I'm working with numerical methods for solving PDEs (Linear Advection/Euler equations with temporal and spatial discretisation) using finite difference/finite volume methods. In these simulations I have to deal with non-smooth solutions which I treat using non-linear spatial discretisation schemes such as ENO/WENO methods.
When one speaks about local and global truncation errors and their link to the global order of accuracy of the solution we usually suppose "smooth" solutions and linear spatial schemes (which don't depend on the solution). However, the definition of "smooth" becomes a bit cumbersome with space discretised solutions. How can one define the "smoothness" of a discretised solution? I suppose that one can apply the Fourier Transform to the discretised solution and observe that the contribution of each Fourier mode is not negligible (opposed to smooth solutions). However, I haven't found yet any formal definition of the smoothness of a discretised solutions, does anyone know if there exist a formal definition of the smoothness of discretised functions?
Moreover, for non-smooth solutions my spatial discretisation schemes become non-linear and therefore the local truncation error at each spatial discretised point of my domain becomes "non-smooth", how can one then link this local truncation error to a general idea of order of accuracy?
Thank you for your help
A discretized solution is neither smooth nor discontinuous. The theoretical solution may be. Nevertheless, one can still introduce numerical smoothness indicators, see Ref. (1).
The definition of the order of accuracy assumes smooth solutions. Therefore, numerical estimation of the order of accuracy should be performed in a configuration where solutions are smooth. For many schemes, the order of accuracy (w.r.t. a given norm) drops in the vicinity of discontinuities. This can be analyzed theoretically, see Sec. 8.7 "Accuracy near discontinuities" of Ref. (2). Note that the order of accuracy isn't everything, as explained in (2).
(1) CW Shu: "High Order Weighted Essentially Nonoscillatory Schemes for Convection Dominated Problems", SIAM Review 51.1 (2009), 82-126. doi:10.1137/070679065
(2) RJ LeVeque: Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, 2002. doi:10.1017/CBO9780511791253