Some numerical methods (such as FDM and FEM) permit the use of high-order schemes for discretization of computational domain yielding higher-order accurate results. The accuracy attainable by a particular numerical method is a function of many variables and the proposed usage of such numerical solutions. From point of view of a Code Developer or Computational Fluid Dynamist, can it be said that high-order discretization method necessarily produce higher-order numerical solution?
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@ Jose Roberto Rasi Thanks for your timely response. Well appreciated.
High order methods provide better accuracy IF the long list of hypotheses on which their superior approximation properties are ALL satistfied. This is not always the case in many applications, so that whenever large gradients, discontinuous solutions and so on arise, more work is needed to take advantage of the full power of higher order methods. Furthermore, in many application areas (one I am more familiar with is river hydraulics) there is such an uncertainty on the data and empirical models used for some aspects of the mathematical modelling that it is a bit useless to worry about high order numerics.
@ Prof. Luca Bonaventura Thanks for your insightful comment. Well appreciated.
Hello to all, Nice question, maybe could be stated slightly better. In any way, if the question is: does using higher order discretization methods lead to more accurate responses, it seems to me the answer is "not necessarily". Even more if the question is: does using higher order discretization methods lead to responses converging with higher order, again it seems to me the answer is "not necessarily". To explain better, for the first case of the question, using discretization methods of higher order can lead to responses with the potential to converge with higher order. Considering the log-log plot of error with respect to the algorithmic parameter (the convergence plot), this means that close to the exact response, responses produced by methods of higher order can change on a straight line steeper (inclined more) than those associated with methods of lower order. However the location of the steeper line depends on other characteristics of the problem as well as the method, and hence, the accuracy produced when using methods of higher order is not necessarily more than when using methods of lower order. Regarding the second case of the question, when the order of discretization method is higher, the produced responses are of the potential to converge with higher order. However, if there are parameters in the problem that are approximations in convergence towards their exact values with some specific order. This specific order can cause the potential not to be materialized and the resulting responses to converge with an order of accuracy smaller than the order provided by the discretization methods. There are several references in this regard. The oldest I remember now is the paper of Penry and Wood on 1985. With many thanks for your kind attention, have a very nice and healthy day and future.
@ Dr. Aram Soroushian Thanks for your insightful contribution and elucidation on the subject matter. Well appreciated.
Dear @Seyi Festus Olatoyinbo. You are most welcome. And thank you for your kind words. Have a nice day and future.
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