What is Truncation Error and How is it Used?

Truncation error analysis is a widely used framework for assessing the accuracy of finite difference schemes. It is defined as the difference between the true (analytical) derivative of a function and its derivative obtained by numerical approximation. In numerical analysis and scientific computation, truncation error is the error caused by approximating a mathematical process. For one-step methods, the local truncation error gives us an indication of how the solution to the differential equation does not solve the difference equation. A high truncation error suggests that the original process LCA model is incomplete, prompting the use of hybrid LCA to replace process LCA when comparing different alternatives.

For example, glycerin fractionation for PSA production has an estimated truncation error much larger than Cap-xylene production. The allowed truncation error, known as tolerance, is typically used to control the accuracy of the integration. In this method of error handling, the primary local truncation error remains less than a prescribed tolerance, so that the allowed error is under control. Therefore, we can conclude that the global truncation error is a smaller order than the local truncation error. Obtaining a global truncation error limit for the second-order Adams-Moulton algorithm applied to a problem requires that the corrector be satisfied exactly after each step.

The truncation error generally increases as the step size increases, while the rounding error decreases as the step size increases. Truncation error is not the only source of error in numerical resolution of initial value problems using difference formats, because there is also a rounding error at each step of the calculation. This is not the correct use of the truncation error; however, calling it by truncating a number may be acceptable. For each iteration (k+th), the step size is usually estimated based on the truncation error in the kth iteration. After estimating the truncation error of integration in (k+)th iteration, a decision can be made whether to accept or reject the step size.