The use of proper orthogonal decomposition (POD) meshless RBF-FD technique to simulate the shallow water equations

Abstract The main aim of this paper is to develop a fast and efficient local meshless method for solving shallow water equations in one- and two-dimensional cases. The mentioned equation has been classified in category of advection equations. The solutions of advection equations have some shock, thus, especial numerical methods should be employed for example discontinuous Galerkin and finite volume methods. Here, based on the proper orthogonal decomposition approach we want to construct a fast meshless method. To this end, we consider shallow water models and obtain a suitable time-discrete scheme based on the predictor-corrector technique. Then by applying the proper orthogonal decomposition technique a new set of basis functions can be built for the solution space in which the size of new solution space is less than the original problem. Thus, by employing the new bases the CPU time will be reduced. Some examples have been studied to show the efficiency of the present numerical technique.

[1]  Gang Li,et al.  High-order well-balanced central WENO scheme for pre-balanced shallow water equations , 2014 .

[2]  Chi-Wang Shu,et al.  High-order finite volume WENO schemes for the shallow water equations with dry states , 2011 .

[3]  Tian Jiang,et al.  Krylov implicit integration factor WENO methods for semilinear and fully nonlinear advection-diffusion-reaction equations , 2013, J. Comput. Phys..

[4]  Xiaohua Zhang,et al.  A fast meshless method based on proper orthogonal decomposition for the transient heat conduction problems , 2015 .

[5]  Ionel M. Navon,et al.  A reduced‐order approach to four‐dimensional variational data assimilation using proper orthogonal decomposition , 2007 .

[6]  Traian Iliescu,et al.  Proper orthogonal decomposition closure models for turbulent flows: A numerical comparison , 2011, 1106.3585.

[7]  Mehdi Dehghan,et al.  A Meshless Method Using Radial Basis Functions for the Numerical Solution of Two-Dimensional Complex Ginzburg-Landau Equation , 2012 .

[8]  Mehdi Dehghan,et al.  Remediation of contaminated groundwater by meshless local weak forms , 2016, Comput. Math. Appl..

[9]  Baodong Dai,et al.  A MOVING KRIGING INTERPOLATION-BASED MESHLESS LOCAL PETROV–GALERKIN METHOD FOR ELASTODYNAMIC ANALYSIS , 2013 .

[10]  F. Gallerano,et al.  Upwind WENO scheme for Shallow Water Equations in contravariant formulation , 2012 .

[11]  C. Vreugdenhil.  Numerical methods for shallow-water flow , 1994 .

[12]  Eugenio Rustico,et al.  Simulation of Nearshore Tsunami Breaking by Smoothed Particle Hydrodynamics Method , 2016 .

[13]  Ionel M. Navon,et al.  An optimizing reduced order FDS for the tropical Pacific Ocean reduced gravity model , 2007 .

[14]  Zhendong Luo,et al.  Reduced-order finite difference extrapolation model based on proper orthogonal decomposition for two-dimensional shallow water equations including sediment concentration , 2015 .

[15]  A. Wazwaz.  Partial Differential Equations and Solitary Waves Theory , 2009 .

[16]  Scott A. Sarra,et al.  Adaptive radial basis function methods for time dependent partial differential equations , 2005 .

[17]  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 .

[18]  Robert A. Dalrymple,et al.  Numerical study on mitigating tsunami force on bridges by an SPH model , 2016 .

[19]  YuanTong Gu,et al.  A meshless local Kriging method for large deformation analyses , 2007 .

[20]  K. M. Liew,et al.  A local Petrov-Galerkin approach with moving Kriging interpolation for solving transient heat conduction problems , 2011 .

[21]  Yulong Xing,et al.  Exactly well-balanced discontinuous Galerkin methods for the shallow water equations with moving water equilibrium , 2014, J. Comput. Phys..

[22]  L. W. Zhang,et al.  An element-free based solution for nonlinear Schrödinger equations using the ICVMLS-Ritz method , 2014, Appl. Math. Comput..

[23]  Xiaolin Li,et al.  A Galerkin boundary node method and its convergence analysis , 2009 .

[24]  Manuel Kindelan,et al.  Optimal variable shape parameter for multiquadric based RBF-FD method , 2012, J. Comput. Phys..

[25]  G. Kerschen,et al.  The Method of Proper Orthogonal Decomposition for Dynamical Characterization and Order Reduction of Mechanical Systems: An Overview , 2005 .

[26]  Fayssal Benkhaldoun,et al.  Slope limiters for radial basis functions applied to conservation laws with discontinuous flux function , 2016 .

[27]  Ionel M. Navon,et al.  2D Burgers equation with large Reynolds number using POD/DEIM and calibration , 2016 .

