Numerical Solutions Applied To Heat Transfer And Fluid Mechanics Problems
Taylor Series Method for forming finite
difference
equation (DEM)
Basic Mathmatical Requirements:
f(x) is a single valued function and if in the interval a<x<b
Its first (N1) derivatives are continuous
Its Nth derivative exists, f^{N}(x)
Taylor Series in Cartesian coordinates:
where
Example:
Heat conduction two dimensional  steady state with internal heat generation
Approximate each differential using Taylor Series:
Add and truncate
thus
A similar expression is obtained for
We've been using the obvious subscript abbreviation T_{o} for T(x_{o}.y_{o}). We'll simplify notation further with the notation:
Then the finite difference equation at a given point becomes
with an error
The error is abbreviated as:
Improvement of the approximation
Simple option is to decrease the mesh size (increase number of mesh points)
Another option is to Obtain a higher order approximation
High order approximation
We must consider more points on the mesh when working with the equations. Introduce extended notation:
We want an approximation to the second derivative that only involves T_{o}, T_{E}, T_{W,} T_{EE}, and T_{WW}. Look at the Taylor expansions giving the values of T at these 4 points.
Now multiply the equation for T_{E} by the unknown coefficient "A", multiply the equation for T_{W} by the unknown "B", multiply the equation for T_{EE} by the unknown "C", and multiply the equation for T_{WW} by the unknown "D". Sum these modified equations. We want the summed equations to provide an approximation to the second derivative of T, so we require that the coefficient of the second derivative in the summed equation be equal to 1. We further require that the coefficients of the first, third, and fourth derivatives be zero. This gives us four equations that can be solved for values of A, B, C, and D. They are:
Replacing these values in the summed equation and rearranging gives:
What is the expression for the error, ?
Cylindrical coordinate system (DEM)
Example: Steady State Heat Conduction
Assume and, are uniform
In the r, direction, define
Write the Taylor Series expansion for each of these variables
In this instance we are free to either deal with all four expansions as a single sum, or group the radial and theta equations separately. I will follow the pattern in the previous higher order example. Multiply the equation for T_{E} by the unknown coefficient "A", multiply the equation for T_{W} by the unknown "B", multiply the equation for T_{N} by the unknown "C", and multiply the equation for T_{S} by the unknown "D". Sum these modified equations. We want the summed equations to provide an approximation to the radial and azimuthal derivatives in the conduction equations, so we require that the coefficient of the first derivative of T with respect to r in the summed equation be equal to 1/r_{o}. The coefficient of the second radial derivative must be one. We require that the coefficient of the first derivative with respect to theta be zero. The coefficient of the second derivative must be 1/r_{o}^{2}. This again gives us four equations that can be solved for values of A, B, C, and D. The solution is:
Thus the finite difference equation is
What is the error?

Or
Special treatment near origin
When symmetry exists at the origin (Geometrical & Thermal)
Indeterminate
Thus
When no symmetry at r = 0, use Cartesian coordinates for the node at the origin:
Everywhere else use cylindrical coordinate system