A new approach of implementing initial and boundary conditions for the lattice Boltzmann method is presented. The basic idea is to calculate the lattice Boltzmann populations at a boundary node from the fluid variables that are specified at this node using the gradients of the fluid velocity. The numerical performance of the lattice Boltzmann method is tested on several problems with exact solutions and is also compared to an explicit finite difference projection method. The discretization error of the lattice Boltzmann method decreases quadratically with finer resolution both in space and in time. The roundoff error of the lattice Boltzmann method creates problems unless double precision arithmetic is used.