Hybrid difference scheme
The hybrid difference scheme[1][2] is a method used in the numerical solution for convection–diffusion problems.
[3][4] Source:[5] Hybrid difference scheme is a method used in the numerical solution for convection-diffusion problems.
These problems play important roles in computational fluid dynamics.
For one-dimensional analysis of convection-diffusion problem in steady state and without the source the equation reduces to, With boundary conditions,
We also define a non-dimensional parameter Péclet number (Pe) as a measure of the relative strengths of convection and diffusion, For a low Peclet number (|Pe|<2) the flow is characterized as dominated by diffusion.
Sources:[3][7] In the above equations (7) and (8), we observe that the values required are at the faces, instead of the nodes.
For example, for the flow to the right (Pe>0)as shown in the diagram, we replace the values as follows; And for Pe < 0, we put the values as shown in the figure 3, By putting these values in equation (7) and rearranging we get the same equation as equation (11), with the following values of the coefficients: Sources:[3][7] The hybrid difference scheme of Spalding (1970) is a combination of the central difference scheme and upwind difference scheme.
It makes use of the central difference scheme, which is second order accurate, for small Peclet numbers (|Pe| < 2).
For large Peclet numbers (|Pe| > 2) it uses the Upwind difference scheme, which first order accurate but takes into account the convection of the fluid.
It produces physically realistic solution and has proved to be helpful in the prediction of practical flows.
The only disadvantage associated with hybrid difference scheme is that the accuracy in terms of Taylor series truncation error is only first order.
