The FLEXBURN neutron transport code is developed by the discrete ordinates (Sn) method to analyze heterogeneous fuel assemblies in light water reactors. The transport equations are formulated with transmission and leakage probabilities in arbitrary convex square meshes. Arbitrary convex square meshes precisely describe fuel assemblies as lattices of cells. The code deals with fuel assemblies including gadolinia doped fuel rods, water rods, or plutonium mixed fuel rods with control blades. The code can make burnup calculation sequentially to high burnup. The results computed by the FLEXBURN code are validated by comparing them with those of the ANISN typical transport code and the KENO-IV Monte Carlo code. The FLEXBURN code provides control blade worth and detailed distributions of flux, power, burnup, and atomic densities in complicated boiling water reactor and pressurized water reactor fuel assemblies.