AI 中文总结
该研究针对格子玻尔兹曼方法模拟多物理现象时手工推导格式的瓶颈，将通量一阶矩构造推广为自动推导，应用于多个输运方程系统，通过自动PDE2LBM编译器实现，验证了收敛性，生成的GPU内核性能良好，且能从PDE本身推导格式。
AI 中文摘要
传统上，用格子玻尔兹曼方法（LBM）模拟多物理现象时，需为每个目标偏微分方程（PDE）手工推导专门格式，这使物理模型重定向成为劳动密集型瓶颈。为此，我们将一类新提出的LBM格式的通量一阶矩构造视为守恒律的离散动力学松弛近似，并将其逐个手工推导的构造推广为对双曲、抛物和混合型守恒形式系统的单一自动推导。这使正交格子与物理输运解耦。我们在十二个输运方程系统中应用此方法，包括可压缩纳维-斯托克斯-傅里叶流、磁流体动力学、非线性弹性和电磁学。非线性通量直接映射到一阶离散矩，空间梯度通过平流-松弛级联逐点跟踪，用局部动力学更新取代有限体积通量重构。我们将此方法封装在自动PDE2LBM符号编译器中，由无坐标领域特定语言（DSL）驱动，将抽象PDE转换为LBM。使用制造解方法（MMS）在所有系统上的验证证实，双精度下收敛接近二阶，参考和平衡移位公式在单精度下保持收敛。针对平台透明框架OpenLB，生成的GPU内核接近内存带宽上限，单精度下可达峰值的96%。与现有需离散格式作为输入的LBM代码生成器不同，此框架从声明的PDE本身推导格式：平衡、梯度跟踪级联和单位缩放均仅由守恒律得出。
英文摘要
Multiphysics simulation with lattice Boltzmann methods (LBM) requires a scheme hand-derived for each partial differential equation (PDE), a labor-intensive, error-prone bottleneck. We recognize our recently proposed class of LBM schemes as a discrete-kinetic relaxation approximation of conservation laws and generalize its hand derivation to an automated one for systems of hyperbolic, parabolic, and mixed-type conservation laws. The derivation splits into three steps: First, the PDE system is equivalently rearranged into a first-order cascade of conservation laws: every spatial derivative in flux or source becomes an auxiliary variable, recursively for higher derivatives, so all fluxes are algebraic and updates stay local. Second, the augmented system is approximated by a discrete-velocity kinetic relaxation model with linear, constant-coefficient transport: all nonlinearity resides in a local equilibrium embedding the flux exactly in its first moment, trading the low-Mach truncation for an a priori checkable sub-characteristic wave-speed bound. Third, the relaxation system is discretized by a standard LBM, yielding collide-and-stream algorithms running unchanged on existing solvers. A symbolic compiler using a domain-specific language encapsulates these steps: unlike existing LBM code generators, which start from the discrete scheme, it automatically derives equilibrium, gradient-tracking cascade, and grid scaling from the declared PDE alone. We exercise it across twelve PDE systems, including compressible Navier--Stokes--Fourier flow, resistive magnetohydrodynamics, and nonlinear elasticity. Manufactured-solution verification confirms convergence at or near second order in double precision, retained in single precision by a reference- and equilibrium-shifted formulation. Targeting OpenLB, the generated GPU kernels reach up to 96% of the memory-bandwidth roofline.
