An immersed boundary method for fluids using the XFEM and the hydrodynamic Boltzmann transport equation

This paper presents a stabilized finite element formulation of the hydrodynamic Boltzmann transport equation (HBTE) to predict nearly incompressible fluid flow. The HBTE is discretized with Hermite polynomials in the velocity variable, and a streamline upwind Petrov–Galerkin formulation is used to discretize the spatial variable. A nonlinear stabilization scheme is presented, from which a simple linear stabilization scheme is constructed. In contrast to the Navier–Stokes (NS) equations, the HBTE is a first order equation and allows for conveniently enforcing Dirichlet conditions along immersed boundaries. A simple and efficient formulation for enforcing Dirichlet boundary conditions is presented and its accuracy is studied for immersed boundaries captured by the extended finite element method (XFEM). Numerical experiments indicate that both the linear and non-linear stabilization methods are sufficiently accurate and stable, but the linear formulation reduces the computational cost significantly. The accuracy of enforcing boundary conditions is satisfactory and shows second order convergence as the mesh is refined. Augmenting the boundary condition formulation with a penalty term increases the accuracy of enforcing the boundary condition constraints, but may degrade the accuracy of the global solution. Comparisons with results of a single relaxation time lattice Boltzmann method show that the proposed finite element method features greater robustness and lesser dependence of the computational costs on the level of mesh refinement.