We provide a general framework to extend the relative entropy method to a class of diffusive relaxation systems with discrete velocities. The methodology is detailed in the toy case of the 1D Jin–Xin model under the diffusive scaling, and provides a direct proof of convergence to the limit parabolic equation in any interval of time, in the regime where the solutions are smooth. Recently, the same approach has been successfully used to show the strong convergence of a vector-BGK model to the 2D incompressible Navier–Stokes equations.

中文翻译：

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一类离散速度BGK模型在扩散弛豫中的相对熵

我们提供了一个通用框架，将相对熵方法扩展到一类具有离散速度的扩散弛豫系统。该方法在扩散标度下的一维Jin-Xin模型的玩具案例中进行了详细介绍，并提供了在任意时间间隔内（在解决方案平滑的情况下）极限抛物线方程收敛的直接证明。最近，成功地使用了相同的方法来显示矢量BGK模型与2D不可压缩Navier–Stokes方程的强收敛性。