[28]  Mehdi Dehghan,et al.  The meshless local collocation method for solving multi-dimensional Cahn-Hilliard, Swift-Hohenberg and phase field crystal equations , 2017 .

[29]  C. Dawson,et al.  Local time-stepping in Runge–Kutta discontinuous Galerkin finite element methods applied to the shallow-water equations , 2012 .

[30]  Adrián Navas-Montilla,et al.  Energy balanced numerical schemes with very high order. The Augmented Roe Flux ADER scheme. Application to the shallow water equations , 2015, J. Comput. Phys..

[31]  L. Lucy.  A numerical approach to the testing of the fission hypothesis. , 1977 .

[32]  Gregor Kosec,et al.  Radial basis function collocation method for the numerical solution of the two-dimensional transient nonlinear coupled Burgers’ equations , 2012 .

[33]  Adrian Sandu,et al.  Comparison of POD reduced order strategies for the nonlinear 2D shallow water equations , 2014, International Journal for Numerical Methods in Fluids.

[34]  Zhu Wang,et al.  Reduced-Order Modeling of Complex Engineering and Geophysical Flows: Analysis and Computations , 2012 .

[35]  Božidar Šarler,et al.  Simulation of laminar backward facing step flow under magnetic field with explicit local radial basis function collocation method , 2014 .

[36]  Alfredo Bermúdez,et al.  Upwind methods for hyperbolic conservation laws with source terms , 1994 .

[37]  Lawrence Sirovich,et al.  Karhunen–Loève procedure for gappy data , 1995 .

[38]  Colin J. Cotter,et al.  A finite element exterior calculus framework for the rotating shallow-water equations , 2012, J. Comput. Phys..

[39]  Mehdi Dehghan,et al.  A numerical method for solution of the two-dimensional sine-Gordon equation using the radial basis functions , 2008, Math. Comput. Simul..

[40]  Ionel M. Navon,et al.  An improved algorithm for the shallow water equations model reduction: Dynamic Mode Decomposition vs POD , 2015 .

[41]  Fayssal Benkhaldoun,et al.  Projection finite volume method for shallow water flows , 2015, Math. Comput. Simul..

[42]  Michael Dumbser,et al.  A high order semi-implicit discontinuous Galerkin method for the two dimensional shallow water equations on staggered unstructured meshes , 2014, Appl. Math. Comput..

[43]  Baodong Dai,et al.  A meshless local moving Kriging method for two-dimensional solids , 2011, Appl. Math. Comput..

[44]  Yulong Xing,et al.  High order finite difference WENO schemes with the exact conservation property for the shallow water equations , 2005 .

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

[46]  Aleksey Marchenko,et al.  Surface wave propagation in shallow water beneath an inhomogeneous ice cover , 1997 .

[47]  T. Driscoll,et al.  Interpolation in the limit of increasingly flat radial basis functions , 2002 .

[48]  Božidar Šarler,et al.  Local radial basis function collocation method for solving thermo-driven fluid-flow problems with free surface , 2015 .

[49]  Philippe G. LeFloch,et al.  Late-time/stiff-relaxation asymptotic-preserving approximations of hyperbolic equations , 2010, Math. Comput..

[50]  R. L. Hardy.  Multiquadric equations of topography and other irregular surfaces , 1971 .

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

[52]  Danny C. Sorensen,et al.  Nonlinear Model Reduction via Discrete Empirical Interpolation , 2010, SIAM J. Sci. Comput..

[53]  Bengt Fornberg,et al.  Stabilization of RBF-generated finite difference methods for convective PDEs , 2011, J. Comput. Phys..

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

[55]  Fabien Marche,et al.  On the well-balanced numerical discretization of shallow water equations on unstructured meshes , 2013, J. Comput. Phys..

[56]  J. Monaghan,et al.  Smoothed particle hydrodynamics: Theory and application to non-spherical stars , 1977 .

[57]  Shmuel Rippa,et al.  An algorithm for selecting a good value for the parameter c in radial basis function interpolation , 1999, Adv. Comput. Math..

[58]  Fabien Marche,et al.  Asymptotic preserving scheme for the shallow water equations with source terms on unstructured meshes , 2015, J. Comput. Phys..

[59]  E. Toro,et al.  Well-balanced high-order centered schemes on unstructured meshes for shallow water equations with fixed and mobile bed , 2010 .

[60]  Danny C. Sorensen,et al.  A State Space Error Estimate for POD-DEIM Nonlinear Model Reduction , 2012, SIAM J. Numer. Anal..

