A Meshless Discrete Boltzmann Solver With the Radial Basis Function Finite Difference Scheme for Fluid Flows With Dirichlet Boundary Conditions

Abstract The discrete Boltzmann equation (DBE), grounded in kinetic theory, has become increasingly attractive for solving transport phenomena. Typically, applications of the DBE employ the lattice Boltzmann method (LBM), which uses a coupled and uniform time–space discretization. However, the fixed lattice structure required by LBM limits flexibility, particularly in handling complex geometries such as porous media. Meshless methods, specifically radial basis function finite difference (RBF-FD), offer promising alternatives by eliminating the need for structured grids. Nevertheless, solving DBE using unstructured discretization introduces challenges, especially in translating macroscale boundary conditions (such as density and velocity) into the particle density function (PDF). This study addresses these challenges by employing an RBF-FD meshless method to solve the DBE for fluid flow simulations. Initially, the Taylor Green vortex flow is analyzed to validate the RBF-FD meshless approach in boundary-free scenarios, highlighting the artificial viscosity effects introduced by the method itself. Subsequently, a novel scheme for resolving unknown PDF components at Dirichlet boundaries is introduced, which separates the PDF into equilibrium and nonequilibrium parts and employs nearest-neighbor interpolation. The effectiveness of this boundary scheme is demonstrated through tests on both flat and curved surfaces, showing strong agreement with existing literature. The proposed method thus provides a viable solution for accurately applying Dirichlet boundary conditions within meshless DBE simulations, significantly enhancing the applicability of DBE to complex flow problems.