Efficiency and accuracy of high-order computations and reduced order modelling in coastal engineering wave propagation problems

Abstract

Several numerical issues have to be considered when solving wave propagation problems, between which there are the artificial boundary conditions, the small geometrical features that can be influential or the variable coefficients. Apart from them, two issues are mainly addressed and discussed. Firstly, low order elements need very high wave resolution for capturing the solution in the area of interest, leading to extremely dense meshes. High-order finite elements are proposed to be an efficient and accurate solution for solving the problem. Secondly, the very large number of test cases. When designing harbour models, a huge number of incident waves, in term of wavelengths and directions, have to be studied. The excessive computational cost to carry out all the possible direct problems prevents the whole data evaluation, inducing the lost of important information. Reduced order models may be an alternative if they are computable, efficient and accurate. The applicability of Proper Generalized Decomposition (PGD) is exploited. Unlike previous PGD contributions, which deal with elliptic problems, the present work is focused on a more challenging scenario for the separable representation due to the loss of the elliptic behaviour. The proposed PGD involves a separable representation of the unknown reflected wave in space, wave number and angle of incidence. Such decomposition appears to be really interesting for practical purpose, where goal-oriented results are critical for a wide range of frequencies and incident waves. Moreover, when accuracy and efficiency are of concern, the number of terms in the reduced model are determined by means of an error estimation based on the dual formulation of the problem.