We present a novel framework for the development of fourth-order lattice Boltzmann schemes to tackle multidimensional nonlinear systems of conservation laws. Our numerical schemes preserve two fundamental characteristics inherent in classical lattice Boltzmann methods: a local relaxation phase and a transport phase composed of elementary shifts on a Cartesian grid. Achieving fourth-order accuracy is accomplished through the composition of second-order time-symmetric basic schemes utilizing rational weights. This enables the representation of the transport phase in terms of elementary shifts. Introducing local variations in the relaxation parameter during each stage of relaxation ensures the entropic nature of the schemes. This not only enhances stability in the long-time limit but also maintains fourth-order accuracy. To validate our approach, we conduct comprehensive testing on scalar equations and systems in both one and two spatial dimensions.

翻译：我们提出了一个新颖的框架，用于构建四阶格子玻尔兹曼格式，以求解多维非线性守恒律系统。我们的数值格式保留了经典格子玻尔兹曼方法的两个基本特征：局部松弛阶段以及由笛卡尔网格上基本平移组成的输运阶段。通过采用有理权重组合二阶时间对称基本格式，可实现四阶精度，从而使输运阶段能够以基本平移形式表示。在每个松弛阶段中引入松弛参数的局部变化，确保了格式的熵特性。这不仅增强了长时间极限下的稳定性，同时保持了四阶精度。为验证该方法，我们在标量方程和系统上进行了单维与二维空间中的全面测试。