2.4 Analysis of Finite Difference Methods | 2.4 Analysis of Finite Difference Methods | Unit 2: Numerical Methods for Partial Differential Equations | Computational Methods in Aerospace Engineering | Aeronautics and Astronautics

2.4.2 Truncation Error of Central Difference Approximation

What is the truncation error and order of accuracy of the central difference approximation \(\delta _{2x}U_ i = \frac{U_{i+1}-U_{i-1}}{2\Delta x}\) of \(U_{x_ i}\)

\(-\frac{1}{3}\Delta x U_{xx_ i} + \mathcal{O}(\Delta x^3)\), first-order accurate

\(-\frac{1}{6}\Delta x U_{xxx_ i} + \mathcal{O}(\Delta x^3)\), first-order accurate

\(-\frac{1}{6}\Delta x^2 U_{xxx_ i} + \mathcal{O}(\Delta x^4)\), second-order accurate

\(-\frac{1}{12}\Delta x^2 U_{xxx_ i} + \mathcal{O}(\Delta x^4)\), second-order accurate

Answer: The truncation error can be computed by inserting the Taylor series into the finite difference approximation and cancelling the appropriate terms. Since the error is \(\mathcal{O}(\Delta x^2)\), the approximation is second-order accurate.