A spectral element shallow water model on spherical geodesic grids

The article of record as published may be located at http://dx.doi.org/10.1002/1097-0363(20010430)35:8<869The spectral element method for the two-dimensional shallow water equations on the sphere is presented. The equations are written in conservation form and the domains are discretized using quadrilateral elements obtained from the generalized icosahedral grid introduced previously (Giraldo FX. Lagrange-Galerkin methods on spherical geodesic grids: the shallow water equations. Journal of Computational Physics 2000; 160: 336-368). The equations are written in Cartesian co-ordinates that introduce an additional momentum equation, but the pole singularities disappear. This paper represents a departure from previously published work on solving the shallow water equations on the sphere in that the equations are all written, discretized, and solved in three-dimensional Cartesian space. Because the equations are written in a three-dimensional Cartesian co-ordinate system, the algorithm simplifies into the integration of surface elements on the sphere from the fully three-dimensional equations. A mapping (Song Ch, Wolf JP. The scaled boundary finite element method-alias consistent infinitesimal finite element cell method-for diffusion. International Journal for Numerical Methods in Engineering 1999; 45: 1403-1431) which simplifies these computations is described and is shown to contain the Eulerian version of the method introduced previously by Giraldo (Journal of Computational Physics 2000; 160: 336-368) for the special case of triangular elements. The significance of this mapping is that although the equations are written in Cartesian co-ordinates, the mapping takes into account the curvature of the high-order spectral elements, thereby allowing the elements to lie entirely on the surface of the sphere. In addition, using this mapping simplifies all of the three-dimensional spectral-type finite element surface integrals because any of the typical two-dimensional planar finite element or spectral element basis functions found in any textbook (for example, Huebner et al. The Finite Element Method for Engineers. Wiley, New York, 1995; Karniadakis GE, Sherwin SJ. Spectral/hp Element Methods for CFD. Oxford University Press, New York, 1999; and Szabo B, Babuska I. Finite Element Analysis. Wiley, New York, 1991) can be used. Results for six test cases are presented to confirm the accuracy and stability of the new method. Published in 2001 by John Wiley & Sons, Ltd

A nodal triangle-based spectral element method for the shallow water equations on the sphere

The article of record as published may be located at http://dx.doi.org/10.1016/j.jcp.2005.01.004A nodal triangle-based spectral element (SE) method for the shallow water equations on the sphere is presented. The original SE method uses quadrilateral elements and high-order nodal Lagrange polynomials, constructed from a tensor-product of the Legendre-Gauss-Lobatto points. In this work, we construct the high-order Lagrange polynomials directly on the triangle using nodal sets obtained from the electrostatics principle [J.S. Hesthaven, From electrostatics to almost optimal nodal sets for polynomial interpolation in a simplex, SIAM Journal on Numerical Analysis 35 (1998) 655-676] and Fekete points [M.A. Taylor, B.A. Wingate, R.E. Vincent, An algorithm for computing Fekete points in the triangle, SIAM Journal on Numerical Analysis 38 (2000) 1707-1720]. These points have good approximation properties and far better Lebesgue constants than any other nodal set derived for the triangle. By employing triangular elements as the basic building-blocks of the SE method and the Cartesian coordinate form of the equations, we can use any grid imaginable including adaptive unstructured grids. Results for six test cases are presented to confirm the accuracy and stability of the method. The results show that the triangle-based SE method yields the expected exponential convergence and that it can be more accurate than the quadrilateral-based SE method even while using 30-60% fewer grid points especially when adaptive grids are used to align the grid with the flow direction. However, at the moment, the quadrilateral-based SE method is twice as fast as the triangle-based SE method because the latter does not yield a diagonal mass matrix

Stability analysis for Eulerian and semi-Lagrangian finite-element formulation of the advection-diffusion equation

The article of record as published may be located at http://dx.doi.org/10.1016/S0898-1221(99)00185-6This paper analyzes the stability of the finite-element approximation to the linearized two-dimensional advection-diffusion equation. Bilinear basis functions on rectangular elements are considered. This is one of the two best schemes as was shown by Neta and Williams [1]. Time is discretized with the theta algorithms that yield the explicit (theta = 0), semi-implicit (theta = 1/2), and implicit (theta = 1) methods. This paper extends the results of Neta and Williams [1] for the advection equation. Giraldo and Neta [2] have numerically compared the Eulerian and semi-Lagrangian finite-element approximation for the advection-diffusion equation. This paper analyzes the finite element schemes used there. The stability analysis shows that the semi-Lagrangian method is unconditionally stable for all values of a while the Eulerian method is only unconditionally stable for 1/2 < theta < 1. This analysis also shows that the best methods are the semi-implicit ones (theta = 1/2). In essence this paper analytically compares a semi-implicit Eulerian method with a semi-implicit semi-Lagrangian method. It is concluded that (for small or no diffusion) the semi-implicit semi-Lagrangian method exhibits better amplitude, dispersion and group velocity errors than the semi-implicit Eulerian method thereby achieving better results. In the case the diffusion coefficient is large, the semi-Lagrangian loses its competitiveness. Published by Elsevier Science Ltd

An element-based spectrally-optimized approximate inverse preconditioner for the Euler equations

