A higher order Finite Volume resolution method for a system related to the inviscid primitive equations in a complex domain

We construct the cell-centered Finite Volume discretization of the two-dimensional inviscid primitive equations in a domain with topography. To compute the numerical fluxes, the so-called Upwind Scheme (US) and the Central-Upwind Scheme (CUS) are introduced. For the time discretization, we use the classical fourth order Runge–Kutta method. We verify, with our numerical simulations, that the US (or CUS) is a robust first (or second) order scheme, regardless of the shape or size of the topography and without any mesh refinement near the topography.

[1]  Roger Temam,et al.  Some Mathematical Problems in Geophysical Fluid Dynamics , 2009 .

[2]  A Study of Cell{center Finite Volume Methods for Diiusion Equations , 1998 .

[3]  R. Temam,et al.  Numerical Approximation of the Inviscid 3D Primitive Equations in a Limited Domain , 2010, 1011.4704.

[4]  A. E. Gill.  Atmosphere-Ocean Dynamics , 1982 .

[5]  Alexander Kurganov,et al.  Central-Upwind Schemes for Two-Layer Shallow Water Equations , 2009, SIAM J. Sci. Comput..

[6]  Michele Coti Zelati,et al.  The Atmospheric Equation of Water Vapor with Saturation , 2012 .

[7]  R. Temam,et al.  The 3D Primitive Equations in the absence of viscosity : Boundary conditions and well-posedness in the linearized case , 2008 .

[8]  R. Temam,et al.  A multilevel method for finite volume discretization of the two-dimensional nonlinear shallow-water equations , 2010 .

[9]  R. Temam,et al.  Numerical Simulations of the Inviscid Primitive Equations in a Limited Domain , 2006 .

[10]  R. LeVeque.  Finite Volume Methods for Hyperbolic Problems: Characteristics and Riemann Problems for Linear Hyperbolic Equations , 2002 .

[11]  E. Tadmor,et al.  Non-oscillatory central differencing for hyperbolic conservation laws , 1990 .

[12]  Alexander Kurganov,et al.  Semidiscrete Central-Upwind Schemes for Hyperbolic Conservation Laws and Hamilton-Jacobi Equations , 2001, SIAM J. Sci. Comput..

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

[14]  Roger Temam,et al.  Open Boundary Conditions for the Primitive and Boussinesq Equations , 2003 .

[15]  R. Temam,et al.  Numerical weather prediction in two dimensions with topography, using a finite volume method , 2014, 1409.4830.

[16]  L. R. Koenig,et al.  A Short Course in Cloud Physics , 1979 .

[17]  Randall J. LeVeque,et al.  A Cartesian Grid Finite-Volume Method for the Advection-Diffusion Equation in Irregular Geometries , 2000 .

[18]  George J. Haltiner,et al.  Numerical weather prediction , 1971 .

[19]  Chuan Yi Tang,et al.  A 2.|E|-Bit Distributed Algorithm for the Directed Euler Trail Problem , 1993, Inf. Process. Lett..

[20]  Weeratunge Malalasekera,et al.  An introduction to computational fluid dynamics - the finite volume method , 2007 .

[21]  Randall J. LeVeque,et al.  Cartesian Grid Methods for Fluid Flow in Complex Geometries , 2001 .

[22]  Doron Levy,et al.  CENTRAL-UPWIND SCHEMES FOR THE SAINT-VENANT SYSTEM , 2002 .

[23]  Roger Temam,et al.  The Barotropic Mode for the Primitive Equations , 2010, J. Sci. Comput..

[24]  Roger Temam,et al.  Some 2.5D Models for the Primitive Equations of the Ocean and the Atmosphere , 2007 .

[25]  J. Lions,et al.  New formulations of the primitive equations of atmosphere and applications , 1992 .

[26]  Roger Temam,et al.  Simulations of the 2.5D inviscid primitive equations in a limited domain , 2008, J. Comput. Phys..

[27]  J. Oliger,et al.  Theoretical and practical aspects of some initial-boundary value problems in fluid dynamics , 1976 .

[28]  Bojan Popov,et al.  Adaptive Semidiscrete Central-Upwind Schemes for Nonconvex Hyperbolic Conservation Laws , 2007, SIAM J. Sci. Comput..

[29]  CELL CENTERED FINITE VOLUME METHODS USING TAYLOR SERIES EXPANSION SCHEME WITHOUT FICTITIOUS DOMAINS , 2010 .

[30]  E. Tadmor,et al.  New High-Resolution Central Schemes for Nonlinear Conservation Laws and Convection—Diffusion Equations , 2000 .

[31]  R. Temam,et al.  Boundary conditions for the 2D linearized PEs of the ocean in the absence of viscosity , 2005 .