Numerical modelling of the mild slope equation using localised differential quadrature method

Although various numerical techniques have been applied over the last few decades to solve the mild slope equation (MSE), each technique has its own limitations, particularly in terms of computational cost, accuracy, and stability. Localised differential quadrature method (LDQM) is here investigated as an alternative new solution to the MSE. Localised DQM, rather than classical DQM, was used to solve the MSE because of its improved performance, lower computational cost and wider range of applicability. To evaluate the proposed method, four examples were studied, covering a range of complexity which included propagation and transformation of waves due to an elliptic shoal, breakwater gap, and non-rectangular harbour resonance. The results were compared with experimental data, analytical solutions, and other numerical methods. The agreement between numerical and benchmark results was good, and in some cases the performance of LDQM exceeded that of other numerical methods. LDQM can lead to accurate results using fewer grid points and lower computational cost if the number of local nodes is optimised. For a large number of local grid points in LDQM, and also for the case of classical DQM, iterative methods such as conjugate gradient should be employed to solve the system of equations.

[1]  Zhi Zong,et al.  Advanced Differential Quadrature Methods , 2009 .

[2]  L. Balas,et al.  Numerical modeling of mild slope equation with finite volume method , 2008 .

[3]  C. Bert,et al.  Differential Quadrature Method in Computational Mechanics: A Review , 1996 .

[4]  A variable order approach to improve differential quadrature accuracy in dynamic analysis , 2003 .

[5]  Bin Li,et al.  A generalized conjugate gradient model for the mild slope equation , 1994 .

[6]  R. Bellman,et al.  DIFFERENTIAL QUADRATURE AND LONG-TERM INTEGRATION , 1971 .

[7]  A. C. Radder,et al.  Verification of numerical wave propagation models for simple harmonic linear water waves , 1982 .

[8]  S. Xiang,et al.  Three-dimensional vibration analysis of isotropic plates by multiquadric and thin-plate spline radial basis functions , 2010 .

[9]  C. Shu.  Differential Quadrature and Its Application in Engineering , 2000 .

[10]  New solutions for the propagation of long water waves over variable depth , 1994 .

[11]  J. Berkhoff,et al.  Computation of Combined Refraction — Diffraction , 1972 .

[12]  Robert A. Dalrymple,et al.  An approximate model for nonlinear dispersion in monochromatic wave propagation models , 1986 .

[13]  Bo Liu,et al.  A differential quadrature analysis of dynamic and quasi-static magneto-thermo-elastic stresses in a conducting rectangular plate subjected to an arbitrary variation of magnetic field , 2010 .

[14]  J. Kirby,et al.  Wave Diffraction Due to Areas of Energy Dissipation , 1984 .

[15]  Changhoon Lee,et al.  Effect of higher-order bottom variation terms on the refraction of water waves in the extended mild-slope equations , 2004 .

[16]  Robert G. Dean,et al.  Wave height variation across beaches of arbitrary profile , 1985 .

[17]  J. Lee,et al.  Wave-induced oscillations in harbours of arbitrary geometry , 1971, Journal of Fluid Mechanics.

[18]  Filipa S. B. F. Oliveira.  Numerical modelling of deformation of multi-directional random waves over a varying topography , 2007 .

[19]  Stanisław Ryszard Massel,et al.  EXTENDED REFRACTION-DIFFRACTION EQUATION FOR SURFACE WAVES , 1993 .

[20]  C. Willmott.  Some Comments on the Evaluation of Model Performance , 1982 .

[21]  Chang Shu,et al.  Vibration analysis of arbitrarily shaped membranes using local radial basis function-based differential quadrature method , 2007 .

[22]  Zeki Demirbilek,et al.  Exterior reflections in elliptic harbor wave models , 1996 .

[23]  Kyung-Duck Suh,et al.  Generation of random waves in time-dependent extended mild-slope equations using a source function method , 2006 .

[24]  M. J. Abedini,et al.  Numerical modeling of long waves in shallow water using Incremental Differential Quadrature Method , 2006 .

[25]  Akbar Alibeigloo,et al.  DIFFERENTIAL QUADRATURE ANALYSIS OF FUNCTIONALLY GRADED CIRCULAR AND ANNULAR SECTOR PLATES ON ELASTIC FOUNDATION , 2010 .

[26]  Asadollah Noorzad,et al.  A coupled boundary element-finite difference solution of the elliptic modified mild slope equation , 2011 .

[27]  An extended time-dependent numerical model of the mild-slope equation with weakly nonlinear amplitude dispersion , 2010 .

[28]  M. Hashemi,et al.  Unsteady seepage analysis using local radial basis function-based differential quadrature method , 2011 .