Non-hydrostatic Unified Model of the Atmosphere (NUMA)The first NUMA papers appeared in 2008. From 2008 through 2010, all the NUMA papers appearing involved the 2D (x-z slice) Euler equations. All the theory and numerical implementations were first developed in 2D.We introduce a method for constructing an element-by-element sparse approximate inverse (SAI) preconditioner designed to be effective in a massively-parallel spectral element modeling environment involving non- symmetric systems. This new preconditioning approach is based on a spectral optimization of a low-resolution pre- conditioned system matrix (PSM). We show that the local preconditioning matrices obtained via this element-based, spectrum-optimized (EBSO) approach may be applied to arbitrarily high-resolution versions of the same system matrix without appreciable loss of preconditioner performance. We demonstrate the performance of the EBSO precondition- ing approach using 2-D spectral element method (SEM) formulations for a simple linear conservation law and for the fully-compressible 2-D Euler equations with various boundary conditions. For the latter model running at suffi- ciently large Courant number, the EBSO preconditioner significantly reduces both iteration count and wall-clock time regardless of whether a generalized minimum residual (GMRES) or a stabilized biconjugate gradient (BICGSTAB) iterative scheme is employed. To assess the value added by this new preconditioning approach, we compare its perfor- mance against two other equally-parallel SAI preconditioning methods: low-order Chebyshev generalized least-squares polynomials and an element-based variant of the well-known Frobenius norm optimization preconditioner which we also develop herein. The EBSO preconditioner significantly out-performs both the Chebyshev polynomials and the element-based Frobenius-norm-optimized (EBFO) preconditioner regardless of whether the GMRES or BICGSTAB iterative scheme is employed. Moreover, when the EBSO preconditioner is combined with the Chebyshev polynomial method dramatic reductions in iterations per time-step can be achieved while still achieving a significant reduction in wall-clock time

Strong and auxiliary forms of the semi-Lagrangian method for incompressible flows

We present a review of the semi-Lagrangian method for advection-diusion and incompressible Navier-Stokes equations discretized with high-order methods. In particular, we compare the strong form where the departure points are computed directly via backwards integration with the auxiliary form where an auxiliary advection equation is solved instead; the latter is also referred to as Operator Integration Factor Splitting (OIFS) scheme. For intermediate size of time steps the auxiliary form is preferrable but for large time steps only the strong form is stable

Lagrange-Galerkin methods on spherical geodesic grids: The shallow water equations

The article of record as published may be located at http://dx.doi.org/10.1006/jcph.2000.6469The weak Lagrange-Galerkin finite element method fur the 2D shallow water equations on the sphere is presented. This method offers stable and accurate solutions because the equations are integrated along thr 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 fur the departure points is also presented for these generalized icosahedral grids which involves very few Boating 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

Strong and weak Lagrange-Galerkin spectral element methods for the shallow water equations

The article of record as published may be located at http://dx.doi.org/10.1016/S0898-1221(03)80010-XThe Lagrange-Galerkin spectral element method for the two-dimensional shallow water equations is presented. The equations are written in conservation form and the domains are discretized using quadrilateral elements. Lagrangian methods integrate the governing equations along the characteristic curves, thus being well suited for resolving the nonlinearities introduced by the advection operator of the fluid dynamics equations. Two types of Lagrange-Galerkin methods are presented: the strong and weak formulations. The strong form relies mainly on interpolation to achieve high accuracy while the weak form relies primarily on integration. Lagrange-Galerkin schemes offer an increased efficiency by virtue of their less stringent CFL condition. The use of quadrilateral elements permits the construction of spectral-type finite-element methods that exhibit exponential convergence as in the conventional spectral method, yet they are constructed locally as in the finite-element method; this is the spectral element method. In this paper, we show how to fuse the Lagrange-Calerkin methods with the spectral element method and present results for two standard test cases in order to compare and contrast these two hybrid schemes. (C) 2003 Published by Elsevier Science Ltd

Lagrange-Galerkin methods on spherical geodesic grids

The article of record as published may be located at http://dx.doi.org/10.1006/jcph.1997.5771Lagrange-Galerkin finite element methods that are high-order accurate, exactly integrable, and highly efficient are presented. This paper derives generalized natural Cartesian coordinates in three dimensions for linear triangles on the surface of the sphere. By using these natural coordinates as the finite element basis functions we can integrate the corresponding integrals exactly thereby achieving a high level of accuracy and efficiency for modeling physical problems on the sphere. The discretization of the sphere is achieved by the use of a spherical geodesic triangular grid. A tree data structure that is inherent to this grid is introduced; this tree data structure exploits the property of the spherical geodesic grid, allowing for rapid searching of departure points which is essential to the Lagrange-Galerkin method. The generalized natural coordinates are also used for determining in which element the departure points lie. A comparison of the Lagrange-Galerkin method with an Euler-Galerkin method demonstrates the impressive lever of high order accuracy achieved by the Lagrange-Galerkin method at computational costs comparable or better than the Euler-Galerkin method. In addition, examples using advancing front unstructured grids illustrate the flexibility of the Lagrange-Galerkin method on different grid types, By introducing generalized natural coordinates and the tree data structure for the spherical geodesic grid, the Lagrange-Galerkin method can be used for solving practical problems on the sphere more accurately than current methods, yet requiring less computer time. (C) 1997 Academic Press