Numerical Simulation of Heat Conduction Problems With the Lattice Boltzmann Method (LBM) and Discrete Boltzmann Method (DBM): A Comparative Study

Abstract

Abstract Unlike Fourier’s law, which is built upon the continuum assumption and constitutive equation of energy conservation, kinetic models study the transport phenomena from a more fundamental level and in a more generalized way. The Boltzmann equation (BE), which is one type of kinetic model, is a generalized transport model that can solve any advection-diffusion problem regardless of whether such a problem is advection-dominated or diffusion-dominated. Although the BE has been successfully applied to model fluid transport, which is an advection-dominated process, in this paper, in order to demonstrate the generality of the BE, heat conduction, which is a diffusion-only process, is simulated by two numerical derivatives of the BE: the lattice Boltzmann method (LBM) and the discrete Boltzmann method (DBM). The DBM model presented in this paper is unique in the way that the BE is solved on complete unstructured grids with the finite volume method. Therefore, it is named the finite volume discrete Boltzmann method (FVDBM). Two two-dimensional heat conduction problems with different domain geometries and boundary conditions are simulated by both the LBM and FVDBM and quantitatively compared. From that comparison, it is found that the FVDBM produces a higher level of accuracy than the LBM for problems with curved boundaries, while maintaining the same accuracy as the LBM for problems with straight boundaries. The advantage displayed by the FVDBM approach is the direct result of a more accurate reconstruction of curved boundaries by the utilization of unstructured grids, versus the Cartesian grids necessary for the LBM.