A Schwarz preconditioner for the cubed-sphere

Stephen J. Thomas, John M. Dennis, Henry M. Tufo, Paul F. Fischer

Research output: Contribution to journalArticlepeer-review


A spectral element formulation of the atmospheric two-dimensional shallow water equations on the cubed-sphere is described. The equations are written in tensor form using the contravariant and covariant velocity components. A semi-implicit time discretization results in a reduced Schur complement system for the pressure. The Laplacian operator is approximated by the L2 pseudo-Laplacian arising in the PN/PN-2 spectral element formulation of the incompressible Stokes problem. The overlapping Schwarz preconditioner of Fischer, Miller, and Tufo [An overlapping Schwarz method for spectral element simulation of three-dimensional incompressible flows, in Parallel Solution of Partial Differential Equations, IMA Vol. Math. Appl. 120, Springer, New York, 2000, pp. 159-180] based on the fast diagonalization method, is extended to generalized curvilinear coordinates. To obtain a separable operator for the linear finite element tensor-product approximation within each spectral element, extrema of the inverse metric tensor and its determinant are employed. Convergence rates and parallel CPU timings are compared against a block-Jacobi preconditioner.

Original languageEnglish (US)
Pages (from-to)442-453
Number of pages12
JournalSIAM Journal on Scientific Computing
Issue number2
StatePublished - Nov 2003
Externally publishedYes


  • Cubed-sphere
  • Semi-implicit
  • Shallow water equations
  • Spectral element

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics


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