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Radial Basis Function-Generated Finite Differences: A Mesh-Free Method for Computational Geosciences

  • Natasha Flyer
  • Grady B. Wright
  • Bengt Fornberg
Reference work entry

Abstract

Radial basis function-generated finite differences (RBF-FD) is a mesh-free method for numerically solving partial differential equations that emerged in the last decade and have shown rapid growth in the last few years. From a practical standpoint, RBF-FD sprouted out of global RBF methods, which have shown exceptional numerical qualities in terms of accuracy and time stability for numerically solving PDEs, but are not practical when scaled to very large problem sizes because of their computational cost and memory requirements. RBF-FD bypass these issues by using local approximations for derivatives instead of global ones. Matrices in the RBF-FD methodology go from being completely full to 99 % empty. Of course, the sacrifice is the exchange of spectral accuracy from the global RBF methods for high-order algebraic convergence of RBF-FD, assuming smooth data. However, since natural processes are almost never infinitely differentiable, little is lost and much gained in terms of memory and runtime. This chapter provides a survey of a group of topics relevant to using RBF-FD for a variety of problems that arise in the geosciences. Particular emphasis is given to problems in spherical geometries, both on surfaces and within a volume. Applications discussed include nonlinear shallow water equations on a sphere, reaction–diffusion equations, global electric circuit, and mantle convection in a spherical shell. The results from the last three of these applications are new and have not been presented before for RBF-FD.

