Stable and convergent approximation of two-dimensional vector fields on unstructured meshes

A new framework is proposed for analyzing staggered-grid finite difference finite volume methods on unstructured meshes. The new framework employs the concept of external approximation of function spaces, and gauge convergence of numerical schemes through the quantities of vorticity and divergence, instead of individual derivatives of the velocity components. The construction of a stable and convergent external approximation of a simple but relevant vector-valued function space is demonstrated, and the new framework is applied to establish the convergence of the MAC scheme for the incompressible Stokes problem on unstructured meshes.

[1]  J. Aubin.  Approximation of Elliptic Boundary-Value Problems , 1980 .

[2]  R. Eymard,et al.  Finite Volume Methods , 2019, Computational Methods for Fluid Dynamics.

[3]  William C. Skamarock,et al.  A Linear Analysis of the NCAR CCSM Finite-Volume Dynamical Core , 2008 .

[4]  A. A. Amsden,et al.  Numerical calculation of almost incompressible flow , 1968 .

[5]  P. Wesseling,et al.  A conservative pressure-correction method for flow at all speeds , 2003 .

[6]  Qingshan Chen,et al.  On staggering techniques and the non-staggered Z-grid scheme , 2016, Numerische Mathematik.

[7]  Max D. Gunzburger,et al.  A co-volume scheme for the rotating shallow water equations on conforming non-orthogonal grids , 2013, J. Comput. Phys..

[8]  Roger Temam,et al.  COMPARISON OF FINITE VOLUME AND FINITE DIFFERENCE METHODS AND APPLICATION , 2006 .

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

[10]  Roger Temam,et al.  Steady-state Navier–Stokes equations , 2001 .

[11]  Daniel Y. Le Roux,et al.  Spurious inertial oscillations in shallow-water models , 2012, J. Comput. Phys..

[12]  Roger Temam,et al.  CONVERGENCE OF A CELL-CENTERED FINITE VOLUME METHOD AND APPLICATION TO ELLIPTIC EQUATIONS , 2014 .

[13]  Thierry Gallouët,et al.  Convergence of the MAC Scheme for the Compressible Stokes Equations , 2010, SIAM J. Numer. Anal..

[14]  J. Thuburn.  Numerical wave propagation on the hexagonal C-grid , 2008, J. Comput. Phys..

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

[16]  A. K. B. Chand,et al.  Positive blending Hermite rational cubic spline fractal interpolation surfaces , 2015 .

[17]  Almut Gassmann,et al.  A global hexagonal C‐grid non‐hydrostatic dynamical core (ICON‐IAP) designed for energetic consistency , 2013 .

[18]  A. D. Gosman,et al.  The computation of compressible and incompressible recirculating flows by a non-iterative implicit scheme , 1986 .

[19]  Robert Eymard,et al.  On MAC schemes on triangular delaunay meshes, their convergence and application to coupled flow problems , 2014 .

[20]  Almut Gassmann.  Inspection of hexagonal and triangular C-grid discretizations of the shallow water equations , 2011, J. Comput. Phys..

[21]  Vivette Girault.  A combined finite element and Marker and cell method for solving Navier-Stokes equations , 1976 .

[22]  David A. Randall,et al.  Geostrophic Adjustment and the Finite-Difference Shallow-Water Equations , 1994 .

[23]  J. Miller.  Numerical Analysis , 1966, Nature.

[24]  E Weinan,et al.  Projection method III: Spatial discretization on the staggered grid , 2002, Math. Comput..

[25]  Hester Bijl,et al.  A Unified Method for Computing Incompressible and Compressible Flows in Boundary-Fitted Coordinates , 1998 .

[26]  S. Patankar,et al.  Pressure based calculation procedure for viscous flows at all speeds in arbitrary configurations , 1988 .

[27]  Vivette Girault,et al.  Finite Element Methods for Navier-Stokes Equations - Theory and Algorithms , 1986, Springer Series in Computational Mathematics.

[28]  Robert Eymard,et al.  MAC schemes on triangular Delaunay meshes , 2012 .

[29]  So-Hsiang Chou,et al.  Analysis and convergence of a covolume method for the generalized Stokes problem , 1997, Math. Comput..

[30]  William C. Skamarock,et al.  Numerical representation of geostrophic modes on arbitrarily structured C-grids , 2009, J. Comput. Phys..

[31]  Thierry Gallouët,et al.  An extension of the MAC scheme to locally refined meshes: convergence analysis for the full tensor time-dependent Navier–Stokes equations , 2015 .

[32]  F. Harlow,et al.  Numerical Calculation of Time‐Dependent Viscous Incompressible Flow of Fluid with Free Surface , 1965 .

[33]  Akio Arakawa,et al.  Computational Design of the Basic Dynamical Processes of the UCLA General Circulation Model , 1977 .

[34]  I. Wenneker,et al.  A Mach‐uniform unstructured staggered grid method , 2002 .

[35]  P. Wesseling.  Principles of Computational Fluid Dynamics , 2000 .

[36]  William C. Skamarock,et al.  A unified approach to energy conservation and potential vorticity dynamics for arbitrarily-structured C-grids , 2010, J. Comput. Phys..

[37]  G. Kanschat.  Divergence‐free discontinuous Galerkin schemes for the Stokes equations and the MAC scheme , 2008 .

[38]  A. A. Amsden,et al.  A numerical fluid dynamics calculation method for all flow speeds , 1971 .

[39]  J. Cea.  Approximation variationnelle des problèmes aux limites , 1964 .