text: Unlike the conduction equation (a finite element solution is used), a numerical solution for the convection–diffusion equation has to deal with the convection part of the governing equation in addition to diffusion. When the Péclet number (Pe) exceeds a critical value, the spurious oscillations result in space and this problem is not unique to finite elements as all other discretization techniques have the same difficulties. In a finite difference formulation, the spatial oscillations are reduced by a family of discretization schemes like upwind scheme. In this method, the basic shape function is modified to obtain the upwinding effect. This method is an extension of Runge–Kutta discontinuous for a convection-diffusion equation.
  For time-dependent equations, a different kind of approach is followed. The finite difference scheme has an equivalent in the finite element method (Galerkin method). Another similar method is the characteristic Galerkin method (which uses an implicit algorithm). For scalar variables, the above two methods are identical.