Numerical validation methods

Abstract

In the last years, numerical simulation has seen a great development thanks to costs reduction and speed increases of the computational systems. With these improvements, the mathematical algorithms are able to work properly with more realistic problems. Nowadays, the solution of a problem using numerical simulation is not just finding a result, but also to ensure the quality. However, can we say that the model results are correct regarding the behaviour of the system? In other words, how could we quantify the similarity between reality and simulations? To answer these questions, it is necessary to establish a validation criterion that allows an objective quantification of the difference between the results and the reality. Another way to say this is, how “true” our results are. In the case of numerical methods, the main objective is to replicate as closely as possible the behaviour of the "real" world through numbers. Normally, the results of the numerical methods are expressed in terms of graphics, pictures, etc. These results represent the view of reality that the chosen method provides. In order to affirm that the result of a numerical solution is fully consistent with the reality, it must be satisfied that: a. The mathematical model must incorporate all aspects of the real world. b. The numerical method has to solve exactly the equations of the mathematical modelling. The problem starts with these two conditions that guarantee the "truth" of the results, since none of them are fully accomplished and it must be admitted that the numerical prediction never completely matches the "real" world behaviour. Then you can only be sure that the numerical solution is a good approximation of the reality. Now, new questions arise: How much does the result obtained by a numerical method resemble the reality? How can we objectively quantify this similarity? The answers to these questions are those that give rise to the validation methods.

Peer Reviewed

Postprint (published version)