High-Order Finite Difference Solution of Euler Equations for Nonlinear Water Waves Available to Purchase

The incompressible Euler equations are solved with a free surface, by applying an Eulerian kinematic boundary condition to capture the position of the free surface. The solution strategy follows that of [1, 2], applying a coordinate-transformation to obtain a time-constant spatial computational domain which is discretized using arbitrary-order finite difference schemes on a staggered grid with arbitrary grid spacing in each coordinate direction. The momentum equations and kinematic free surface condition are integrated in time using the classic fourth-order four-stage Runge-Kutta scheme. Mass conservation is satisfied implicitly, at the end of each time stage, by constructing the pressure from a discrete Poisson equation. This is derived from the discrete continuity and momentum equations and taking the time-dependent physical domain into account. An efficient Preconditioned Defect Correction (PDC) solution of the discrete Poisson equation for the pressure is presented, in which the preconditioning step is based on an order-multigrid formulation with a direct solution on the lowest order-level. This ensures fast convergence of the PDC method with a computational effort which scales linearly with the problem size. Results obtained with a two-dimensional implementation of the model are compared with highly accurate stream function solutions to the nonlinear wave problem, which show the approximately expected convergence rates and a clear advantage of using high-order finite difference schemes in combination with the Euler equations.