Total views : 731

Radial Basis Function Methods for Solving Partial Differential Equations-A Review

Affiliations

  • Department of Mathematics, Lovely Professional University, Punjab, 144411,, India

Abstract


Background/Objectives: The approximation using radial basis function (RBF) is an extremely powerful method to solve partial differential equations (PDEs). This paper presents different types of RBF methods to solve PDEs. Methods/ Statistical Analysis: Due to their meshfree nature, ease of implementation and independence of dimension, RBF methods are popular to solve PDEs. In this paper we examine different generalized RBF methods, including Kansa method, Hermite symmetric approach, localized and hybrid methods. We also discussed the preference of using meshfree methods like RBF over the mesh based methods. Findings: This paper presents a state-of-the-art review of the RBF methods. Some recent development of RBF approximation in solving PDEs is also discussed. The mathematical formulation of different RBF methods are discussed for better understanding. RBF methods have been actively developed over the years from global to local approximation and then to hybrid methods. Hybrid RBF methods help in reduction of computational cost and become very effective in solving large scale problems. Application/Improvements: RBF methods have been applied to various diverse fields like image processing, geo-modeling, pricing option and neural network etc.

Keywords

Differential Quadrature Radial Basis Function, Kansa Collocation Method, Partition of unity, Partial Differential Equation, Radial Basis Function.

Full Text:

 |  (PDF views: 1026)

