Finite Difference method

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: u_j = u(x_j), u_j+1, u_j-1, where x_j = 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 1^st order accurate, meanwhile (1.5) is 2^nd 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 2^nd order accurate and higher order terms are resulting in lower accuracy, therefore the 2^nd 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.

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Reference: Computational Fluid Dynamics, Lecture notes - Jacek Rokicki, 2014