研究了基本线性一维晶格玻尔兹曼格式的稳定性概念，该概念与最大原理有关。我们试图表征该方案的参数，以保证保留粒子分布函数的非负性。在相对速度方案的背景下，我们推导了非负性保持性质的必要和充分条件。然后，当相对速度减小到零时，可以用简单的方式表示这些条件。对于一般情况，我们提出了一些关于松弛参数的简单必要条件，并在数值上证明了非负性保留区域。