References


  • Hardy RL. Multiquadric equations of topography and other irregular surfaces. J.Geo phys Res, 1971; 76
  • Franke R. Scattered data interpolation: Tests of some methods. Math Comput, 1982; 38: 181–200
  • Micchelli C A. Interpolation of scattered data: Distance matrices and conditionally positive defnite functions. Constructive Approximation. 1986; 2:11–22
  • Kansa EJ. Multiquadrics – A scattered data approximation scheme with applications to computational fluid dynamics I: Surface approximations and partial derivative estimates. Computers and Mathematics with Applications. 1990; 19:127–145
  • Kansa EJ. Multiquadrics -A scattered data approximation scheme with applications to computational fluid-dynamics II: Solutions to parabolic, hyperbolic and elliptic partial differential equations. Computers and Mathematics with Applications. 1990; 19:147–161
  • Fasshauer GE. Solving partial differential equations by collocation with radial basis functions, In: Mehaute A, Rabut C, Schumaker L L, editors. Surface Fitting and Multiresolution Methods, Vanderbilt University Press, Nashville. 1997; 131–138.
  • Power H, Barraco V. A comparison analysis between unsymmetric and symmetric radial basis function collocation methods for the numerical solution of partial differential equations. Computers and Mathematics with Applications. 2002; 43:551–583
  • Larsson E, Fornberg B. A numerical study of some radial basis function based solution methods for elliptic PDEs. Computers and Mathematics with Applications. 2003;46: 891–902
  • Ling L, Kansa EJ. A least-squares preconditioner for radial basis functions collocation methods. Advances in Computational Mathematics. 2005; 23:31–54
  • Ling L, Kansa EJ. Preconditioning for radial basis functions with domain decomposition methods. Mathematical and Computer Modelling. 2004; 40(13):1413–1427
  • Jichun L, Hon YC. Domain decomposition for radial basis meshless methods. Numerical Methods Partial Differential Equations. 2004; 20(3):450–462
  • Chantasiriwan S. Investigation of the use of radial basis functions in local collocation method for solving diffusion problems. International Communications in Heat Mass Transfer. 2004; 31(8):1095–1104
  • Šarler B, Vertnik R. Meshfree explicit local radial basis function collocation method for diffusion problems. Computers and Mathematics with Applications. 2006; 51(8):1269–1282
  • Šarler B, Vertnik R. Meshless local radial basis function collocation method for convective diffusive solid-liquid phase change problems. International Journal of Numerical Methods for Heat Fluid Flow. 2006; 16(5):617–640.
  • Divo E, Kassab AJ. An efficient localized radial basis function Meshless method for fluid flow and conjugate heat transfer. Journal Heat Transfer Transfer. ASME, 2007; 129(2):124–136
  • Šarler B. From global to local radial basis function collocation method for transport phenomena. Advance Meshfree Technique. 2007; 5:257–282
  • Kosec G, Šarler B. Local RBF collocation method for Darcy flow. Computer Modeling Engineeering and Sciences. 2008; 25(3):197–207
  • Sanyasiraju Y, Chandhini G. Local radial basis function based gridfree scheme for unsteady incompressible viscous flows. Journal of Computational Physics. 2008; 227(20):8922–8948
  • Lee CK, Liu X, Fan SC. Local multiquadric approximation for solving boundary value problems. Computational Mechanics. 2003; 30:396–409.
  • Yao GM. Local radial basis function methods for solving partial differential equations. Dissertation for the Doctoral Degree, [PhD thesis].University of Southern Mississippi, 2010
  • Yao GM, Kolibal J, Chen CS. A localized approach for the method of approximate particular solutions. Computers and Mathematics with Applications. 2011; 61:2376–2387
  • Fornberg B, Wright G. Stable computation of multiquadric interpolants for all values of the shape parameter. Computers and Mathematics with Applications. 2004; 48:853–867
  • Fornberg B, Piret C. A stable algorithm for flat radial basis functions on a sphere. SIAM Journal on Scientific Computing. 2007; 30:60–80
  • Shu C, Ding H, Yeo KS. Local Radial Basis Function-based Differential Quadrature Method and Its Application to Solve Two-dimensional Incompressible Navier-Stokes Equations. Computers Methods in Applied Mechanics and Engineering. 2003; 192: 941–954
  • Tolstykh A, Shirobokov D. On using radial basis functions in a finite difference mode with applications to elasticity problems. Computational Mechanics. 200; 33(1):68–79
  • Roque CMC, Cunha D, Shu C, et al. A Local Radial Basis Functions-Finite Differences Technique for the Analysis of Composite Plates. Engineering Analysis with Bound Elements. 2011; 35(3):363–374
  • Rodrigues JD, Roque CMC, Ferreira AJM, et al. Radial Basis Functions-Finite Differences Collocation and a Unified Formulation for Bending, Vibration and Buckling Analysis of Laminated Plates, According to Murakami’s Zig-Zag Theory. Composite Structures. 2011; 93(7):1613–1620.
  • Rodrigues JD, Roque CMC, Ferreira AJM. Analysis of Thick Plates by Local Radial Basis Functions-Finite Differences Method. Mecc, 2012; 47(5):1157–1171
  • Bollig EF, Flyer N, Erlebacher G. Solution to PDEs Using Radial Basis Function Finite-Differences (RBF-FD) on Multiple GPUs. Journal of Computational Physics. 2012; 231(21):7133–7151
  • Shu C, Ding H, Zhao N. Numerical Comparison of Least Square-Based Finite-Difference (LSFD) and Radial Basis Function-Based Finite-Difference (RBFFD) Methods. Computers and Mathematics Applications. 2006; 51(8):1297–131.
  • Wendland H. Fast evaluation of radial basis functions methods based on partition of unity. In: Approximation Theory X (St. Louis, MO, 2001), Vanderbilt University Press, Nashville, TN, 2002; 473–483.
