7th World Congress on Computational Mechanics (Los Angeles, CA)

A Hybrid Meshless/Spectral -Element Method for Geophysical Fluid Dynamics on the Sphere

Christopher D. Blakely

Center for Scientific Computation and Mathematical Modeling at University of Maryland, College Park

Abstract: Two fundamental problems in numerically simulating Global Climate Models (GCMs) which have challenged researchers during the past few decades are the long-term stability requirements in the underlying numerical approximations of the climate model along with the sensitivity to small changes in regional scales of the model. Most GCMs in the past have had difficulty being able to simulate regional meteorological phenomena such as tropical storms, which play an important part in the latitudinal transfer of energy and momentum. This is partly due to the fact that climate models have traditionally employed spectral methods using spherical harmonics which are global and require excessively large resolution in order to inherit any properties which can be used to study regional scale phenomenon. This is a massive computational burden since the increase in resolution must be done globally due to the nature of the numerical method. Furthermore, coupling the climate model to physics forcing packages and to data provided by remote sensing requires local interpolation and thus the exponential convergence of the spectral method is lost. In this presentation, we introduce a hybrid numerical technique that couples spectral element approximation methods for global approximation with a meshless collocation method developed in [Blakely (2006)] for regional approximation for large-scale geophysical fluid problems on the sphere. The interest in constructing such a hybrid numerical method for large-scale problems is namely to capture the robust high-order approximation properties of spectral element methods with the versatile approximation properties of meshless collocation. We demonstrate that this hybrid method allows for high-order approximation at a global scale along with regional approximation on smaller scales using the meshless collocation technique without the need for a traditional computational mesh. An efficient parallel formulation of the hybrid method will be presented along with numerical examples using standard test problems of the shallow-water equations on the sphere from Williamson et al [Williamson et al. (1994)], validating the robustness and accuracy of the method.   
 

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