Application of weighted-least-square local polynomial approximation to 2D shallow water equation problems

Abstract In this study, a numerical model based on shallow water equations (SWE) is developed. An explicit predictor–corrector approach is adopted for the time marching process. Using the leap-frog formulae, the three unknowns in SWE, which are the water depth h , and the water fluxes uh , vh , are firstly estimated directly by their values and their spatial derivatives in the previous time step. Then they are corrected by the Crank-Nicolson formulation. The spatial derivatives of h , uh and vh for the further time marching processes are calculated by using the Weighted Least Square (WLS) local polynomial approximation, which is a kind of meshless method. This model is applied to the simulations of dam break flows and tidal currents.

[1]  Nan-Jing Wu,et al.  Generation of stable solitary waves by a piston-type wave maker , 2014 .

[2]  A. I. Delis,et al.  Numerical solution of the two-dimensional shallow water equations by the application of relaxation methods , 2005 .

[3]  Satya N. Atluri,et al.  The basis of meshless domain discretization: the meshless local Petrov–Galerkin (MLPG) method , 2005, Adv. Comput. Math..

[4]  M. Ooe,et al.  Ocean Tide Models Developed by Assimilating TOPEX/POSEIDON Altimeter Data into Hydrodynamical Model: A Global Model and a Regional Model around Japan , 2000 .

[5]  E. Kansa.  MULTIQUADRICS--A SCATTERED DATA APPROXIMATION SCHEME WITH APPLICATIONS TO COMPUTATIONAL FLUID-DYNAMICS-- II SOLUTIONS TO PARABOLIC, HYPERBOLIC AND ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS , 1990 .

[6]  S. C. Fan,et al.  Local multiquadric approximation for solving boundary value problems , 2003 .

[7]  V. Soulis,et al.  Experimental investigation of two-dimensional dam-break induced flows , 1992 .

[8]  J. S. Wang,et al.  FINITE-DIFFERENCE TVD SCHEME FOR COMPUTATION OF DAM-BREAK PROBLEMS , 2000 .

[9]  Utku Senturk,et al.  Modeling nonlinear waves in a numerical wave tank with localized meshless RBF method , 2011 .

[10]  Juntao Zhou,et al.  MLPG Method Based on Rankine Source Solution for Modelling 3D Breaking Waves , 2010 .

[11]  Tae Hoon Yoon,et al.  Finite volume model for two-dimensional shallow water flows on unstructured grids , 2004 .

[12]  Benny Y. C. Hon,et al.  Compactly supported radial basis functions for shallow water equations , 2002, Appl. Math. Comput..

[13]  J. Sládek,et al.  Extrapolated local radial basis function collocation method for shallow water problems , 2015 .

[14]  Nam Mai-Duy,et al.  Approximation of function and its derivatives using radial basis function networks , 2003 .

[15]  T. Belytschko,et al.  Element‐free Galerkin methods , 1994 .

[16]  D. Ouazar,et al.  RBF Based Meshless Method for Large Scale Shallow Water Simulations: Experimental Validation , 2010 .

[17]  Hong-Ming Kao,et al.  Numerical modeling of dambreak-induced flood and inundation using smoothed particle hydrodynamics , 2012 .

[18]  Rainald Löhner,et al.  A finite point method for compressible flow , 2002 .

[19]  N. Aluru.  A point collocation method based on reproducing kernel approximations , 2000 .

[20]  Derek M. Causon,et al.  HIGH-RESOLUTION FINITE-VOLUME METHOD FOR SHALLOW WATER FLOWS , 1998 .

[21]  Qingwei Ma,et al.  Meshless local Petrov-Galerkin method for two-dimensional nonlinear water wave problems , 2005 .

[22]  S. Liang,et al.  Least-squares finite-element method for shallow-water equations with source terms , 2009 .

[23]  S. Atluri,et al.  A new Meshless Local Petrov-Galerkin (MLPG) approach in computational mechanics , 1998 .

[24]  R. Franke.  Scattered data interpolation: tests of some methods , 1982 .

[25]  E. Oñate,et al.  A FINITE POINT METHOD IN COMPUTATIONAL MECHANICS. APPLICATIONS TO CONVECTIVE TRANSPORT AND FLUID FLOW , 1996 .

[26]  R. Farwig,et al.  Multivariate interpolation of arbitrarily spaced data by moving least squares methods , 1986 .

[27]  Gang Li,et al.  New approximations and collocation schemes in the finite cloud method , 2005 .

[28]  T. Driscoll,et al.  Observations on the behavior of radial basis function approximations near boundaries , 2002 .

[29]  Rainald Löhner,et al.  Adaptive methodology for meshless finite point method , 2008, Adv. Eng. Softw..

[30]  J. Oden,et al.  H‐p clouds—an h‐p meshless method , 1996 .

[31]  N. Aluru,et al.  Finite cloud method: a true meshless technique based on a fixed reproducing kernel approximation , 2001 .

[32]  P. Liu,et al.  An analysis of 2004 Sumatra earthquake fault plane mechanisms and Indian Ocean tsunami , 2006 .

[33]  S. Liang,et al.  Solution of shallow-water equations using least-squares finite-element method , 2008 .

[34]  Pilar García-Navarro,et al.  A HIGH-RESOLUTION GODUNOV-TYPE SCHEME IN FINITE VOLUMES FOR THE 2D SHALLOW-WATER EQUATIONS , 1993 .

[35]  D. L. Young,et al.  Solution of Maxwell's Equations Using the MQ Method , 2005 .

[36]  Hong-Ming Kao,et al.  Numerical simulation of shallow-water dam break flows in open channels using smoothed particle hydrodynamics , 2011 .

[37]  Nan-Jing Wu,et al.  A robust local polynomial collocation method , 2013 .

[38]  Kwok Fai Cheung,et al.  Multiquadric Solution for Shallow Water Equations , 1999 .

[39]  E. Oñate,et al.  A stabilized finite point method for analysis of fluid mechanics problems , 1996 .

[40]  D. Young,et al.  Application of localized meshless methods to 2D shallow water equation problems , 2013 .

[41]  Scott F. Bradford,et al.  Finite-Volume Model for Shallow-Water Flooding of Arbitrary Topography , 2002 .

[42]  Mourad Heniche,et al.  A two-dimensional finite element drying-wetting shallow water model for rivers and estuaries , 2000 .

[43]  Ching-Shyang Chen,et al.  A numerical method for heat transfer problems using collocation and radial basis functions , 1998 .

[44]  Kuang-An Chang,et al.  Simulation of free‐surface waves in liquid sloshing using a domain‐type meshless method , 2011 .

[45]  E. Kansa.  Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics—I surface approximations and partial derivative estimates , 1990 .