Use Taylor series expansion to determine the truncation erro

Use Taylor series expansion to determine the truncation error associated with the up-wind scheme: (rho u)_theta T_theta = max[(rho u)_theta, 0]T_p - max[(rho u)_theta, 0]T_E

Solution

solution;

1)1 D unsteady diffusion equation is given as

e=density,u=velocity,T= diffusing temperature

d/dx(euT)=d/dx(D*T)

D=conductance of surface

by applying finite difference approximation we get

k=e*u*t

m=DT

hence equation becomes

K_i+1ⁿ -k_iⁿ/dx=m _i+1ⁿ -m_iⁿ/dx

on applying taylor series we get

where by applying upwind scheme means considering higher order derivatives we get

[(K_iⁿ +dx(kx)+((dx)^2/2)(kxx)+((dx)^3/6)kxxx+-----)-k_iⁿ]/dx=((m_iⁿ +dx(mx)+((dx)^2/2)(mxx)+((dx)^3/6)mxxx+-----)-m_iⁿ)/dx

on solving we get on left side is diffusionequation and on right side is truncation error due to considering higher order derivatives that is upwind scheme

kx-mx=[(((dx)^2/2)(kxx)+((dx)^3/6)kxxx+-----)-(((dx)^2/2)(mxx)+((dx)^3/6)mxxx+-----)]

4)hence truncation error is

error=kx-mx=[(((dx)^2/2)(kxx)+((dx)^3/6)kxxx+-----)-(((dx)^2/2)(mxx)+((dx)^3/6)mxxx+-----)]