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Heat conduction in three two-dimensional (2D) nonlinear lattices, including the purely quartic lattice, the FPU-β lattice and the FPU-αβ lattice, are numerically calculated via both non-equilibrium heat-bath and equilibrium Green-Kubo algorithms.Our simulations for the purely quartic lattice confirm the theoretical expectation that the thermal conductivity κ in such a 2D momentum-conserving lattice diverges with lattice length N logarithmically.