Central differencing scheme

In applied mathematics, the central differencing scheme is a finite difference method that optimizes the approximation for the differential operator in the central node of the considered patch and provides numerical solutions to differential equations.^[1] It is one of the schemes used to solve the integrated convection–diffusion equation and to calculate the transported property Φ at the e and w faces, where e and w are short for east and west (compass directions being customarily used to indicate directions on computational grids). The method's advantages are that it is easy to understand and implement, at least for simple material relations; and that its convergence rate is faster than some other finite differencing methods, such as forward and backward differencing. The right side of the convection-diffusion equation, which basically highlights the diffusion terms, can be represented using central difference approximation. To simplify the solution and analysis, linear interpolation can be used logically to compute the cell face values for the left side of this equation, which is nothing but the convective terms. Therefore, cell face values of property for a uniform grid can be written as:^[2]

$${\displaystyle \Phi _{e}={\tfrac {1}{2}}(\Phi _{P}+\Phi _{E})}$$ $${\displaystyle \Phi _{w}={\tfrac {1}{2}}(\Phi _{W}+\Phi _{P})}$$