Laplace-Heat-equation-with-periodic-boundary-conditions-

Discretization Scheme :An explicit finite difference method is used to discretize the governing equation.

At any interior grid point $(i,j)$ the following equation is valid : $$T_{i,j}^{'} = T_{i,j}^{t} + \frac{ \epsilon}{4}(T_{i+1,j}^{t} + T_{i,j+1}^{t} +T_{i-1,j}^{t} +T_{i,j-1}^{t} -4 T_{i,j}^{t} )$$ At the left boundary using periodic boundary condition : $$(i = 0, j = 1 \xrightarrow[]{} N_y -1) $$ Then $T_{0,j}^{t} = T_{N_x - 1,j}^{t}$ in equation 2d thermal equation $$T_{0,j}^{'} = T_{0,j}^{t} + \frac{ \epsilon}{4}(T_{1,j}^{t} + T_{0,j+1}^{t} +T_{N_x - 1,j}^{t} +T_{0,j-1}^{t} -4 T_{0,j}^{t})$$ At the right boundary using periodic boundary condition : $$(i = N_x - 1, j = 1 \xrightarrow[]{} N_y -1) $$ Then $T_{0,j}^{t} = T_{N_x - 1,j}^{t}$ gives $$T_{N_x - 1,j}^{'} = T_{N_x - 1,j}^{t} + \frac{ \epsilon}{4}(T_{0,j}^{t} + T_{N_x - 1,j+1}^{t} +T_{N_x - 1,j}^{t} +T_{N_x - 1,j-1}^{t} -4 T_{N_x - 1,j}^{t})$$ Note that $(-1,j)$ and ($N_x,j$) are fictitious points outside the computational domain placed at distance of $\Delta x$ from the left and horizontal boundaries