The Truncation error is the difference between a truncated value and the actual value. A truncated quantity is represented by a number with a fixed number of allowed digits, with any excess digits trimmed (hence the expression is truncated). In numerical analysis and scientific computation, the truncation error is the error caused by the approximation of a mathematical process. This error occurs when an infinite sum is truncated and approximated to a finite sum.
For example, if we approximate the sine function by the first two non-zero terms of its Taylor series, as in small, the resulting error is a truncation error. It is present even with infinite precision arithmetic, because it is caused by the truncation of Taylor's infinite series to form the algorithm. Occasionally, the rounding error is also called a truncation error, especially if the number is rounded by truncation. The truncation error can also include the discretization error, which is the error that arises when taking a finite number of steps in a calculation to approximate an infinite process.
This type of error can be seen in numerical methods for ordinary differential equations, where a continuously varying function, which is the solution of the differential equation, is approximated by a process that progresses step by step. The power (or exponent) to which the grid spacing is raised in the initial order error term is also known as the order of the truncation error. This order can be different for each nodal arrangement and form of differential equation. For example, if S9 %3D x, the truncation error would be zero because the fourth derivative would be zero.
On the other hand, if S9 %3D ex, the truncation error would not only be non-zero, but it would also vary from node to node. Truncation errors are inevitable in any discretization scheme, unless the higher derivatives are zero. For typical structural dynamics problems, the step size is governed by the precision required by the engineer and the truncation error of the time integration method. As discussed above, one way to achieve this goal is to refine the mesh to a point where the truncation error becomes negligible.