Computational Fluid Structure Interaction - Computational Hemodynamics: Theory, Modelling and Applications - page 119

∂∂ and / x ∂∂ , / y Fig. 5.22 Finite difference approximations for the first order derivatives

2 2

∂∂ . White circles represent a negative value, while shaded
circles represent a positive value, at a specified point each of which contribute towards the finite
difference approximation

/ x and second order derivatives

Example: 1D Convection-Diffusion Momentum Flow Through a Pipe
We consider momentum equation from Eq. (5.13) in the form of a 1D convection-
diffusion equation as

2

∂
∂

u
t

∂
∂

u
x

∂
∂

u

+ A =+ D

(5.41)

2

x

local convection

diffusion

where A is treated as a constant velocity, and D is /µρ , the kinematic viscosity.
The velocity u is a function of time, t and distance, x . The local acceleration term is
discretised with a forward difference, the convection term with a central differenc-
ing, and the diffusion term with a second difference summarised as

n + 1 n n n 2 n n n

u − u uu

x

− uuu −+

∂
∂

u
t

∂
∂

u
x

∂
∂

u

i i i + 1 i − 1 i + 1 i i − 1

(5.42)

= = =

2

∆ x 2

∆ t 2 ∆

x

Substituting Eq. (5.42) into (5.41) and rearranging for the new value forward in
time we get