Efficient enforcement of far-field boundary conditions in the Transformed Field Expansions method

On the statistical properties of surface elevation, velocities and accelerations in multi-directional irregular water waves

“…The first part treats the time integration of and under the assumption that is known. The second part deals with the computation of from and and is inspired by the work of Nicholls (2011), as it utilizes the same strategy to reduce the size of the computational domain when solving the Laplace equation (2.1 a ). In what follows we will briefly describe these two parts, as well as how the method may be used for the computation of the velocity and acceleration fields of the fluid.…”

“…By solving the Laplace equation analytically on the lower part of the domain, it can be shown (see e.g. § 5 in the paper of Nicholls (2011)) that the full Laplace problem is equivalent to solving the reduced problem on the upper part of the domain. Here, the linear operator is defined by the action for all , and we note that this definition of differs slightly from the definition used in Klahn et al.…”

On the statistical properties of inertia and drag forces in nonlinear multi-directional irregular water waves

“…As shown by Klahn et al (2021a), this strategy enables the solution of the Laplace equation with an error 916 A59-5 that decreases exponentially with the spatial resolution for practically all values of the water depth and the wave steepness when considering steady nonlinear waves. In order to remain efficient for large problems, the method utilizes an artificial boundary condition as used by Nicholls (2011) and a preconditioning strategy inspired by the work of Fuhrman & Bingham (2004). Denoting the total number of grid points by N, these techniques ensure that the computational effort per time step grows in proportion to N log(N) when increasing the horizontal resolution.…”

Numerical Simulation of Grating Structures Incorporating Two-Dimensional Materials: A High-Order Perturbation of Surfaces Framework

“…and from these we can compute, e.g., the exact surface current We make the physical parameter choices and compute approximations to ν ex by the FE and TFE algorithms delivering ν FE and ν TFE , respectively. (The parameters a and b specify locations of artificial boundaries in the TFE formulation, while N z gives the spatial discretization in the vertical direction; please see [55,44] for full details.) We measure the relative error…”