[29]  D. L. Young,et al.  The localized differential quadrature method for two-dimensional stream function formulation of Navier–Stokes equations , 2011 .

[30]  Jian-an Sun,et al.  Upwind local differential quadrature method for solving incompressible viscous flow , 2000 .

[31]  Alper Korkmaz,et al.  Solitary wave simulations of Complex Modified Korteweg-de Vries Equation using differential quadrature method , 2009, Comput. Phys. Commun..

[32]  P. Malekzadeh,et al.  Optimization of non-symmetric convective–radiative annular fins by differential quadrature method , 2007 .

[33]  D. L. Young,et al.  Local Differential Quadrature Method for 2-D Flow and Forced-Convection Problems in Irregular Domains , 2009 .

[34]  Song-Ping Zhu,et al.  A general DRBEM model for wave refraction and diffraction , 1993 .

[35]  Zhi Zong,et al.  A localized differential quadrature (LDQ) method and its application to the 2D wave equation , 2002 .

[36]  Quan Shen.  Local RBF-based differential quadrature collocation method for the boundary layer problems , 2010 .

[37]  Y. Agnon,et al.  Nonlinear refraction–diffraction of water waves: the complementary mild-slope equations , 2009, Journal of Fluid Mechanics.

[38]  Yvon Ouellet,et al.  A new kind of nonlinear mild-slope equation for combined refraction-diffraction of multifrequency waves , 1997 .

[40]  Per A. Madsen,et al.  An efficient finite-difference approach to the mild-slope equation , 1987 .

[41]  Münevver Tezer-Sezgin.  Solution of magnetohydrodynamic flow in a rectangular duct by differential quadrature method , 2004 .

[42]  R. Bellman,et al.  DIFFERENTIAL QUADRATURE: A TECHNIQUE FOR THE RAPID SOLUTION OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS , 1972 .

[43]  C. Shu,et al.  APPLICATION OF GENERALIZED DIFFERENTIAL QUADRATURE TO SOLVE TWO-DIMENSIONAL INCOMPRESSIBLE NAVIER-STOKES EQUATIONS , 1992 .

[44]  High order numerical code for hyperbolic mild-slope equations with nonlinear dispersion relation , 2007 .

[45]  James T. Kirby,et al.  A note on linear surface wave‐current interaction over slowly varying topography , 1984 .

[46]  M. Bouhadef,et al.  Modified mild slope equation and open boundary conditions , 2004 .

[47]  M. J. Abedini,et al.  Tidal and surge modelling using differential quadrature: a case study in the Bristol Channel. , 2008 .

[48]  Shu-xue Liu,et al.  Self-adaptive FEM numerical modeling of the mild-slope equation , 2008 .

[49]  Solving the sod shock tube problem using localized differential quadrature (LDQ) method , 2011 .

[50]  Zhu Zheng-you,et al.  Upwind local differential quandrature method for solving coupled viscous flow and heat transfer equations , 2004 .

[51]  Wei Chen,et al.  On the modeling of wave-current interaction using the elliptic mild-slope wave equation , 2005 .

[52]  Ge Wei,et al.  Solution of the mild-slope wave problem by iteration , 1991 .

[53]  O. C. Zienkiewicz,et al.  Diffraction and refraction of surface waves using finite and infinite elements , 1977 .

[54]  Jun Tang,et al.  An efficient and flexible computational model for solving the mild slope equation , 2004 .

[55]  M. J. Abedini,et al.  A differential quadrature analysis of unsteady open channel flow , 2007 .

[56]  Hongzhi Zhong,et al.  Triangular differential quadrature and its application to elastostatic analysis of Reissner plates , 2001 .

[57]  P. Troch,et al.  Numerical implementation and sensitivity analysis of a wave energy converter in a time-dependent mild-slope equation model , 2010 .

[58]  P. Malekzadeh.  THREE-DIMENSIONAL FREE VIBRATION ANALYSIS OF THICK FUNCTIONALLY GRADED PLATES ON ELASTIC FOUNDATIONS , 2009 .

[59]  Hang Ma,et al.  Local differential quadrature method using irregularly distributed nodes for solving partial differential equations , 2008 .

[60]  Changhoon Lee,et al.  Internal Generation of Waves for Time-Dependent Mild-Slope Equations , 1998 .

[61]  Robert A. Dalrymple,et al.  Verification of a parabolic equation for propagation of weakly-nonlinear waves , 1984 .

[62]  C. Shu,et al.  Local radial basis function-based differential quadrature method and its application to solve two-dimensional incompressible Navier–Stokes equations , 2003 .

[63]  Complex differential quadrature method for two-dimensional potential and plane elastic problems , 2007 .

[64]  R. E. Rossi,et al.  The method of differential quadrature and its application to the approximate solution of ocean engineering problems , 1994 .