Lagrange-Galerkin Methods on Spherical Geodesic Grids

The weak Lagrange?Galerkin finite element method for the 2D shallow water equations on the sphere is presented. This method offers stable and accurate solutions because the equations are integrated along the characteristics. The equations are written in 3D Cartesian conservation form and the domains are discretized using linear triangular elements. The use of linear triangular elements permits the construction of accurate (by virtue of the second-order spatial and temporal accuracies of the scheme) and efficient (by virtue of the less stringent CFL condition of Lagrangian methods) schemes on unstructured domains. Using linear triangles in 3D Cartesian space allows for the explicit construction of area coordinate basis functions thereby simplifying the calculation of the finite element integrals. The triangular grids are constructed by a generalization of the icosahedral grids that have been typically used in recent papers. An efficient searching strategy for the departure points is also presented for these generalized icosahedral grids which involves very few floating point operations. In addition a high-order scheme for computing the characteristic curves in 3D Cartesian space is presented: a general family of Runge?Kutta schemes. Results for six test cases are reported in order to confirm the accuracy of the scheme.

[1]  David L. Williamson,et al.  Integration of the barotropic vorticity equation on a spherical geodesic grid , 1968 .

[2]  R. Glowinski,et al.  Computing Methods in Applied Sciences and Engineering , 1974 .

[3]  T. Chung,et al.  Finite Element Analysis in Fluid Dynamics , 1978 .

[4]  J. P. Benque,et al.  A new finite element method for Navier-Stokes equations coupled with a temperature equation , 1982 .

[5]  Charles S. Peskin,et al.  On the construction of the Voronoi mesh on a sphere , 1985 .

[6]  Harold Ritchie,et al.  Semi-Lagrangian advection on a Gaussian grid , 1987 .

[7]  Endre Süli,et al.  Stability of the Lagrange-Galerkin method with non-exact integration , 1988 .

[8]  J. Côté,et al.  A Lagrange multiplier approach for the metric terms of semi‐Lagrangian models on the sphere , 1988 .

[9]  J. R. Bates,et al.  Semi-Lagrangian Integration of a Gridpoint Shallow Water Model on the Sphere , 1989 .

[10]  A. Priestley,et al.  The Taylor–Galerkin Method for the Shallow-Water Equations on the Sphere , 1992 .

[11]  P. Swarztrauber,et al.  A standard test set for numerical approximations to the shallow water equations in spherical geometry , 1992 .

[12]  A. Priestley.  A Quasi-Conservative Version of the Semi-Lagrangian Advection Scheme , 1993 .

[13]  A comparison of Eulerian-Lagrangian methods for the solution of the transport equation , 1994 .

[14]  A. Priestley,et al.  Exact projections and the Lagrange-Galerkin method: a realistic alternative to quadrature , 1994 .

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

[16]  A. Baptista,et al.  A comparison of integration and interpolation Eulerian‐Lagrangian methods , 1995 .

[17]  F. Giraldo,et al.  A Comparison of a Family of Eulerian and Semi-Lagrangian Finite Element Methods for the Advection-Diffusion Equations , 1997 .

[18]  M. Taylor.  The Spectral Element Method for the Shallow Water Equations on the Sphere , 1997 .

[19]  Francis X. Giraldo,et al.  Analysis of the Turkel-Zwas Scheme for the Two-Dimensional Shallow Water Equations in Spherical Coordinates , 1997 .