[61]  Mehdi Dehghan,et al.  A meshless technique based on the local radial basis functions collocation method for solving parabolic–parabolic Patlak–Keller–Segel chemotaxis model , 2015 .

[62]  MATHEMATICAL ANALYSIS OF A SAINT-VENANT MODEL WITH VARIABLE TEMPERATURE , 2010 .

[63]  Christophe Berthon,et al.  Efficient well-balanced hydrostatic upwind schemes for shallow-water equations , 2012, J. Comput. Phys..

[64]  Ionel M. Navon,et al.  Non-intrusive reduced order modelling of the Navier-Stokes equations , 2015 .

[65]  Wang Ling-hui,et al.  A moving Kriging interpolation-based boundary node method for two-dimensional potential problems , 2010 .

[66]  P. Holmes,et al.  The Proper Orthogonal Decomposition in the Analysis of Turbulent Flows , 1993 .

[67]  C. Shu,et al.  An upwind local RBF-DQ method for simulation of inviscid compressible flows , 2005 .

[68]  Evan F. Bollig,et al.  Solution to PDEs using radial basis function finite-differences (RBF-FD) on multiple GPUs , 2012, J. Comput. Phys..

[69]  A. Cheng.  Multiquadric and its shape parameter—A numerical investigation of error estimate, condition number, and round-off error by arbitrary precision computation , 2012 .

[70]  Saifon Chaturantabut.  Dimension reduction for unsteady nonlinear partial differential equations via empirical interpolation methods , 2009 .

[71]  Manuel Kindelan,et al.  Optimal constant shape parameter for multiquadric based RBF-FD method , 2011, J. Comput. Phys..

[72]  S. Sarra,et al.  A linear system‐free Gaussian RBF method for the Gross‐Pitaevskii equation on unbounded domains , 2012 .

[73]  Mehdi Dehghan,et al.  The method of variably scaled radial kernels for solving two-dimensional magnetohydrodynamic (MHD) equations using two discretizations: The Crank-Nicolson scheme and the method of lines (MOL) , 2015, Comput. Math. Appl..

[74]  Scott A. Sarra,et al.  A local radial basis function method for advection-diffusion-reaction equations on complexly shaped domains , 2012, Appl. Math. Comput..

[75]  Wen Chen,et al.  Recent Advances in Radial Basis Function Collocation Methods , 2013 .

[76]  Martin D. Buhmann,et al.  Radial Basis Functions: Theory and Implementations: Preface , 2003 .

[77]  Erik Lehto,et al.  A guide to RBF-generated finite differences for nonlinear transport: Shallow water simulations on a sphere , 2012, J. Comput. Phys..

[78]  Juan Du,et al.  Reduced order modeling based on POD of a parabolized Navier-Stokes equations model II: Trust region POD 4D VAR data assimilation , 2013, Comput. Math. Appl..

[79]  D. Young,et al.  Localized radial basis function scheme for multidimensional transient generalized newtonian fluid dynamics and heat transfer , 2016 .

[80]  Vinod Kumar,et al.  High-order finite volume shallow water model on the cubed-sphere: 1D reconstruction scheme , 2015, Appl. Math. Comput..

[81]  Michael Dumbser,et al.  A staggered semi-implicit spectral discontinuous Galerkin scheme for the shallow water equations , 2013, Appl. Math. Comput..

[82]  R. Dalrymple,et al.  SPH modeling of dynamic impact of tsunami bore on bridge piers , 2015 .

[83]  Ionel M. Navon,et al.  Non‐intrusive reduced order modelling with least squares fitting on a sparse grid , 2017 .

[84]  Jing Tang Xing,et al.  Shape adaptive RBF-FD implicit scheme for incompressible viscous Navier–Strokes equations , 2014 .

[85]  Chang Shu,et al.  Numerical comparison of least square-based finite-difference (LSFD) and radial basis function-based finite-difference (RBFFD) methods , 2006, Comput. Math. Appl..

[86]  Benjamin Boutin,et al.  Shock profiles for the Shallow-water Exner models , 2015 .

[87]  A. I. Tolstykh,et al.  On using radial basis functions in a “finite difference mode” with applications to elasticity problems , 2003 .

[88]  M. Piggott,et al.  A POD reduced order unstructured mesh ocean modelling method for moderate Reynolds number flows , 2009 .

[89]  Manuel Kindelan,et al.  Laurent series based RBF-FD method to avoid ill-conditioning , 2015 .

[90]  Ionel M. Navon,et al.  Reduced‐order modeling based on POD of a parabolized Navier–Stokes equation model I: forward model , 2012 .

[91]  S. S. Ravindran,et al.  Reduced-Order Adaptive Controllers for Fluid Flows Using POD , 2000, J. Sci. Comput..