  • Li J C, Chen CS. Some observations on unsymmetric radial basis function collocation methods for convection-diffusion problems. International Journal for Numerical Methods in Engineering. 2003; 57(8):1085–1094
  • Kovacevic I, Poredos A, Sarler B. Solving the Stefan problem with the radial basis function collocation mehod. Numerical Heat Transfer Part B Fundamentals. 2003; 44(6):575–599
  • Zhou X, Hon YC, Cheung KF. A grid-free, nonlinear shallow-water model with moving boundary. Engineering Analysis Boundary Elements. 2004; 28(8):967–973
  • Chantasiriwan S. Multiquadric collocation method for time-dependent heat conduction problems with temperature-dependent thermal properties. Journal of Heat Transfer ASME. 2007; 129(2):109–113
  • Duan Y, Tang PF, Huang T Z, et.al. Coupling projection domain decomposition method and Kansa’s method in electrostatic problems. Computer Physics Communications. 2009; 180(2):209–214
  • Chen W, Ye LJ, Sun HG. Fractional diffusion equations by the Kansa method. Computers and Mathematics with Applications. 2010; 59(5):1614–1620
  • Leitao VMA. RBF-based meshless methods for 2D elastostatic problems. Engineering Analysis with Bound Elements. 2004; 28(10):1271–1281
  • Rocca A La, Rosales AH, Power H. Symmetric radial basis function meshless approach for time dependent PDEs. Boundary Elements Xxvi, 2004; 19:81–90
  • Rocca A La, Power H, Rocca V La, Morale M. A meshless approach based upon radial basis function hermite collocation method for predicting the cooling and the freezing times of foods. CMC Computers Materials and Continua. 2005; 2(4):239–250
  • Rocca A La, Rosales AH, Power H. Radial basis function Hermite collocation approach for the solution of time dependent convection-diffusion problems. Engineering Analysis with Bound Elements. 2005; 29(4):359–370
  • Rosales AH, Rocca A La, Power H. Radial basis function Hermite collocation approach for the numerical simulation of the effect of precipitation inhibitor on the crystallization process of an over-saturated solution. Numerical Methods for Partial Differential Equations. 2006; 22(2):361–380
  • Naffa M, Al-Gahtani H J. RBF-based meshless method for large deflection of thin plates. Eng Anal Bound Elem, 2007; 31(4): 311–317
  • Chen W. New RBF collocation schemes and kernel RBFs with applications. Lecture Notes in Computational Science and Engineering. 2002; 26:75–86
  • Brown D, Ling L, Levesley J, et.al. On approximate cardinal preconditioning methods for solving PDEs with radial basis functions. Engineering Analysis with Boundary Elements. 2005; 29(4):343–353
  • Chen W, Fu ZJ, Chen CS. Recent advances in radial basis function collocation methods. Springer briefs in applied sciences and technology, 2014
  • Siraj-ul-Islam, Vertnik R, Šarlera B. Local radial basis function collocation method along with explicit time stepping for hyperbolic partial differential equations. Applied Numerical Mathematics. 2013; 67:136–151
  • Mavric B, Šarler B. Local radial basis function collocation method for linear thermoelasticity in two dimensions. International Journal of Numerical Methods for Heat and Fluid Flow. 2015; 25(6):1488–1511
  • Hon YC, Šarler B, Yun DF. Local radial basis function collocation method for solving thermo-driven fluid-flow problems with free surface. Engineering Analysis with Boundary Elements. (2015) http://dx.doi.org/10.1016/j.enganabound.2014.11.006i
  • Dehghan M, Abbaszadeh M, Mohebbi A. A meshless technique based on the local radial basis functions collocation method for solving parabolic–parabolic Patlak–Keller–Segel chemotaxis model. Engineering Analysis Boundary Elements. 2015; 56: 129–144
  • Fornberg B, Larsson E, Flyer N. Stable computations with Gaussian radial basis functions. SIAM Journal of Scientific Computing. 2011; 33(2):869–892
  • Piret C, Hanert E. A radial basis functions method for fractional diffusion equations. Journal of Computational Physics. 2013; 238:71–81
  • Dehghan M, Ilati M. The use of radial basis functions (RBFs) collocation and RBF-QR methods for solving the coupled nonlinear sine-Gordon equations. Engineering Analysis with Boundary Elements. 2015; 52:99–109.
  • Dehghan M, Abbaszadeh M, Mohebbi A. Solution of two-dimensional modified anomalous fractional sub-diffusion equation via radial basis functions (RBF) meshless method. Engineering Analysis with Boundary Elements. 2013; 38:72–82
  • Dehghan M, Abbaszadeh M, Mohebbi A. An implicit RBF meshless approach for solving the time fractional nonlinear sine-Gordon and Klein–Gordon equations. Engineering Analysis with Boundary Elements. 2015; 50:412–434
  • Karageorghis A, Chen CS, Kansa A-RBF method for Poisson problems in annular domains. In: Brebbia C A, Cheng A H-D, eds. Boundary Elements and Other Mesh Reduction Methods XXXVII, WIT Press, Southampton, 2014; 77–83.
  • Bellman RE, Kashef BG, Casti J. Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations. Journal of Computational Physics. 1972; 10:40–52.
  • Shu C. Differential quadrature and its application in engineering. Springer Verlag, London, 2000.
  • Liew K, Huang YQ, Reddy JN. Moving least squares differential quadrature method and its application to the analysis of sheared formable plates. International Journal for Numerical Methods in Engineering. 2003; 56:2331–2351.
  • Wu YL, Shu C. Development of RBF-DQ method for derivative approximation and its application to simulate natural convection in concentric annuli. Computational Mechanics. 2002; 29(6):477–485
  • Shu C, Ding H, Yeo KS. Solution of partial differential equations by a global radial basis function-based differential quadrature method. Engineering Analysis with Boundary Elements. 2004; 28:1217–1226
