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It is well known that the lattice Boltzmann Method (LBM) has been very successful in many computational physics fields. In contrast, numerical theory in the lattice Boltzmann method has made little headway. In order to solve a 2D Burgers equation, a lattice Boltzmann scheme with a BGK model is constructed. The maximum value principle is proved and stability is also obtained. Numerical experiments show the second order convergent accuracy of the scheme.