Automated Derivation of Lattice Boltzmann Schemes for Systems of Conservation Laws

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.