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 of digits trimmed (hence the truncated expression). In numerical analysis and scientific computation, the error of truncation is the error caused by the approximation of a mathematical process. This error is present even with infinite precision arithmetic, as it is caused by the truncation of Taylor's infinite series to form the algorithm.
Truncation errors are inevitable in any discretization scheme unless the higher derivatives are zero, which rarely occurs in problems of practical interest. 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. Additionally, the discretization error, which is the error that arises when taking a finite number of steps in a calculation to approximate an infinite process, is often included in the truncation error. 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.
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. The name, truncation error, comes from the fact that Taylor's infinite series was truncated after a few terms to derive the expression of the approaching derivative. For example, if S 9% 3D x, the truncation error would be zero because the fourth derivative would be zero. On the other hand, if S 9% 3D ex, the truncation error would not only be non-zero, but would also vary from node to node.
In addition, an error of 10% occurs in an integrating nephelometer that truncates the range of the scattering angle to ~150°. The local truncation error gives us a measure to determine how the solution to the differential equation does not solve the difference equation. The introduction of this type of error sacrifices accuracy for numerical solutions to differential equations. Obtain a global truncation error limit for the second-order Adams-Moulton algorithm applied to a problem and assume that corrector is satisfied exactly after each step.
Therefore, we can conclude that global truncation errors are smaller than local ones.
