We propose a formal expansion of multiple relaxation times lattice Boltzmann schemes in terms of a single infinitesimal numerical variable. The result is a system of partial differential equations for the conserved moments of the lattice Boltzmann scheme. The expansion is presented in the nonlinearcase up to fourth order accuracy. The asymptotic corrections of the nonconserved moments are developed in terms of equilibrium values and partial differentials of the conserved moments. Both expansions are coupled and conduct to explicit compact formulas. The new algebraic expressions are validated with previous results obtained with this framework. The example of isothermal D2Q9 lattice Boltzmann scheme illustrates the theoretical framework.

中文翻译：

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格子玻尔兹曼格式的非线性四阶泰勒展开

我们根据单个无穷小数值变量提出了多个弛豫时间晶格Boltzmann方案的形式扩展。结果是一个格子Boltzmann格式的守恒矩的偏微分方程组。扩展在非线性情况下达到了四阶精度。根据守恒矩的平衡值和偏微分，发展了非守恒矩的渐近校正。两种扩展都耦合并执行明确的紧凑公式。新的代数表达式已通过该框架获得的先前结果进行了验证。等温D2Q9格子Boltzmann方案的示例说明了理论框架。