I have a problem when I want to use the high order center difference approximation:
$$\left(\frac{-u_{i+2,j}+16u_{i+1,j}-30u_{i,j}+16u_{i-1,j}-u_{i-2,j}}{12}\right)$$
for the Poisson equation
$$(u_{xx}+u_{yy}=0)$$ in a square domain in which the boundary conditions are:
$$u(0,y)=u(x,0)=u(x,1)=0,u(1,y)=\sin \pi y$$ $$\Delta{x}=\Delta{y}=0.1$$
When I want to obtain the value of inside points of domain, considering this approximation some points depend on the outside points of boundary. For example, $u_{1,1}$ needs to have the value of $u_{i-2,j}=u_{-1,0}$ a point which is outside of boundary. Can anybody please help me in this case?