Finite-Difference Method for Potential Flow

Abstract

For a numerical solution of the potential equation, the region in the (x, y) plane is covered by a grid with sizes Δx, Δy (not necessarily equal); an example is given in Fig. 20.1. If you limit yourself to regular and rectangular grids, the boundaries of the region have to be represented by broken lines through the nearest grid points; the same difficulty was met in Chapter 18. Here, there is no question of staggered grids, as there is only one unknown. If you replace each term in the potential equation (19.4) by the straightforward finite-difference expression (there is not too much of a choice), you get for the special case of Δx= Δy:

$${\phi _k}_{ + 1}{,_j}{\text{ }}{\phi _k} + {\phi _{k - 1,j}} + {\phi _{k,j + 1}} + {\phi _{k,j - 1}} - 4{\phi _{k,j}} = 0$$

((20.1))

If you mark the weighting coefficients for each grid point in Fig. 20.1, you obtain a “finite-difference molecule” connecting each grid point to its four direct neighbours. Each equation (20.1) contains five unknowns, so that the method is implicit and you will have to solve the set of such equations for all grid points together.