Comparison of unstructured, staggered grid methods for the shallow water equations

Unstructured grid models are receiving increased attention mainly because of their ability to provide a flexible spatial discretization. Hence, some areas can be resolved in great detail while not over-resolving other areas. Development of these models is an ongoing process with significant longstanding issues with spurious computational modes, efficiency, advection and Coriolis approximations, and so forth. However, many of these problems have been solved with the current generation of models which have much promise for coastal to global scale ocean modelling. Our purpose is to intercompare a class of unstructured grid models where the continuity equation reduces to a finite volume approximation. The momentum equations can be approximated with finite difference, finite element, or finite volume methods. Each of these methods can have advantages and disadvantages in different classes of problems that range from hydraulics to coastal and global ocean flows. Some of the more important differences are restrictions on grid irregularity and stability of the Coriolis term. The finite element version of the model has important advantages in the discretization of the Coriolis term and does not require a reconstruction of a tangential velocity component. The comparison is illustrated with a simple test case. (C) 2008 Elsevier Ltd. All rights reserved.

[1]  Andrew Staniforth,et al.  An accurate interpolating scheme for semi-Lagrangian advection on an unstructured mesh for ocean modelling , 1997 .

[2]  David R. Owen,et al.  Computational methods for non-linear problems , 1987 .

[3]  Jean-François Remacle,et al.  Dispersion Analysis of Discontinuous Galerkin Schemes Applied to Poincaré, Kelvin and Rossby Waves , 2008, J. Sci. Comput..

[4]  V. Legat,et al.  A three-dimensional unstructured mesh finite element shallow-water model, with application to the flows around an island and in a wind-driven, elongated basin , 2008 .

[5]  A Quantitative Analysis of the Dissipation Inherent in Semi-Lagrangian Advection , 1988 .

[6]  Dale R. Durran,et al.  The Third-Order Adams-Bashforth Method: An Attractive Alternative to Leapfrog Time Differencing , 1991 .

[7]  Jens Schröter,et al.  A finite-element ocean model: principles and evaluation , 2004 .

[8]  Vincenzo Casulli,et al.  Semi-implicit numerical modeling of nonhydrostatic free-surface flows for environmental problems , 2002 .

[9]  Roy A. Walters,et al.  Geometrically based, automatic generator for irregular triangular networks , 1993 .

[10]  R. Walters,et al.  Operational forecasting of sea level height using an unstructured grid ocean model , 2009 .

[11]  Changsheng Chen,et al.  An Unstructured Grid, Finite-Volume, Three-Dimensional, Primitive Equations Ocean Model: Application to Coastal Ocean and Estuaries , 2003 .

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

[13]  Roy A. Walters,et al.  A robust, finite element model for hydrostatic surface water flows , 1998 .

[14]  Graham F. Carey,et al.  Analysis of spurious oscillation modes for the shallow water and Navier-Stokes equations , 1983 .

[15]  Eric Deleersnijder,et al.  A comparison of three finite elements to solve the linear shallow water equations , 2003 .

[16]  Daniel Y. Le Roux,et al.  Analysis of Numerically Induced Oscillations in 2D Finite-Element Shallow-Water Models Part I: Inertia-Gravity Waves , 2007, SIAM J. Sci. Comput..

[17]  B. Perot.  Conservation Properties of Unstructured Staggered Mesh Schemes , 2000 .

[18]  T. Ringler,et al.  Analysis of Discrete Shallow-Water Models on Geodesic Delaunay Grids with C-Type Staggering , 2005 .

[19]  Useful time-stepping methods for the Coriolis term in a shallow water model , 2009 .

[20]  David A. Ham,et al.  The symmetry and stability of unstructured mesh C-grid shallow water models under the influence of Coriolis , 2007 .

[21]  G. Yeh.  Numerical solutions of Navier-Stokes equations with an integrated compartment method (ICM) , 1981 .

[22]  Roy A. Walters,et al.  A semi‐implicit finite element model for non‐hydrostatic (dispersive) surface waves , 2005 .

[23]  J. Maître,et al.  Connection between finite volume and mixed finite element methods , 1996 .

[24]  J. J. Leendertse,et al.  Aspects of a computational model for long-period water-wave propagation , 1967 .

[25]  Guus S. Stelling,et al.  A staggered conservative scheme for every Froude number in rapidly varied shallow water flows , 2003 .

[26]  H. Hasumi,et al.  Developments in ocean climate modelling , 2000 .

[27]  Daniel Y. Le Roux,et al.  Analysis of Numerically Induced Oscillations in Two-Dimensional Finite-Element Shallow-Water Models Part II: Free Planetary Waves , 2008, SIAM J. Sci. Comput..

[28]  Emmanuel Hanert,et al.  Impact of mass lumping on gravity and Rossby waves in 2D finite-element shallow-water models , 2009 .

[29]  Emmanuel Hanert,et al.  A tale of two elements: P1NC-P1 and RT0 , 2009 .

[30]  Roy A. Walters,et al.  Semi-Lagrangian methods for a finite element coastal ocean model , 2007 .

[31]  Alfio Quarteroni,et al.  Finite element approximation of Quasi-3D shallow water equations , 1999 .

[32]  Eric Deleersnijder,et al.  An efficient Eulerian finite element method for the shallow water equations , 2005 .

[33]  A. Staniforth,et al.  Semi-Lagrangian integration schemes for atmospheric models - A review , 1991 .

[34]  Circulation in the Coastal Ocean , 1982 .

[35]  Rainald Löhner,et al.  An advancing front point generation technique , 1998 .

[36]  Andrew Staniforth,et al.  A Semi-implicit Semi-Lagrangian Finite-Element Shallow-Water Ocean Model , 2000 .

[37]  Roy A. Walters,et al.  An unstructured grid, three‐dimensional model based on the shallow water equations , 2000 .

[38]  Conservation of energy for schemes applied to the propagation of shallow‐water inertia‐gravity waves in regions with varying depth , 2000 .

[39]  Richard M. Iverson,et al.  Flow of variably fluidized granular masses across three‐dimensional terrain: 2. Numerical predictions and experimental tests , 2001 .

[40]  Guus S. Stelling,et al.  On the accurate and stable reconstruction of tangential velocities in C-grid ocean models , 2009 .

[41]  C.R.E. de Oliveira,et al.  Three-dimensional unstructured mesh ocean modelling , 2005 .

[42]  M. Gerritsen,et al.  An unstructured-grid, finite-volume, nonhydrostatic, parallel coastal ocean simulator , 2006 .