Generalized Finite Difference Method For Mild-slope Equation

The mild-slope equation(MSE)is an approximate form of three-dimensional Laplace equation based on potential wave theory,reduces a three-dimensional problem to two-dimensional by integrating wave velocity potential along vertical direction and can be used to describe combined wave refraction and diffraction.These characteristics make it a fundamental equation to research the basic law of near-shore wave transformation based on the linear wave theory and widely used in the study of wave motion.But it's difficult to solve the original mild-slope equation directly because of its non-separable character.To find out a simple,accurate and steady method for MSE,the generalized finite difference method(GFDM),an emerging meshless method,is proposed to solve the MSE based on Matlab proragramming and several numerical cases are adopted to verify its feasibility.The GFDM can get rid of time-consuming meshing generation and numerical quadrature.The numerical procedures of the GFDM are simple because the partial derivatives can be expressed as linear combinations of nearby function values based on the moving-least-squares method and the concept of subdomain.The main contens and results are as follows,(1)Applying the GFDM to solve the original MSE.Four numerical models,paraboloidal shoal,combined paraboloidal shoal and cylinder,combined mild slope and elliptic shoal and combined breakwater and mild slope,are simulated maching with corresponding boundary conditions.Simulated results are compared with experimental data,numerical results and analytical solutions to verify the feasibility of applying the GFDM to solve the MSE,satisfactory results are obtained.Influences of nonlinear term are analysed by changing the control equation to nonlinear MSE.It's also verified that the GFDM can be applied in nonlinear wave motion.In addition,steability and convergency of the GFDM are demonstrated by comparing simulated results with different total points.(2)GFDM and Houbolt finite difference method are respectively applied to discretize temporal item and special item of the time-dependent MSE,which contains second derivative of both time and space.Four numerical cases,propagation of long and short wave in constant depth water,wave shoaling from deep to shallow water,wave propagate over combined mild slope and elliptic shoal,as well as a asymmetric region with a circular shoal,are simulated.The feasibility of applying the GFDM to solve the time-dependent MSE is verified by comparing the simulated results with experimental data,numerical results and analytical solutions.(3)GFDM is used to simulate a practical engineering after the reliability of solving the MSE is proved.Wave propagation in the Suizhong harbor district of Huludao harbor locate in Suizhong County is simulated by applyling the GFDM to solve the original MSE.Factors such as incident wave period,direction and width of breakwater gap,influencing wave distribution and wave height in the harbor are discussed.Application of the GFDM in practical engineering proves that the GFDM can be used to solve the MSE and provide reference for practical engineering.