A New Parallel Domain-Decomposed Chebyshev Collocation Method for Atmospheric and Oceanic Modeling

Abstract

Spectral methods seek the solution to a differential equation in terms of series of known smooth function. The Chebyshev series possesses the exponential-convergence property regardless of the imposed boundary condition, and therefore is suited for the regional modeling. We propose a new domain-decomposed Chebyshev collocation method which facilitates an efficient parallel implementation. The boundary conditions for the individual sub-domains are exchanged through one grid interval overlapping. This approach is validated using the one dimensional advection equation and the inviscid Burgers' equation. We further tested the vortex formation and propagation problems using two-dimensional nonlinear shallow water equations. The domain decomposition approach in general gave more accurate solutions compared to that of the single domain calculation. Moreover, our approach retains the exponential error convergence and conservation of mass and the quadratic quantities such as kinetic energy and enstrophy. The efficiency of our method is greater than one and increases with the number of processors, with the optimal speed up of 29 and efficiency 3.7 in 8 processors. Efficiency greater than one was obtained due to the reduction the degrees of freedom in each sub-domain that reduces the spectral operational count and also due to a larger time step allowed in the sub-domain method. The communication overhead begins to dominate when the number of processors further increases, but the method still results in an efficiency of 0.9 in 16 processors. As a result, the parallel domain-decomposition Chebyshev method may serve as an efficient alternative for atmospheric and oceanic modeling.