[92]  Fabien Marche,et al.  An efficient scheme on wet/dry transitions for shallow water equations with friction , 2011 .

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

[94]  Fayssal Benkhaldoun,et al.  A stabilized meshless method for time-dependent convection-dominated flow problems , 2017, Math. Comput. Simul..

[95]  Chih‐Tsung Hsu,et al.  Iterative explicit simulation of 1D surges and dam‐break flows , 2002 .

[96]  Gregory E. Fasshauer,et al.  Meshfree Approximation Methods with Matlab , 2007, Interdisciplinary Mathematical Sciences.

[97]  INTERPOLATION TECHNIQUES FOR SCATTERED DATA BY LOCAL RADIAL BASIS FUNCTION DIFFERENTIAL QUADRATURE METHOD , 2013 .

[98]  Xiaolin Li,et al.  Meshless Galerkin algorithms for boundary integral equations with moving least square approximations , 2011 .

[99]  C. Covelli,et al.  The analytic solution of the Shallow-Water Equations with partially open sluice-gates: The dam-break problem , 2015 .

[100]  Jirí Felcman,et al.  Adaptive finite volume approximation of the shallow water equations , 2012, Appl. Math. Comput..

[101]  Mehdi Dehghan,et al.  Weighted finite difference techniques for the one-dimensional advection-diffusion equation , 2004, Appl. Math. Comput..

[102]  Siraj-ul-Islam,et al.  Local radial basis function collocation method along with explicit time stepping for hyperbolic partial differential equations , 2013 .

[103]  Moving Kriging Interpolation‐based Meshfree Method for Dynamic Analysis of Structures , 2011 .

[104]  Mehdi Dehghan,et al.  A meshless local Petrov-Galerkin method for the time-dependent Maxwell equations , 2014, J. Comput. Appl. Math..

[105]  C. Pain,et al.  Non‐intrusive reduced‐order modelling of the Navier–Stokes equations based on RBF interpolation , 2015 .

[106]  S. Ravindran.  A reduced-order approach for optimal control of fluids using proper orthogonal decomposition , 2000 .

[107]  Yulong Xing,et al.  Positivity-preserving high order well-balanced discontinuous Galerkin methods for the shallow water equations , 2010 .

[108]  L. Gu,et al.  Moving kriging interpolation and element‐free Galerkin method , 2003 .

[109]  C. T. Wu,et al.  A novel upwind-based local radial basis function differential quadrature method for convection-dominated flows , 2014 .

[110]  R. LeVeque.  Finite Volume Methods for Hyperbolic Problems: Characteristics and Riemann Problems for Linear Hyperbolic Equations , 2002 .

[111]  Juan Du,et al.  Non-linear model reduction for the Navier-Stokes equations using residual DEIM method , 2014, J. Comput. Phys..

[112]  K. Liew,et al.  An element-free IMLS-Ritz framework for buckling analysis of FG–CNT reinforced composite thick plates resting on Winkler foundations , 2015 .

[113]  Fayssal Benkhaldoun,et al.  Exact solutions to the Riemann problem of the shallow water equations with a bottom step , 2001 .

[114]  Gui-Rong Liu,et al.  An Introduction to Meshfree Methods and Their Programming , 2005 .

[115]  Ionel M. Navon,et al.  Non-linear Petrov-Galerkin methods for reduced order modelling of the Navier-Stokes equations using a mixed finite element pair , 2013 .

[116]  F. Benkhaldoun,et al.  A non-homogeneous Riemann solver for shallow water equations in porous media , 2016 .

[117]  Jing Chen,et al.  Mixed Finite Element Formulation and Error Estimates Based on Proper Orthogonal Decomposition for the Nonstationary Navier-Stokes Equations , 2008, SIAM J. Numer. Anal..

[118]  Yulong Xing,et al.  High-order well-balanced finite volume WENO schemes for shallow water equation with moving water , 2007, J. Comput. Phys..

[119]  R. Dalrymple,et al.  SPH Modeling of Short-crested Waves , 2017, 1705.08547.

[120]  R. Dalrymple,et al.  SPH MODELING OF VORTICITY GENERATION BY SHORT-CRESTED WAVE BREAKING , 2017 .

[121]  Fayssal Benkhaldoun,et al.  A simple finite volume method for the shallow water equations , 2010, J. Comput. Appl. Math..

[122]  Bengt Fornberg,et al.  Stable calculation of Gaussian-based RBF-FD stencils , 2013, Comput. Math. Appl..

[123]  Razvan Stefanescu,et al.  POD/DEIM nonlinear model order reduction of an ADI implicit shallow water equations model , 2012, J. Comput. Phys..