Keywords

Nusselt Number Spherical Shell Discontinuous Galerkin Shallow Water Equation Discontinuous Galerkin Method 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. Ascher UM, Ruuth SJ, Wetton BTR (1995) Implicit-explicit methods for time-dependent partial differential equations. SIAM J Numer Anal 32:797–823MathSciNetCrossRefzbMATHGoogle Scholar
  2. Backus GE (1966) Potentials for tangent tensor fields on spheroids. Arch Ration Mech Anal 22:210–252MathSciNetCrossRefzbMATHGoogle Scholar
  3. Blaise S, St-Cyr A (2012) A dynamic hp-adaptive discontinuous Galerkin method for shallow water flows on the sphere with application to a global tsunami simulation. Mon Wea Rev 140(3):978–996CrossRefGoogle Scholar
  4. Bochner S (1933) Monotine Functionen, Stieltjes Integrale und Harmonische Analyse. Math Ann 108:378–410MathSciNetCrossRefzbMATHGoogle Scholar
  5. Bollig E, Flyer N, Erlebacher G (2012) Solution to PDEs using radial basis function finite-differences (RBF-FD) on multiple GPUs. J Comput Phys 231:7133–7151MathSciNetCrossRefGoogle Scholar
  6. Chandrasekhar S (1981) Hydrodynamic and hydromagnetic stability. Dover, New YorkzbMATHGoogle Scholar
  7. Collatz L (1960) The numerical treatment of differential equations. Springer, BerlinCrossRefzbMATHGoogle Scholar
  8. Driscoll TA, Fornberg B (2002) Interpolation in the limit of increasingly flat radial basis functions. Comput Math Appl 43:413–422MathSciNetCrossRefzbMATHGoogle Scholar
  9. Driscoll TA, Heryudono A (2007) Adaptive residual subsampling methods for radial basis function interpolation and collocation problems. Comput Math Appl 53:927–939MathSciNetCrossRefzbMATHGoogle Scholar
  10. Fasshauer GE (2007) Meshfree approximation methods with MATLAB. Interdisciplinary mathematical sciences, vol 6. World Scientific, SingaporeGoogle Scholar
  11. Flyer N, Lehto E, Blaise S, Wright GB, St-Cyr A (2012) A guide to RBF-generated finite differences for nonlinear transport: shallow water simulations on a sphere. J Comput Phys 231:4078–4095MathSciNetCrossRefzbMATHGoogle Scholar
  12. Fornberg B (1996) A practical guide to pseudospectral methods. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  13. Fornberg B, Lehto E (2011) Stabilization of RBF-generated finite difference methods for convective PDEs. J Comput Phys 230:2270–2285MathSciNetCrossRefzbMATHGoogle Scholar
  14. Fornberg B, Piret C (2007) A stable algorithm for flat radial basis functions on a sphere. SIAM J Sci Comput 30:60–80MathSciNetCrossRefzbMATHGoogle Scholar
  15. Fornberg B, Wright G (2004) Stable computation of multiquadric interpolants for all values of the shape parameter. Comput Math Appl 48:853–867MathSciNetCrossRefzbMATHGoogle Scholar
  16. Fornberg B, Zuev J (2007) The Runge phenomenon and spatially variable shape parameters in RBF interpolation. Comput Math Appl 54:379–398MathSciNetCrossRefzbMATHGoogle Scholar
  17. Fornberg B, Driscoll TA, Wright G, Charles R (2002) Observations on the behavior of radial basis functions near boundaries. Comput Math Appl 43:473–490MathSciNetCrossRefzbMATHGoogle Scholar
  18. Fornberg B, Wright G, Larsson E (2004) Some observations regarding interpolants in the limit of flat radial basis functions. Comput Math Appl 47:37–55MathSciNetCrossRefzbMATHGoogle Scholar
  19. Fornberg B, Larsson E, Flyer N (2011) Stable computations with Gaussian radial basis functions. SIAM J Sci Comput 33(2):869–892MathSciNetCrossRefzbMATHGoogle Scholar
  20. Fornberg B, Lehto E, Powell C (2013) Stable calculation of Gaussian-based RBF-FD stencils. Comput Math Appl 65:627–637MathSciNetCrossRefzbMATHGoogle Scholar
  21. Fox L (1947) Some improvements in the use of relaxation methods for the solution of ordinary and partial differential equations. Proc R Soc A 190:31–59CrossRefMathSciNetzbMATHGoogle Scholar
  22. Fuselier EJ, Wright GB (2013) A high-order kernel method for diffusion and reaction-diffusion equations on surfaces. J Sci Comput 1–31. doi:10.1007/s10915-013-9688-xGoogle Scholar
  23. Galewsky J, Scott RK, Polvani LM (2004) An initial-value problem for testing numerical models of the global shallow-water equations. Tellus 56A:429–440CrossRefGoogle Scholar
  24. Gottlieb D, Orszag SA (1977) Numerical analysis of spectral methods. SIAM, PhiladelphiaCrossRefzbMATHGoogle Scholar
  25. Gupta MM (1991) High accuracy solutions of incompressible Navier-Stokes equations. J Comput Phys 93:343–359MathSciNetCrossRefzbMATHGoogle Scholar
  26. Harder H (1998) Phase transitions and the three-dimensional planform of thermal convection in the Martian mantle. J Geophys Res 103:16,775–16,797CrossRefGoogle Scholar
  27. Hardy RL (1971) Multiquadric equations of topography and other irregular surfaces. J Geophys Res 76:1905–1915CrossRefGoogle Scholar
  28. Hesthaven JS, Gottlieb S, Gottlieb D (2007) Spectral methods for time-dependent problems. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  29. Kameyama MC, Kageyama A, Sato T (2008) Multigrid-based simulation code for mantle convection in spherical shell using Yin-Yang grid. Phys Earth Planet Inter 171:19–32CrossRefGoogle Scholar
  30. Lele SK (1992) Compact finite difference schemes with spectral-like resolution. J Comput Phys 103:16–42MathSciNetCrossRefzbMATHGoogle Scholar
