一维非线性波的双熵稳定格子玻尔兹曼格式
Stable lattice Boltzmann schemes with a dual entropy approach for monodimensional nonlinear waves
AI总结：
本文采用Bouchut提出的数学框架，通过双熵方法确定格子玻尔兹曼格式的平衡态，用于Burgers方程和非线性声学系统，并证明了非线性稳定性条件，同时指出该方法不适用于浅水方程。
摘要：
我们遵循Bouchut提出的数学框架，在本文中提出了一种用于确定格子玻尔兹曼格式平衡态的双熵方法。该方法用相对于底层守恒律的数学熵的对偶来表达。它似乎为建立离散速度方程组的“H定理”提供了一个良好的数学框架。双熵方法与Burgers方程的D1Q3格子玻尔兹曼格式一起使用。它导致三种不同的粒子平衡分布的显式化，并自然导出了一个非线性稳定性条件。给出了强非线性激波和稀疏波的令人满意的数值结果。我们还证明了双熵方法可以应用于线性和非线性声学系统的D1Q3格子玻尔兹曼格式，并给出了非线性声学中强非线性波的数值结果。我们还建立了一个否定结果：在当前框架下，双熵方法不能用于浅水方程。
英文摘要：
We follow the mathematical framework proposed by Bouchut and present in this contribution a dual entropy approach for determining equilibrium states of a lattice Boltzmann scheme. This method is expressed in terms of the dual of the mathematical entropy relative to the underlying conservation law. It appears as a good mathematical framework for establishing a "H-theorem" for the system of equations with discrete velocities. The dual entropy approach is used with D1Q3 lattice Boltzmann schemes for the Burgers equation. It conducts to the explicitation of three different equilibrium distributions of particles and induces naturally a nonlinear stability condition. Satisfactory numerical results for strong nonlinear shocks and rarefactions are presented. We prove also that the dual entropy approach can be applied with a D1Q3 lattice Boltzmann scheme for systems of linear and nonlinear acoustics and we present a numerical result with strong nonlinear waves for nonlinear acoustics. We establish also a negative result: with the present framework, the dual entropy approach cannot be used for the shallow water equations.
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