Finite Volume Methods on Spheres and Spherical Centroidal Voronoi Meshes

We study in this paper a finite volume approximation of linear convection-diffusion equations defined on a sphere using the spherical Voronoi meshes, in particular the spherical centroidal Voronoi meshes. The high quality of spherical centroidal Voronoi meshes is illustrated through both theoretical analysis and computational experiments. In particular, we show that the L2 error of the approximate solution is of quadratic order when the underlying mesh is given by a spherical centroidal Voronoi mesh. We also demonstrate numerically the high accuracy and the superconvergence of the approximate solutions.

[1]  G. R. Stuhne,et al.  New icosahedral grid-point discretization of the shallow water equation on the sphere , 1999 .

[2]  Zhiqiang Cai,et al.  On the finite volume element method , 1990 .

[3]  Panayot S. Vassilevski,et al.  Finite Difference Schemes on Triangular Cell-Centered Grids with Local Refinement , 1992, SIAM J. Sci. Comput..

[4]  Emmanuel Hebey,et al.  Sobolev Spaces on Riemannian Manifolds , 1996 .

[5]  M. Gunzburger,et al.  Voronoi-based finite volume methods, optimal Voronoi meshes, and PDEs on the sphere ☆ , 2003 .

[6]  Franz Aurenhammer,et al.  Voronoi diagrams—a survey of a fundamental geometric data structure , 1991, CSUR.

[7]  E. Saff,et al.  Discretizing Manifolds via Minimum Energy Points , 2004 .

[8]  Panayot S. Vassilevski,et al.  Finite volume methods for convection-diffusion problems , 1996 .

[9]  R. Nicolaides.  Analysis and convergence of the MAC scheme. I : The linear problem , 1992 .

[10]  X. Wu,et al.  Analysis and convergence of the MAC scheme. II. Navier-Stokes equations , 1996, Math. Comput..

[11]  Todd D. Ringler.  Comparing Truncation Error to Partial Differential Equation Solution Error on Spherical Voronoi Tesselations , 2003 .

[12]  P. Vassilevski,et al.  Local refinement techniques for elliptic problems on cell-centered grids , 1991 .

[13]  Qiang Du,et al.  Probabilistic methods for centroidal Voronoi tessellations and their parallel implementations , 2002, Parallel Comput..

[14]  R. Herbin,et al.  An Error Estimate for a Nite Volume Scheme for a Diiusion Convection Problem on a Triangular Mesh , 1995 .

[15]  Qiang Du,et al.  Centroidal Voronoi Tessellations: Applications and Algorithms , 1999, SIAM Rev..

[16]  Yves Coudi,et al.  DISCRETE SOBOLEV INEQUALITIES AND L p ERROR ESTIMATES FOR FINITE VOLUME SOLUTIONS OF CONVECTION DIFFUSION EQUATIONS , 2001 .

[17]  Qian Li,et al.  Error estimates in L2, H1 and Linfinity in covolume methods for elliptic and parabolic problems: A unified approach , 1999, Math. Comput..

[18]  Anita T. Layton.  Cubic Spline Collocation Method for the Shallow Water Equations on the Sphere , 2002 .

[19]  Andreas Hense,et al.  The shallow water equations on the sphere and their Lagrange-Galerkin-solution , 2002 .

[20]  Hirofumi Tomita,et al.  Shallow water model on a modified icosahedral geodesic grid by using spring dynamics , 2001 .

[21]  P. Vassilevski,et al.  Local refinement techniques for elliptic problems on cell-centered grids. I. Error analysis , 1991 .

[22]  Ilya D. Mishev,et al.  FINITE VOLUME METHODS ON VORONOI MESHES , 1998 .

[23]  Qiang Du.  Convergence Analysis of a Numerical Method for a Mean Field Model of Superconducting Vortices , 2000, SIAM J. Numer. Anal..

[24]  G. Dziuk.  Finite Elements for the Beltrami operator on arbitrary surfaces , 1988 .

[25]  Ronghua Li.  Generalized Difference Methods for Differential Equations: Numerical Analysis of Finite Volume Methods , 2000 .

[26]  Yves Coudière,et al.  Discrete Sobolev inequalities and Lp error estimates for finite volume solutions of convection diffusion equations , 2001 .

[27]  Qiang Du,et al.  Numerical simulations of the quantized vortices on a thin superconducting hollow sphere , 2004 .

[28]  Randall J. LeVeque,et al.  A Wave Propagation Algorithm for the Solution of PDEs on the Surface of a Sphere , 2001 .

[29]  R. Heikes,et al.  Numerical Integration of the Shallow-Water Equations on a Twisted Icosahedral Grid , 1995 .

[30]  O. C. Zienkiewicz,et al.  An ‘upwind’ finite element scheme for two‐dimensional convective transport equation , 1977 .

[31]  Robert J. Renka,et al.  Algorithm 772: STRIPACK: Delaunay triangulation and Voronoi diagram on the surface of a sphere , 1997, TOMS.

[32]  Qiang Du,et al.  Analysis and Convergence of a Covolume Approximation of the Ginzburg--Landau Model of Superconductivity , 1998 .

[33]  Thierry Gallouët,et al.  Error Estimates on the Approximate Finite Volume Solution of Convection Diffusion Equations with General Boundary Conditions , 2000, SIAM J. Numer. Anal..

[34]  R. Nicolaides.  Direct discretization of planar div-curl problems , 1992 .

[35]  Qiang Du,et al.  Constrained Centroidal Voronoi Tessellations for Surfaces , 2002, SIAM J. Sci. Comput..

[36]  Li Ronghua,et al.  Generalized difference methods for a nonlinear Dirichlet problem , 1987 .

[37]  P. Frederickson,et al.  Icosahedral Discretization of the Two-Sphere , 1985 .

[38]  Zhimin Zhang,et al.  Ultraconvergence of ZZ patch recovery at mesh symmetry points , 2003, Numerische Mathematik.

[39]  Noel J. Walkington,et al.  Co-volume methods for degenerate parabolic problems , 1993 .

[40]  Qiang Du,et al.  Approximations of a Ginzburg-Landau model for superconducting hollow spheres based on spherical centroidal Voronoi tessellations , 2004, Math. Comput..