🤖 AI Summary
This work addresses the limitation of traditional lattice Boltzmann methods (LBM), which require manual construction of specialized schemes for each class of partial differential equations (PDEs), hindering efficient multiphysics modeling. The authors propose a unified framework that treats systems of conservation laws as discrete kinetic relaxation approximations, enabling automatic derivation of complete LBM formulations directly from conservative PDEs. By leveraging a direct flux-to-moment-space mapping and a convection-relaxation cascade mechanism, the approach intrinsically captures spatial gradients without ad hoc discretization. Built upon a coordinate-free domain-specific language (DSL), the symbolic compiler PDE2LBM automatically generates high-performance GPU kernels compatible with the OpenLB platform. Validation across twelve transport equations demonstrates second-order convergence in double precision, stable convergence in single precision, and GPU implementations achieving up to 96% of peak memory bandwidth.
📝 Abstract
The simulation of multiphysics phenomena with Lattice Boltzmann Methods (LBM) traditionally requires a specialized scheme hand-derived for each targeted Partial Differential Equation (PDE), making the retargeting of physical models a labor-intensive bottleneck. To resolve this, we recognize the flux-in-first-moment construction of a recently proposed class of LBM schemes as a discrete-kinetic relaxation approximation of conservation laws, and generalize their case-by-case, hand-derived construction into a single automated derivation for conservation-form systems of hyperbolic, parabolic, and mixed type. This decouples the quadrature lattice from physical transport, and we exercise the approach across twelve transport-equation systems, including compressible Navier--Stokes--Fourier flow, magnetohydrodynamics, nonlinear elasticity, and electromagnetics. Nonlinear fluxes map directly onto the first-order discrete moments, while spatial gradients are tracked point-wise via advection-relaxation cascades, replacing finite-volume flux reconstruction with local kinetic updates. We encapsulate the approach in an automated PDE2LBM symbolic compiler, driven by a coordinate-free Domain-Specific Language (DSL) that transforms abstract PDEs into LBMs. Validation across all systems using a Method of Manufactured Solutions (MMS) confirms convergence at or near second order in double precision, and the reference- and equilibrium-shifted formulation retains convergence in single precision. Targeting the platform-transparent framework OpenLB, the generated GPU kernels approach the memory-bandwidth roofline, reaching up to 96% of peak in single precision. Unlike existing LBM code generators, which require the discrete scheme as input, this framework derives the scheme from the declared PDE itself: the equilibrium, gradient-tracking cascade, and unit scaling all follow from the conservation law alone.