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<p>This paper presents a new Lattice Boltzmann formulation for the numerical solution of the Eikonal equation. The method incorporates a nonlinear local constraint within a kinetic relaxation framework and reproduces the Eikonal behavior at the macroscopic level through fully explicit and strictly local updates. In contrast to classical Fast Marching and Fast Sweeping techniques, the proposed approach eliminates sorting procedures, directional sweeps, and global data dependencies. A detailed complexity analysis shows that each iteration performs a constant amount of work per grid node, leading to an overall linear computational cost in practice, since the required number of iterations remains moderate. The memory footprint is likewise linear. Numerical experiments conducted in one-, two-, and three-dimensional settings demonstrate that the method delivers accurate and robust approximations while significantly reducing computational cost. Owing to its locality and explicit structure, the proposed Lattice Boltzmann framework offers an efficient, scalable, and easily parallelizable alternative for large-scale Eikonal simulations.</p>

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Eikonal equation solved with a novel Lattice Boltzmann method framework

  • 期刊论文

  • 发表日期：

    2026年4月20日

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  • Yassine Douich,
  • Hassan Silkan,
  • Youssef Hanyf

摘要

This paper presents a new Lattice Boltzmann formulation for the numerical solution of the Eikonal equation. The method incorporates a nonlinear local constraint within a kinetic relaxation framework and reproduces the Eikonal behavior at the macroscopic level through fully explicit and strictly local updates. In contrast to classical Fast Marching and Fast Sweeping techniques, the proposed approach eliminates sorting procedures, directional sweeps, and global data dependencies. A detailed complexity analysis shows that each iteration performs a constant amount of work per grid node, leading to an overall linear computational cost in practice, since the required number of iterations remains moderate. The memory footprint is likewise linear. Numerical experiments conducted in one-, two-, and three-dimensional settings demonstrate that the method delivers accurate and robust approximations while significantly reducing computational cost. Owing to its locality and explicit structure, the proposed Lattice Boltzmann framework offers an efficient, scalable, and easily parallelizable alternative for large-scale Eikonal simulations.