In this paper, an attempt has been made to developed a numerical scheme
for numerical approximation of the convection-diffusion problem in
convection dominant situations. Applying Lagrange interpolation
technique, new expressions are obtained to approximate the variable at
spatial interfaces of the computational domain. Subsequently, these
interface approximations are used to developed a numerical scheme based
on upwind approach in the finite volume method. Crank-Nicolson technique
is used for the approximation along temporal direction. This newly
constructed numerical scheme is unconditionally stable with second order
accuracy along space and time both. The numerical experiments are
performed using the proposed upwind approach and numerical results
confirm the theoretical algorithm. Numerical results produce by our
constructed numerical are compared with conventional finite volume
method. This comparison indicates for convection dominant phenomena the
numerical solution of conventional finite volume method contains with
non-physical oscillations where are our proposed numerical schemes give
a high accurate and stable solution.