Finite difference methods moving grid

The described differential equations were solved by the finite difference method of Patankar and Spalding (26). The boundary-layer nature of the problem permits us to solve the equations by moving in the direction of flow for discrete grid points, starting with known initial conditions. 

A number of standard numerical techniques are available to solve the resultant partial differential equation derived from Darcy s law, including the finite difference method (FDM), the boundary element method (BEM) and the finite element method (FEM) [16]. Since mould filling is a moving boundary problem, these numerical techniques can be broadly divided into two approaches into moving grid or fixed grid [17-18] ... 

Now we have to move down one grid step in the z direction, and in order to do this we must use the finite difference versions of conservation equations (i) and (ii). There are two ways to do this, using either the so-called explicit or implicit approximations. We will use the explicit method for illustration here. 

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