A high-order harmonic polynomial method for solving the Laplace equation with complex boundaries and its application to free-surface flows. Part I: Two-dimensional cases

A high-order harmonic polynomial method (HPM) is developed for solving the Laplace equation with complex boundaries. The “irregular cell” is proposed for the accurate discretization of the Laplace equation, where it is difficult to construct a high-quality stencil. An advanced discretization scheme is also developed for the accurate evaluation of the normal derivative of potential functions on complex boundaries. Thanks to the irregular cell and the discretization scheme for the normal derivative of the potential functions, the present method can avoid the drawback of distorted stencils, that is, the possible numerical inaccuracy/instability. Furthermore, it can involve stationary or moving bodies on the Cartesian grid in an accurate and simple way. With the proper free-surface tracking methods, the HPM has been successfully applied to the accurate and stable modeling of highly nonlinear free-surface potential flows with and without moving bodies, that is, sloshing, water entry, and plunging breaker.