In this post, you can see how the
analysis of the accuracy of the given finite-difference formula is achieved for
a first order derivative case.
In order to solve ODE problems or Partial Differential
Equations (PDE) by system of algebraic equations, there are certain methods
available. The Finite Difference method is probably the oldest numerical method
that is used.
 |
| Figure 1. Numerical solution flowchart |
It is recommended to choose a
uniformly distributed grid size, having the size of X and Y components the
same, due to memory limitations.
Suppose that function U(x) is given as
such:
 |
| Figure 2. Selection of points on a function |
One would like to estimate the first derivative
of the function U(x) at some point x(j). The value of the
neighboring nodes are given: uj = u(xj), uj+1,
uj-1, where xj = j*h.
 |
| Figure 3. 2D final difference grid |
Having a differential equation for a 2D,
compressible flow, non-viscous, non-stationary:
One takes the definition of the first
derivative:
If the discretization is small enough
(Δx), it will approximate the value of the function as:
Similarly to Equation (1.2) one can propose
different algebraic formulas for determining the determinant for a given point
of the function.
It is important to notice that Equations (1.2) and
(1.4) are 1st order accurate, meanwhile (1.5) is 2nd
order accurate.
 |
| Figure 4. Different approximations |
From the above plot, it is clearly
visible that out of the 3 different formulas for finding the derivative at a
given point Xj, the line that is closest to the tangent point at that point is
III. This is the so called central
difference and is more accurate that the other, forward difference one. The
errors can be determined simply by the Taylor expansion.
Similarly to Equation (1.3) the
following algebraic equations can be written:
One can conclude that the finite
difference formula has order of accuracy n and is proportional to hn for small values of step
size h. The central difference is 2nd order accurate and higher
order terms are resulting in lower accuracy, therefore the 2nd order
formula works best for calculating ODE-s and PDE-s.
Other method for deriving finite
difference formulas (with different accuracy) for a given differential order
remains a problem to solve.
To download full text in PDF: click here
Reference: Computational Fluid Dynamics, Lecture notes - Jacek Rokicki, 2014
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ReplyDeleteReally interesting write-up on the Finite Difference method! The visual comparison in Figure 4 makes it so clear why central difference approximations hit that sweet spot of second-order accuracy compared to forward differences. It’s always fascinating how a smaller step size $h$ changes everything in numerical stability when setting up these grids. Coming across reliable resources or solid assignment help for Australian students dealing with complex CFD coursework can make a huge difference, especially when trying to translate Taylor expansion errors into actual code. Thanks for taking the time to lay out the derivation step by step definitely bookmarking this for my next numerical methods review!
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ReplyDeleteReally neat breakdown of the finite difference method! It is always satisfying seeing how the Taylor expansion nicely proves that the central difference scheme achieves second-order accuracy compared to standard forward differences. Seeing the geometric visual with line III sitting closest to the actual tangent really makes the concept click. I usually code these numerical solvers in C++, where managing specific syntax and Language Features helps keep matrix operations running efficiently across uniform grid spaces. It makes me wonder though in complex 2D CFD problems with sharp boundary gradients, how much finer do you usually need to make grid size $h$ before computational overhead outweighs the accuracy gains of central differencing? Thanks for putting this together, fantastic explanation!
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