Abstract
A semi-implicit spectral element shallow water model on the cubed-sphere is described and numerical results are compared with an explicit formulation and the traditional spectral transform method using the standard test cases proposed by Williamson et al. (1992). The explicit time step is limited by the phase speed of the fastest gravity waves. Semi-implicit time integration schemes remove this stability restriction but require the solution of an elliptic problem. A weak variational formulation of the governing equations leads to a symmetric Helmholtz operator. The resulting implicit problem for the geopotential is then solved using a block-Jacobi preconditioned conjugate gradient solver. The simulation rate of the semi-implicit model is accelerated relative to the explicit model for practical climate resolutions by a factor of two.
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Thomas, S.J., Loft, R.D. Semi-Implicit Spectral Element Atmospheric Model. Journal of Scientific Computing 17, 339–350 (2002). https://doi.org/10.1023/A:1015129420882
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DOI: https://doi.org/10.1023/A:1015129420882