  • Shu C, Ding H, Chen HQ, et al. An upwind local RBF-DQ method for simulation of inviscid compressible flows. Computer Methods in Applied Mechanics and Engineering. 2005; 194:2001–2017
  • Shen Q. Local RBF-based differential quadrature collocation method for the boundary layer problems. Engineering Analysis with Boundary Elements. 2010; 34:213–228
  • Soleimani S, Jalaal M, Bararnia H, et al. Local RBF-DQ method for two dimensional transient heat conduction problems. International Communications in Heat and Mass Transfer. 2010; 37:1411–1418
  • Shu C, Wu YL. Integrated radial basis functions-based differential quadrature method and its performance. International Journal for Numerical Methods in Fluids. 2007; 53:969–984
  • Hashemi MR, Hatam F. Unsteady seepage analysis using local radial basis function-based differential quadrature method. Applied Mathematical Modelling. 2011; 35: 4934–4950
  • Dehghan M, Nikopour A. Numerical solution of the system of second-order boundary value problems using the local radial basis functions based differential quadrature collocation method. Applied Mathematical Modelling. 2013; 37:8578–8599
  • Homayoon L, Abedini MJ, Hashemi SMR. RBF-DQ Solution for Shallow Water Equations. Journal of Waterway, Port, Coastal and Ocean Engineering. 2013; 139(1): 45–60
  • Viola E, Tornabene F, Ferretti E, et al. Radial basis functions based on differential quadrature method for the free vibration analysis of laminated composite arbitrarily shaped plates. Composites Part B, 2015; 78:65–78
  • Dehghan M, Mohammadi V. The numerical solution of CahneHilliard (CH) equation in one, two and three-dimensions via globally radial basis functions (GRBFs) and RBFs-differential quadrature (RBFs-DQ) methods. Engineering Analysis with Boundary Elements. 2015; 51:74–100
  • Parand K, Hashemi S. RBF-DQ Method for Solving Non-linear Differential Equations of Lane-Emden type. arXiv preprint arXiv:1510.06619, 2015
  • Richardson L F. The approximate arithmetical solution by Finite differences of physical problems involving differential equations, with an application to the stresses in a masonry dam. Philosophical Transactions of the Royal Society of London. 1911; 210: 307–357
  • Tolstykh AI. On using RBF-based differencing formulas for unstructured and mixed structured unstructured grid calculations. Proceedings of the 16th IMACS World Congress 228. Lausanne 2000, 4606-4624.
  • Wang JG, Liu GR. A point interpolation meshless method based on radial basis functions. Journal for Numerical Methods in Engineering. 2002; 54:1623–1648
  • Wright GB. Radial basis function interpolation: Numerical and analytical developments, Dissertation for the Doctoral Degree.[PhD thesis] Boulder: University of Colorado, 2003
  • Chandhini G, Sanyasiraju YVSS. Local RBF-FD solutions for steady convection-diffusion problems. Int J Num Meth Eng, 2007; 72: 352–378
  • Chinchapatnam P P, Djidjeli K, Nair P B, et.al. A compact RBF- fd based meshless method for the incompressible navier-stokes equations. Journal of Engineering for the Maritime Environment. 2009; 223:275–290
  • Wright GB, Fornberg B. Scattered node compact finite difference-type formulas generated from radial basis functions. Journal of Computational Physics. 2006; 212:99–123
  • Fornberg B, Lehto E. Stabilization of RBF-generated finite difference methods for convective PDEs. Journal of Computational Physics. 2011; 230:2270–2285
  • Flyer N, Lehto E, Blaise S, et al. A guide to RBF-generated finite differences for nonlinear transport: Shallow water simulations on a sphere. Journal of Computational Physics. 2012; 231:4078–4095
  • Shankar V, Wright GB, Fogelson AL, et al. A radial basis function (rbf) finite difference method for the simulation of reaction diffusion equations on stationary platelets within the augmented forcing method. International Journal for Numerical Methods in Fluids. 2014; 75:1–12
  • Shankar V, Wright GB, Fogelson AL, et al. A radial basis function (RBF)-Finite difference (FD) method for diffusion and reaction-diffusion equations on surfaces. Journal of Scientific Computing. 2014; 63:745–768
  • Avazzadeh Z, Chen W, Hosseini VR. Numerical solution of fractional telegraph equation by using radial basis functions. Engineering Analysis with Boundary Elements. 2014; 38:31–39
  • Barnett GA, Flyer N, Wicker LJ. An RBF-FD polynomial method based on polyharmonic splines for the Navier-Stokes equations: Comparisons on different node layouts. arXiv:1509.02615 [physics], 2015
  • Kumar A, Tripathi LP, Kadalbajoo MK. A numerical study of Asian option with radial basis functions based finite differences method. Engineering Analysis with Boundary Elements. 2015; 50:1–7.
  • Kumar A, Tripathi LP, Kadalbajoo MK. A radial basis functions based finite differences method for wave equation with an integral condition. Applied Mathematics and Computation. 2015; 253:8–16
  • Babuška I, Melenk JM. The partition of unity method. International Journal for Numerical Methods in Engineering. 1997; 40:727–758.
  • Cavoretto R, Rossi A De. Spherical interpolation using the partition of unity method: an efficient and flexible algorithm. Applied Mathematics Letters. 2012; 25(10):1251–1256
  • Cavoretto R, Rossi A De. A meshless interpolation algorithm using a cell-based searching procedure. Computers and Mathematics with Applications. 2014; 67(5):1024–1038.
  • Safdari-Vaighani A, Heryudono A, Larsson E. A radial basis function partition of unity collocation method for convection–diffusion equations arising in financial applications. Journal of Scientific Computing. 2014; 1–27.
  • Heryudono A, Larsson E, Ramage A, et al. Preconditioning for Radial Basis Function Partition of Unity Methods. Journal of Scientific Computing., 2015; 1–15

Refbacks

  • There are currently no refbacks.


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.