What is the Truncation Error and How to Calculate it?

The truncation error is the difference between a truncated value and the actual value. In numerical analysis and scientific computation, it is the error caused by the approximation of a mathematical process. It is defined as the difference between the true (analytical) derivative of a function and its derivative obtained by numerical approximation. The accuracy of a finite difference formula is a fundamental issue when discretizing differential equations.

Truncation error can be estimated at each iteration by calculating the difference between the true solution and the numerical approximation. After the integration truncation error has been estimated, a decision can be made as to whether to accept or reject the step size. An advantage of truncation error analysis compared to empirical estimation of convergence rates is that it reveals the accuracy of the various building blocks in the numerical method and how each building block affects overall accuracy. Given an infinite series, one can calculate the truncation error by using only a finite number of terms.

The “truncation error T (x, h) associated with the “corrective stage” of the “Milne—Hamming” method is given by. It seems that the truncation error is relatively simple to calculate without specifying the differential equation and the discrete model in a special case. As an example of truncation error, consider the speed of light in a vacuum. The official value is 299, 792, 458 meters per second. In scientific notation (power of notation), this quantity is expressed as 2, 9979-2458 x 108. Obtain a global truncation error limit for the second-order Adams-Moulton algorithm applied to the problem of Example 13, 13, assuming that the corrector is satisfied exactly after each step.

This is not the correct use of the truncation error; however, calling it by truncating a number may be acceptable. The global truncation error is the agglomeration of the local truncation error in all iterations, assuming a perfect knowledge of the true solution in the initial time step.