一维标量守恒律的双松弛时间动力学近似向解的收敛性
Convergence of a two-relaxation-times kinetic approximation towards the solution of a scalar conservation law
AI总结：
研究一维标量非线性守恒律的双松弛时间动力学近似，通过格子玻尔兹曼格式建立其解的全局存在性并证明收敛性，还通过查普曼-恩斯考格展开等对松弛参数和平衡系数作用进行定性分析。
AI中文摘要：
我们引入了一种用于一维标量非线性守恒律的双松弛时间（TRT）动力学近似，探讨松弛近似向熵解的收敛性。所提出的TRT系统源自格子玻尔兹曼格式，推广了经典的BGK（单松弛时间）框架。通过要求松弛算子的拟单调性，我们建立了TRT系统解的全局存在性，并证明当松弛时间趋于零时，其通过格子玻尔兹曼格式收敛到原始守恒律的熵解。此外，我们通过查普曼-恩斯考格展开和研究行波解，对松弛参数和平衡系数在解的形成中的作用进行了定性分析。
英文摘要：
We introduce a two-relaxation-times (TRT) kinetic approximation for scalar non-linear conservation laws in one space dimension, addressing the convergence of relaxation approximations to entropy solutions. The proposed TRT system, derived from a lattice Boltzmann scheme, generalizes the classical BGK (single-relaxationtime) framework. Requesting quasi-monotonicity of the relaxation operator, we establish global existence of solutions for the TRT system and prove their convergence, using a lattice Boltzmann scheme, to the entropy solution of the original conservation law as the relaxation times vanish. Moreover, we propose a qualitative analysis, clarifying the role of relaxation parameters and equilibrium coefficients in shaping solutions, via Chapman-Enskog expansion and looking at travelling wave solutions.
