The hybrid difference scheme is a method used in the numerical solution for convection–diffusion problems. It was introduced by Spalding (1970). It is a combination of central difference scheme and upwind difference scheme as it exploits the favorable properties of both of these schemes.
Introduction Source: 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:
∂ ∂ t ( ρ ϕ ) + ∇ ( ρ u ϕ ) = ∇ ( Γ ⋅ grad ϕ ) + S ϕ {\displaystyle {\frac {\partial }{\partial t}}(\rho \phi )+\nabla (\rho \mathbf {u} \phi )\,=\nabla (\Gamma \cdot \operatorname {grad} \phi )+S_{\phi }}
{\displaystyle \;} (1) Where, ρ {\displaystyle \rho } is density, u {\displaystyle \mathbf {u} } is the velocity vector, Γ {\displaystyle \Gamma } is the diffusion coefficient and S ϕ {\displaystyle S_{\phi }} is the source term. In this equation property, ϕ {\displaystyle \phi } can be temperature, internal energy or component of velocity vector u {\displaystyle \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,
∂ ∂ x ( ρ u ϕ ) = ∂ ∂ x ( Γ ∂ ϕ ∂ x ) , 0 < x < L {\displaystyle {\frac {\partial }{\partial x}}(\rho u\phi )\,={\frac {\partial }{\partial x}}\left(\Gamma {\frac {\partial \phi }{\partial x}}\right),\quad 0<x<L\;} {\displaystyle \;} (2) With boundary conditions, ϕ ( 0 ) = ϕ 0 {\displaystyle \phi (0)\,=\phi _{0}} and ϕ ( L ) = ϕ L {\displaystyle \phi (L)\,=\phi _{L}} , where L is the length, ϕ 0 {\displaystyle \phi _{0}} and ϕ L {\displaystyle \phi _{L}} are the given values.
Grid generation Integrating equation 2 over the control volume containing node N, and using Gauss’ theorem i.e.,
∫ C V ∇ ( ρ u ϕ ) d V = ∫ A n ⋅ ( ρ u ϕ ) d A {\displaystyle \int _{CV}\nabla (\rho \mathbf {u} \phi )dV\,=\int _{A}\mathbf {n} \cdot (\rho \mathbf {u} \phi )dA} {\displaystyle \;} (3) Yields the following result,
( ρ u A ϕ ) r − ( ρ u A ϕ ) l {\displaystyle \left(\rho uA\phi \right)_{r}-\left(\rho uA\phi \right)_{l}} = ( Γ A ∂ ϕ ∂ x ) r − ( Γ A ∂ ϕ ∂ x ) l {\displaystyle \left(\Gamma A{\frac {\partial \phi }{\partial x}}\right)_{r}-\left(\Gamma A{\frac {\partial \phi }{\partial x}}\right)_{l}} {\displaystyle \;} (4) Where, A is the cross-sectional area of the control volume. The equation must also satisfy the continuity equation, i.e.,
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