Comparisons of lattice Boltzmann and finite difference methods for a two-dimensional viscous Burgers equation

Lattice Boltzmann methods have been proposed as a computational means for solving various partial differential equations. We look at three lattice Boltzmann methods and an analogous finite difference method for solving a two-dimensional scalar, viscous Burgers equation with periodic boundary conditions. First, using monotonicity arguments we prove that in the $\ell _1 $ norm the lattice Boltzmann methods converge first-order temporally and second-order spatially. Then we provide some computational results which substantiate the theoretical results. Finally, we compare the lattice Boltzmann and finite difference methods. The basis of comparison includes numerical results with regard to accuracy, order of convergence, performance, and timing measurements; memory requirements; and conservations laws.