The authors discuss applications of implicit difference methods to the well-known nonlinear stream function equation with special attention to the stationary case when these methods can be considered as iterative ones. Two basic steps in passing from \(t_n\) to \(t_{n+1}\) are chosen by analogy with the ADI methods and by using a linearization. The steps contain rather involved difference equations in each direction (three parallel lines are used). Possible simplifications of these systems are suggested. The main numerical results are obtained for stationary driven cavity flow and for a flow in a sudden expansion with grids like \(257\times 257\) (\(\operatorname{Re}=100,\;400, \;1000\)). A comparison with known results is made.
It should be noted that theoretically effective iterative methods for nonlinear difference systems (for the stationary stream function equation) were investigated fairly thoroughly (especially for rectangular regions) even in the sixties. Some of such methods, dealing for example with linearization and two-stage iterations, and the corresponding references can be found in [E. G. D’yakonov, Optimization in solving elliptic problems. Boca Raton, FL: CRC Press. xxviii (1996; Zbl 0852.65087)].