Introduction

The Lattice Boltzmann Method (LBM) has emerged as a powerful alternative to traditional computational fluid dynamics (CFD) approaches. Unlike conventional methods that directly solve the Navier-Stokes equations, LBM is based on kinetic theory and statistical physics, offering a unique perspective on fluid dynamics simulation.

Theoretical Foundation

The Boltzmann Equation

At the heart of LBM lies the Boltzmann equation, which describes the evolution of a particle distribution function f(x, ξ, t):

Introduction to Fast Multipole Method

The Fast Multipole Method (FMM) is a revolutionary algorithm that reduces the computational complexity of N-body problems from O(N²) to O(N). This breakthrough enables efficient simulation of large-scale particle systems and field calculations.

Key Concepts

1. Multipole Expansion

   • Hierarchical representation of particle interactions
   • Far-field approximations
   • Error-controlled truncation

2. Tree-based Structure

   • Octree decomposition
   • Near-field direct interactions
   • Far-field approximations

Integration with Direct Solvers

FASTSolver combines FMM with advanced direct solvers to provide a comprehensive solution for large-scale scientific computing:

Vision

FASTSolver aims to revolutionize scientific computing by providing a seamless, high-performance framework that bridges the gap between ease of use and computational efficiency. Our roadmap outlines the strategic development path toward achieving this vision.

Current State (2025 Q1)

Core Features

• Hybrid Python/C++ architecture

  • Python frontend with NumPy-compatible API:
    • Full compatibility with NumPy’s ndarray operations
    • Support for broadcasting and advanced indexing
    • Array operations with minimal overhead (<5% compared to native NumPy)
    • Transparent data type conversion and memory management
    • Just-in-time compilation for critical code paths
  • C++ computational backend:
    • Optimized computational core using modern C++20 features
    • SIMD vectorization with AVX-512 instruction set
    • Cache-friendly data structures with aligned memory access
    • Template metaprogramming for compile-time optimizations
    • Performance >80% of pure C++ implementations
  • Automated type conversion and memory management:
    • Zero-copy data transfer between Python and C++ layers using Pybind11
    • Intelligent memory pooling for frequent allocations
    • Custom allocator with thread-local caching
    • Reference counting with cycle detection
    • Automatic memory defragmentation

• Advanced numerical solvers