Implicit Upwind Discretization of the Scalar Transport Equation

8 Implicit Upwind Discretization of the Scalar Transport Equation 

The governing transport equation for a scalar property tp in an unsteady flow has the general form  

To illustrate the principles of the finite volume method, as a first approach, the implicit upwind differencing scheme is used for a multi-dimensional problem. Although the upwind differencing scheme is very diffusive, this scheme is frequently recommended on the grounds of its stability as the preferred method for treatment of convection terms in multiphase flow and determines the basis for the implementation of many higher order upwinding schemes. 

Introducing the standard finite volume space integrations and integrating over the time interval to using the implicit Euler approximation, we get  

The upwind scheme is obtained by substituting the UDS approximation of the convective terms and central difference approximation of the diffusive terms into the equation. 

---