A compatible higher-order interpolation method for BEM-MEL simulations of water waves

• A pseudo-quadratic interpolation compatible with linear element mesh is proposed and validated. • The method is validated using experimental data of water surface waves. • Wave attenuation is primarily caused by oversmoothing in linear ALE interpolation. • The proposed method improves BEM-MEL-ALE simulation accuracy with minimal code modification. • The scheme potentially enhances a wide range of simulations based on linear ALE frameworks. Recent studies have indicated that the Boundary Element Method with Mixed Eulerian-Lagrangian (BEM-MEL), when combined with the Arbitrary Lagrangian-Eulerian (ALE) approach, tends to cause undesirable attenuation of progressive waves when linear interpolation is employed. Although linear interpolation is suspected to be a primary cause of this attenuation, the specific mechanism behind it remains unclear. This study investigates two potential causes of this attenuation: (1) the interpolation used in BEM-MEL for both the discretization of the boundary integral equation and the calculation of nodal flow velocities, and (2) the interpolation used in the ALE procedure. To isolate and evaluate the effects of linear interpolation, we develop a higher-order interpolation method designed to be compatible with linear elements. This compatibility, made possible by using the same nodal configuration, allows for a direct comparative evaluation of interpolation schemes without altering the mesh resolution. The method can also replace linear elements on a per-element basis, supports standard mesh operations like edge-flipping. Our findings reveal that the linear interpolation in the ALE procedure is the primary source of wave attenuation, whereas the interpolation in BEM-MEL has only a minor influence on the free surface. Moreover, the proposed higher-order interpolation can be introduced into BEM computations without increasing the number of nodes, making it an effective and practical approach.