The weighting function scheme proposed previously has shown great success in solving physical problems without a conservative form such as the wave instability problems and the non-similarity boundary layer flow equations. However, in the previous formulation for the weighting function scheme, the grid is restricted to uniform step size and a modification must be made to force the scheme to obey an important numerical rule. In the present investigation, a new formulation is proposed to reformulate the weighting function scheme such that the constraint of uniform grid can be removed. In addition, the new formulation guarantees the weighting function scheme to satisfy the numerical rule without the need of any further assumption. When applied to conservation equations, the weighting function scheme is seen to become Patankar's exponential scheme for uniform thermal conductivity cases. For cases of variable thermal conductivity, however, the weighting function scheme has a performance superior to that of Patankar's exponential scheme. The weighting function scheme thus is expected to be good for use in solving a turbulent flow near a wall where the eddy viscosity possesses a sharp variation due to the existence of a viscous sublayer adjacent to the solid surface.