Numerical solutions of mild slope equation by generalized finite difference method

The mild slope equation (MSE) has been widely used to describe combined wave refraction and diffraction in the field of coastal and offshore engineering owing to its applicability for a wide range of wave frequencies. In this paper, a meshless numerical algorithm, based on the generalized finite difference method (GFDM), is firstly proposed to efficiently and accurately solve the MSE. As a newly-developed domain-type meshless method, the GFDM can truly get rid of time-consuming meshing generation and numerical quadrature. The partial differential terms of the MSE for each point in the computational domain can be discretized into linear combinations of nearby function values with the moving-least-squares method of the GFDM, so the numerical implementation is very convenient and efficient. To evaluate the accuracy and capability of the proposed scheme for MSE, a series of numerical tests were conducted, covering a range of complexity that included propagation and transformation of waves due to a parabolic shoal, a circular island mounted on a paraboloidal shoal and elliptic shoal situated on a slope, as well as breakwater gap. The results were compared with experimental data, analytical solutions and other numerical methods, and reasonable agreements have been achieved.