Discrete effects on the forcing term for the lattice Boltzmann modeling of steady hydrodynamics

•The exact steady solution of LBM under a non-uniform force field is derived.•Two types of discrete force errors are identified.•One, represented as the discrete Laplacian of force and intrinsic to LBM.•Other, given by the gradient of the second-order force moment.•The former is corrected by TRT scheme while the latter by first order force scheme.