Journal of Scientific Computing
Journal of Computational Physics
Journal of Computational Physics
The spectral element method for the shallow water equations on the sphere
Journal of Computational Physics
Lagrange-Galerkin methods on spherical geodesic grids
Journal of Computational Physics
Convergence Analysis for a Class of High-Order Semi-Lagrangian Advection Schemes
SIAM Journal on Numerical Analysis
The Lagrange-Galerkin spectral element method on unstructured quadrilateral grids
Journal of Computational Physics
Lagrange—Galerkin methods on spherical geodesic grids: the shallow water equations
Journal of Computational Physics
A semi-Lagrangian high-order method for Navier-Stokes equations
Journal of Computational Physics
Nodal high-order discontinuous Galerkin methods for the spherical shallow water equations
Journal of Computational Physics
Parallel Implementation Issues: Global versus Local Methods
Computing in Science and Engineering
Weather and climate numerical algorithms: an efficient, parallel solution scheme for the shallow water equations
Parallel Implementation Issues: Global versus Local Methods
Computing in Science and Engineering
Nonlinear operator integration factor splitting for the shallow water equations
Applied Numerical Mathematics
The NCAR Spectral Element Climate Dynamical Core: Semi-Implicit Eulerian Formulation
Journal of Scientific Computing
Hybrid Eulerian-Lagrangian Semi-Implicit Time-Integrators
Computers & Mathematics with Applications
Semi-Lagrangian Runge-Kutta Exponential Integrators for Convection Dominated Problems
Journal of Scientific Computing
Nonlinear operator integration factor splitting for the shallow water equations
Applied Numerical Mathematics
A Fourier-Legendre spectral element method in polar coordinates
Journal of Computational Physics
Method of Moving Frames to Solve Conservation Laws on Curved Surfaces
Journal of Scientific Computing
Multiphysics simulations: Challenges and opportunities
International Journal of High Performance Computing Applications
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A spectral element semi-Lagrangian (SESL) method for the shallow water equations on the sphere is presented. The sphere is discretized using a hexahedral grid although any grid imaginable can be used as long as it is comprised of quadrilaterals. The equations are written in Cartesian coordinates to eliminate the pole singularity which plagues the equations in spherical coordinates. In a previous paper [Int. J. Numer. Methods Fluids 35 (2001) 869] we showed how to construct an explicit Eulerian spectral element (SE) model on the sphere; we now extend this work to a semi-Lagrangian formulation. The novelty of the Lagrangian formulation presented is that the high order SE basis functions are used as the interpolation functions for evaluating the values at the Lagrangian departure points. This makes the method not only high order accurate but quite general and thus applicable to unstructured grids and portable to distributed memory computers. The equations are discretized fully implicitly in time in order to avoid having to interpolate derivatives at departure points. By incorporating the Coriolis terms into the Lagrangian derivative, the block LU decomposition of the equations results in a symmetric positive-definite pseudo-Helmholtz operator which we solve using the generalized minimum residual method (GMRES) with a fast projection method [Comput. Methods Appl. Mech. Eng. 163 (1998) 193]. Results for eight test cases are presented to confirm the accuracy and stability of the method. These results show that SESL yields the same accuracy as an Eulerian spectral element semi-implicit (SESI) while allowing for time-steps 10 times as large and being up to 70% more efficient.