Numerical Solution

In a finite difference solution of the partial differential equation, an equally spaced grid is set ( h_x = h_y for simplicity ) to cover the domain. The number of grid points is $N_x \times N_y$. The grid coordinates are x_i = h_x i and y_j = h_y j. The unknown field and the source are written on this grid.

u(x,y) = u_i,j (11)

S(x,y) = S_i,j (12)

Using the formula for the partial derivative the partial differential equation becomes

$$\frac{1}{h^2} \left( u_{i+1,j} + u_{i-1,j} + u_{i,j+1} + u_{i,j-1} - 4*u_{i,j} \right) = S_{i,j}$$ (13)