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Hybrid difference scheme is a method used in the numerical solution for convection-diffusion problems.

Introduction

Hybrid difference scheme is a method used in the numerical solution for convection-diffusion problems. These problems play important roles in computational fluid dynamics. It can be described by the general partial equation as follows:

${\frac {\partial }{\partial t}}(\rho \phi )+\nabla (\rho \mathbf {u} \phi )\,=\nabla (\Gamma \cdot {\text{grad}}\phi )+S_{\phi }$$\;$ (1)

Where, $\rho$ is density, $\mathbf {u}$ is the velocity vector, $\Gamma$ is the diffusion coefficient and $S_{\phi }$ is the source term. In this equation property , $\phi$ can be temperature, internal energy or component of velocity vector $\mathbf {u}$ in x, y and z directions.

For one-dimensional analysis of convection-diffusion problem in steady state and without the source the equation reduces to,

${\frac {\partial }{\partial x}}(\rho u\phi )\,={\frac {\partial }{\partial x}}\left(\Gamma {\frac {\partial \phi }{\partial x}}\right),\quad 0<x<L\;$ $\;$ (2)

With boundary conditions, $\phi (0)\,=\phi _{0}$ and $\phi (L)\,=\phi _{L}$ , where L is the length, $\phi _{0}$ and $\phi _{L}$ are the given values.

Grid Generation

Integrating equation (2) over the control volume containing node N, and using the Gauss’ theorem i.e.,

$\int _{CV}\nabla (\rho \mathbf {u} \phi )dV\,=\int _{A}\mathbf {n} \cdot (\rho \mathbf {u} \phi )dA$ $\;$ (3)