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：我们提出了一种新颖的框架，用于开发四阶格子玻尔兹曼格式，以解决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固有的两个基本特性：局部松弛阶段和基于笛卡尔网格上基本平移的传输阶段。通过利用有理权重组合二阶时间对称基本格式，实现了四阶精度。这使得传输阶段可以用基本平移来表示。在每一层松弛过程中引入局部变化的松弛参数，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，还保持了四阶精度。为验证我们的方法，我们在一维和二维空间中对标量方程及方程组进行了全面测试。