When finite-difference methods are used to solve the benchmark problem of natural convection in a square cavity, a very fine grid is required to obtain predictions that are accurate to 1-2%. The derivation of the finite-difference equations requires the introduction of many approximations; this study systematically evaluates these approximations to establish which are mainly responsible for the fine-grid requirement. The poorest approximations are then improved one by one, resulting in a scheme that yields highly accurate predictions using a relatively coarse grid. The method of evaluating the accuracy of the approximations, the improved approximations themselves, and the solution method used all contain novel features. Storage and computing time requirements for a new sparse matrix solver, which was used in the current study to simultaneously solve for stream function and vorticity, are presented.