Analysis of modified Godunov type schemes for the two-dimensional linear wave equation with Coriolis source term on cartesian meshes

Abstract The study deals with collocated Godunov type finite volume schemes applied to the two-dimensional linear wave equation with Coriolis source term. The purpose is to explain the wrong behaviour of the classic scheme and to modify it in order to avoid accuracy issues around the geostrophic equilibrium and in geostrophic adjustment processes. To do so, a Hodge-like decomposition is introduced. Then three different well-balanced strategies are introduced. Some properties of the associated modified equations are proven and then extended to the semi-discrete case. Stability of fully discrete schemes under a suitable CFL condition is established thanks to a Von Neumann analysis. Some numerical results reinforce the purpose and exhibit the concrete improvements achieved by the application of these new techniques in both linear and nonlinear cases.

[1]  H. Guillard,et al.  On the behaviour of upwind schemes in the low Mach number limit , 1999 .

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

[3]  Hamed Zakerzadeh.  The RS-IMEX scheme for the rotating shallow water equations with the Coriolis force , 2017 .

[4]  Stéphane Dellacherie,et al.  Construction of modified Godunov-type schemes accurate at any Mach number for the compressible Euler system , 2016 .

[5]  H. Guillard,et al.  On the behavior of upwind schemes in the low Mach number limit: II. Godunov type schemes , 2004 .

[6]  R. Temam,et al.  The primitive equations on the large scale ocean under the small depth hypothesis , 2002 .

[7]  Dirk Olbers,et al.  Ocean Dynamics , 2012 .

[8]  Felix Rieper,et al.  A low-Mach number fix for Roe's approximate Riemann solver , 2011, J. Comput. Phys..

[9]  R. F. Warming,et al.  The modified equation approach to the stability and accuracy analysis of finite-difference methods , 1974 .

[10]  Francisco Guillén,et al.  Mathematical Justification of the Hydrostatic Approximation in the Primitive Equations of Geophysical Fluid Dynamics , 2001, SIAM J. Math. Anal..

[11]  Ami Harten,et al.  Self adjusting grid methods for one-dimensional hyperbolic conservation laws☆ , 1983 .

[12]  Stéphane Dellacherie,et al.  The influence of cell geometry on the Godunov scheme applied to the linear wave equation , 2010, J. Comput. Phys..

[13]  E. Toro.  Riemann Solvers and Numerical Methods for Fluid Dynamics , 1997 .

[14]  Minh Hieu Do,et al.  Analysis of Apparent Topography Scheme for the Linear Wave Equation with Coriolis Force , 2017 .

[15]  Colin J. Cotter,et al.  A Framework for Mimetic Discretization of the Rotating Shallow-Water Equations on Arbitrary Polygonal Grids , 2012, SIAM J. Sci. Comput..

[16]  Stéphane Dellacherie,et al.  Analysis of Godunov type schemes applied to the compressible Euler system at low Mach number , 2010, J. Comput. Phys..

[17]  Emmanuel Audusse,et al.  Conservative discretization of Coriolis force in a finite volume framework , 2009, J. Comput. Phys..

[18]  Emmanuel Audusse,et al.  A Fast and Stable Well-Balanced Scheme with Hydrostatic Reconstruction for Shallow Water Flows , 2004, SIAM J. Sci. Comput..

[19]  Alexander F. Shchepetkin,et al.  The regional oceanic modeling system (ROMS): a split-explicit, free-surface, topography-following-coordinate oceanic model , 2005 .

[20]  Mária Lukácová-Medvid'ová,et al.  Well-balanced finite volume evolution Galerkin methods for the shallow water equations , 2015, J. Comput. Phys..

[21]  R. Klein.  Semi-implicit extension of a Godunov-type scheme based on low Mach number asymptotics , 1995 .

[22]  V. Zeitlin,et al.  Nonlinear dynamics of rotating shallow water : methods and advances , 2007 .

[24]  Christophe Chalons,et al.  A simple well-balanced and positive numerical scheme for the shallow-water system , 2015 .

[25]  F. Bouchut.  Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws: and Well-Balanced Schemes for Sources , 2005 .

[26]  P. Roe.  Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes , 1997 .

[27]  Manuel Jesús Castro Díaz,et al.  Finite Volume Simulation of the Geostrophic Adjustment in a Rotating Shallow-Water System , 2008, SIAM J. Sci. Comput..

[28]  Vladimir Zeitlin,et al.  Frontal geostrophic adjustment and nonlinear wave phenomena in one-dimensional rotating shallow water. Part 2. High-resolution numerical simulations , 2004, Journal of Fluid Mechanics.

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

[30]  Stefan Vater,et al.  Stability of a Cartesian grid projection method for zero Froude number shallow water flows , 2009, Numerische Mathematik.