  31. Li M, Tang T, Fornberg B (1995) A compact fourth-order finite difference scheme for the steady incompressible Navier-Stokes equations. Int J Numer Methods Fluids 20:1137–1151MathSciNetCrossRefzbMATHGoogle Scholar
  32. Madych WR, Nelson SA (1992) Bounds on multivariate polynomials and exponential error estimates for multiquadric interpolation. J Approx Theory 70:94–114MathSciNetCrossRefzbMATHGoogle Scholar
  33. Mairhuber JC (1956) On Haar’s theorem concerning Chebyshev approximation problems having unique solutions. Proc Am Math Soc 7(4):609–615MathSciNetzbMATHGoogle Scholar
  34. Micchelli CA (1986) Interpolation of scattered data: distance matrices and conditionally positive definite functions. Constr Approx 2:11–22MathSciNetCrossRefzbMATHGoogle Scholar
  35. Peyret R (2002) Spectral methods for incompressible viscous flow. Springer, New YorkCrossRefzbMATHGoogle Scholar
  36. Piret C (2012) The orthogonal gradients method: a radial basis functions method for solving partial differential equations on arbitrary surfaces. J Comput Phys 231:4662–4675MathSciNetCrossRefzbMATHGoogle Scholar
  37. Powell MJD (1992) The theory of radial basis function approximation in 1990. In: Light W (ed) Advances in numerical analysis, vol II: wavelets, subdivision algorithms and radial functions. Oxford University Press, Oxford, pp 105–210Google Scholar
  38. Ratcliff JT, Schubert G, Zebib A (1996) Steady tetrahedral and cubic patterns of spherical shell convection with temperature-dependent viscosity. J Geophys Res 101:473–484Google Scholar
  39. Schaback R (1995) Error estimates and condition numbers for radial basis function interpolants. Adv Comput Math 3:251–264MathSciNetCrossRefzbMATHGoogle Scholar
  40. Schaback R (2005) Multivariate interpolation by polynomials and radial basis functions. Constr Approx 21:293–317MathSciNetCrossRefzbMATHGoogle Scholar
  41. Schoenberg IJ (1938) Metric spaces and completely monotone functions. Ann Math 39:811–841MathSciNetCrossRefzbMATHGoogle Scholar
  42. Spotz WF, Taylor MA, Swarztrauber PN (1998) Fast shallow water equation solvers in latitude-longitude coordinates. J Comput Phys 145:432–444MathSciNetCrossRefzbMATHGoogle Scholar
  43. St-Cyr A, Jablonowski C, Dennis JM, Tufo HM, Thomas SJ (2008) A comparison of two shallow-water models with nonconforming adaptive grids. Mon Weather Rev 136:1898–1922CrossRefGoogle Scholar
  44. Stemmer K, Harder H, Hansen U (2006) A new method to simulate convection with strongly temperature-dependent and pressure-dependent viscosity in spherical shell. Phys Earth Planet Inter 157:223–249CrossRefGoogle Scholar
  45. Taylor M, Tribbia J, Iskandarani M (1997) The spectral element method for the shallow water equations on the sphere. J Comput Phys 130:92–108CrossRefzbMATHGoogle Scholar
  46. Trefethen LN (2000) Spectral methods in MATLAB. SIAM, PhiladelphiaCrossRefzbMATHGoogle Scholar
  47. Turing A (1952) The chemical basis of morphogenesis. Philos Trans R Soc B 237:37–52CrossRefGoogle Scholar
  48. van der Vorst H (1992) BI-CGSTAB: a fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems. SIAM J Sci Stat Comput 13(2):631–644. doi:10.1137/0913035CrossRefMathSciNetzbMATHGoogle Scholar
  49. Varea C, Aragon J, Barrio R (1999) Turing patterns on a sphere. Phys Rev E 60:4588–4592CrossRefGoogle Scholar
  50. Wilson CTR (1920) Investigations on lightning discharges and on the electric field of thunderstorms. Philos Trans R Soc Lond A 221:73–115CrossRefGoogle Scholar
  51. Wilson CTR (1929) Some thundercloud problems. J Frankl Inst 208:1–12CrossRefGoogle Scholar
  52. Womersley RS, Sloan IH (2003/2007) Interpolation and cubature on the sphere. Website, http://web.maths.unsw.edu.au/~rsw/Sphere/Google Scholar
  53. Wright GB, Fornberg B (2006) Scattered node compact finite difference-type formulas generated from radial basis functions. J Comput Phys 212:99–123MathSciNetCrossRefzbMATHGoogle Scholar
  54. Wright GB, Flyer N, Yuen DA (2010) A hybrid radial basis function – pseudospectral method for thermal convection in a 3D spherical shell. Geophys Geochem Geosyst 11(7):Q07,003CrossRefGoogle Scholar
  55. Yoon J (2001) Spectral approximation orders of radial basis function interpolation on the Sobolev space. SIAM J Math Anal 33(4):946–958MathSciNetCrossRefzbMATHGoogle Scholar
  56. Yoshida M, Kageyama A (2004) Application of the Ying-Yang grid to a thermal convection of a Boussinesq fluid with infinite Prandtl number in a three-dimensional spherical shell. Geophys Res Lett 31:L12,609CrossRefGoogle Scholar
  57. Zhai S, Feng X, He Y (2013) A family of fourth-order and sixth-order compact difference schemes for the three-dimensional Poisson equation. J Sci Comput 54:97–120MathSciNetCrossRefzbMATHGoogle Scholar
  58. Zhong S, McNamara A, Tan E, Moresi L, Gurnis M (2008) A benchmark study on mantle convection in a 3-D spherical shell using CitcomS. Geochem Geophys Geosyst 9:Q10,017CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Natasha Flyer
    • 1
  • Grady B. Wright
    • 2
  • Bengt Fornberg
    • 3
  1. 1.Institute for Mathematics Applied to GeosciencesNational Center for Atmospheric ResearchBoulderUSA
  2. 2.Department of MathematicsBoise State UniversityBoiseUSA
  3. 3.Department of Applied MathematicsUniversity of ColoradoBoulderUSA

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