A High-Order Radial Basis Function (RBF) Leray Projection Method for the Solution of the Incompressible Unsteady Stokes Equations

A new projection method based on radial basis functions (RBFs) is presented for discretizing the incompressible unsteady Stokes equations in irregular geometries. The novelty of the method comes from the application of a new technique for computing the Leray-Helmholtz projection of a vector field using generalized interpolation with divergence-free and curl-free RBFs. Unlike traditional projection methods, this new method enables matching both tangential and normal components of divergence-free vector fields on the domain boundary. This allows incompressibility of the velocity field to be enforced without any time-splitting or pressure boundary conditions. Spatial derivatives are approximated using collocation with global RBFs so that the method only requires samples of the field at (possibly scattered) nodes over the domain. Numerical results are presented demonstrating high-order convergence in both space (between 5th and 6th order) and time (up to 4th order) for some model problems in two dimensional irregular geometries.

[1]  Joseph D. Ward,et al.  Divergence-Free RBFs on Surfaces , 2007 .

[2]  Jian-Guo Liu,et al.  Stable and accurate pressure approximation for unsteady incompressible viscous flow , 2010, J. Comput. Phys..

[3]  Xiu Ye,et al.  A discrete divergence-free basis for finite element methods , 1997, Numerical Algorithms.

[4]  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 .

[5]  Grady B. Wright,et al.  Stability and Error Estimates for Vector Field Interpolation and Decomposition on the Sphere with RBFs , 2009, SIAM J. Numer. Anal..

[6]  Qinghai Zhang,et al.  A fourth-order approximate projection method for the incompressible Navier-Stokes equations on locally-refined periodic domains , 2014 .

[7]  Elisabeth Larsson,et al.  Stable Computations with Gaussian Radial Basis Functions , 2011, SIAM J. Sci. Comput..

[8]  Edward J. Fuselier,et al.  Sobolev-type approximation rates for divergence-free and curl-free RBF interpolants , 2008, Math. Comput..

[9]  G. Wright,et al.  A hybrid radial basis function–pseudospectral method for thermal convection in a 3‐D spherical shell , 2010 .

[10]  Bengt Fornberg,et al.  A primer on radial basis functions with applications to the geosciences , 2015, CBMS-NSF regional conference series in applied mathematics.

[11]  Jean Leray,et al.  Étude de diverses équations intégrales non linéaires et de quelques problèmes que pose l'Hydrodynamique. , 1933 .

[12]  Natasha Flyer,et al.  Transport schemes on a sphere using radial basis functions , 2007, J. Comput. Phys..

[13]  Jie Shen,et al.  An overview of projection methods for incompressible flows , 2006 .

[14]  Tobin A. Driscoll,et al.  Eigenvalue stability of radial basis function discretizations for time-dependent problems , 2006, Comput. Math. Appl..

[15]  Natasha Flyer,et al.  A radial basis function method for the shallow water equations on a sphere , 2009, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

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

[17]  Gene H. Golub,et al.  Numerical solution of saddle point problems , 2005, Acta Numerica.

[18]  Michael J. McCourt,et al.  Stable Evaluation of Gaussian Radial Basis Function Interpolants , 2012, SIAM J. Sci. Comput..

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

[20]  Souleymane Kadri Harouna,et al.  Divergence-free Wavelet Projection Method for Incompressible Viscous Flow , 2012 .

[21]  L. Trefethen.  Spectral Methods in MATLAB , 2000 .

[22]  Cécile Piret,et al.  The orthogonal gradients method: A radial basis functions method for solving partial differential equations on arbitrary surfaces , 2012, J. Comput. Phys..

[23]  R. Temam.  Sur l'approximation de la solution des équations de Navier-Stokes par la méthode des pas fractionnaires (II) , 1969 .

[24]  Roger Temam,et al.  Navier-Stokes Equations and Turbulence. Encyclopedia of Math and its Applications, Vol. 83 , 2002 .

[25]  B. Fornberg,et al.  A numerical study of some radial basis function based solution methods for elliptic PDEs , 2003 .

[26]  N. SIAMJ.,et al.  DIVERGENCE-FREE KERNEL METHODS FOR APPROXIMATING THE STOKES PROBLEM , 2009 .

[27]  Quoc Thong Le Gia,et al.  Approximation of parabolic PDEs on spheres using spherical basis functions , 2005, Adv. Comput. Math..

[28]  Svenja Lowitzsch.  Error estimates for matrix-valued radial basis function interpolation , 2005, J. Approx. Theory.

[29]  Grady B. Wright,et al.  A High-Order Kernel Method for Diffusion and Reaction-Diffusion Equations on Surfaces , 2012, Journal of Scientific Computing.

[30]  G. Wright,et al.  A Radial Basis Function Method for Computing Helmholtz-Hodge Decompositions , 2015, 1502.01575.

[31]  Bengt Fornberg,et al.  Fast generation of 2-D node distributions for mesh-free PDE discretizations , 2015, Comput. Math. Appl..

[32]  Per-Olof Persson,et al.  A Simple Mesh Generator in MATLAB , 2004, SIAM Rev..

[33]  M. Minion,et al.  Accurate projection methods for the incompressible Navier—Stokes equations , 2001 .

[34]  Elisabeth Larsson,et al.  A Radial Basis Function Partition of Unity Collocation Method for Convection–Diffusion Equations Arising in Financial Applications , 2015, J. Sci. Comput..

[35]  Robert L. Pego,et al.  Stability and convergence of efficient Navier‐Stokes solvers via a commutator estimate , 2007 .

[36]  Steven J. Ruuth,et al.  Implicit-explicit methods for time-dependent partial differential equations , 1995 .

[37]  F. J. Narcowich,et al.  Generalized Hermite interpolation via matrix-valued conditionally positive definite functions , 1994 .

[38]  Steven J. Ruuth,et al.  Implicit-Explicit Methods for Time-Dependent PDE''s , 1993 .