High-Order Quasi-Uniform Approximation on the Sphere Using Fourier-Finite-Elements

Dubos, Thomas

doi:10.1007/978-3-642-15337-2_14

Cited by 2 publications

(8 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This should be slightly more economical since the required smoothness, and hence polynomial degree, is less. Work is under way to explore both possibilities (Dubos 2009). quadrature points provide exact quadrature for d = 1 (resp.…”

Section: Discussionmentioning

confidence: 99%

A conservative Fourier‐finite‐element method for solving partial differential equations on the whole sphere

Dubos

2009

Quart J Royal Meteoro Soc

Self Cite

View full text Add to dashboard Cite

Solving transport equations on the whole sphere using an explicit time stepping and an Eulerian formulation on a latitude-longitude grid is relatively straightforward but suffers from the pole problem: due to the increased zonal resolution near the pole, numerical stability requires unacceptably small time steps. Commonly used workarounds such as near-pole zonal filters affect the qualitative properties of the numerical method. Rigorous solutions based on spherical harmonics have a high computational cost.The numerical method we propose to avoid this problem is based on a Galerkin formulation in a subspace of a Fourierfinite-element spatial discretization. The functional space we construct provides quasi-uniform resolution and high-order accuracy, while the Galerkin formalism guarantees the conservation of linear and quadratic invariants. For N 2 degrees of freedom, the computational cost is O (N 2 log N), dominated by the zonal Fourier transforms. This is more than with a finite-difference or finite-volume method, which costs O(N 2 ), and less than with a spherical harmonics method, which costs O(N 3 ). Differential operators with latitude-dependent coefficients are inverted at a cost of O(N 2 ).We present experimental results and standard benchmarks demonstrating the accuracy, stability and efficiency of the method applied to the advection of a scalar field by a prescribed velocity field and to the incompressible rotating Navier-Stokes equations. The steps required to extend the method towards compressible flows and the Saint-Venant equations are described.

show abstract

Section: Discussionmentioning

confidence: 99%

A conservative Fourier‐finite‐element method for solving partial differential equations on the whole sphere

Dubos

2009

Quart J Royal Meteoro Soc

Self Cite

View full text Add to dashboard Cite

show abstract

“…This can be alleviated to a large extent by introducing the quasi-uniform resolution to the zonal Fourier coefficients in terms of either the high-order spectral filtering (Cheong et al 2004;Cheong 2006) or the reduced number of grid points (or zonal Fourier components) near the poles (Dubos 2009(Dubos , 2011. In particular, for a given horizontal resolution, the highorder spectral filtering was shown to provide almost the same time step size as that for the spherical harmonics model.…”

Section: Introductionmentioning

confidence: 89%

“…The FFEM can be used not only for differential equations in the spherical coordinates, but also for those in other coordinates as was demonstrated by previous studies (e.g., Heinrich 1996;Kim and Kweon 2009). Dubos (2009) has applied the FFEM to a variety of problems including simple derivatives of first and second order, the advection equation, and shallow-water models on a sphere. In Dubos (2009), the error (or accuracy) convergence rate of the FFEM with the B splines of degree d turned out to be the dth order; the quadratic and cubic splines, for example, provided the discretization errors proportional to D 2 and D 3 , respectively, where D is the grid size.…”

Section: Introductionmentioning

confidence: 92%

“…The pole singularity for the gridpoint-based models is avoidable by adopting the off-pole grids (e.g., the interior grids as is used in DFS spectral model). Dubos (2009Dubos ( , 2011 presented that for the FFEM, which should include the poles, the singularity can be avoided by setting the vanishing values at the poles for the zonal Fourier coefficients other than the zonal-mean components. This kind of simple pole condition, however, does not seem sufficient to avoid the discontinuity at poles, as will be demonstrated in this study.…”

Section: Introductionmentioning

confidence: 98%

“…Recently, the Fourier finite-element method (FFEM), another kind of Galerkin method of O(N 2 log 2 N) operations for the spherical coordinate system, has been studied (Dubos 2009(Dubos , 2011, which incorporates the Fourier spectral method in longitude and the finiteelement method (FEM) in latitude using the B-spline functions of degree up to three. The FFEM can be used not only for differential equations in the spherical coordinates, but also for those in other coordinates as was demonstrated by previous studies (e.g., Heinrich 1996;Kim and Kweon 2009).…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Fourier Finite-Element Method with Linear Basis Functions on a Sphere: Application to Elliptic and Transport Equations

Cheong

Kong

Kang

et al. 2015

Monthly Weather Review

View full text Add to dashboard Cite

The Fourier finite-element method (FFEM) on the sphere, which performs with an operation count of O(N 2 log 2 N) for 2N 3 N grids in spherical coordinates, was developed using linear basis functions. Dependent field variables are expanded with the Fourier series in the longitude, and the Fourier coefficients are represented with a series of first-order finite elements. Different types of pole conditions were incorporated into the Fourier coefficients of the scalar and vector variables in order to avoid discontinuity at the poles. For the Laplacian operator, the linear element was defined as a function of the sine of latitude instead of the latitude. The FFEM was applied to the derivatives of the first-and second-order elliptic equations and the transport equations. The scale-selective high-order Laplacian-type filter was implemented as a hyperviscosity. For the first-order derivative the fourth-order convergence rate of the accuracy, as is expected from the theoretical analysis, was achieved. Elliptic equations were found to be solved accurately without pole discontinuity, and the convergence rate turned out to be second order. The cosine bell advection, time-differenced with a thirdorder Runge-Kutta method, showed that the squared-norm error convergence rate was slightly above second order. Both the Gaussian bell advection and the deformational flow produced the theoretical convergence rate of fourth order. The high-order filter was found to be effective in maintaining a quasi-uniform resolution over the sphere, and thus allowed a large time step size. Sensitivity experiments of cosine bell advection over the poles revealed that the CFL number, as defined with the maximum grid size on the global domain, can be taken to be as large as unity.

show abstract

High-Order Quasi-Uniform Approximation on the Sphere Using Fourier-Finite-Elements

Cited by 2 publications

References 18 publications

A conservative Fourier‐finite‐element method for solving partial differential equations on the whole sphere

A conservative Fourier‐finite‐element method for solving partial differential equations on the whole sphere

Fourier Finite-Element Method with Linear Basis Functions on a Sphere: Application to Elliptic and Transport Equations

Contact Info

Product

Resources

About