Chebyshev Collocation Method for Shallow Water Models with Domain Decomposition

The spectral methods seek the numerical solutions by a set of known polynomials. The main advantage of using spectral methods for solving atmospheric problems is the high efficiency and conservations of important quadratic quantities such as kinetic energy and enstrophy. Namely, we can get very high accuracy through the exponential convergence. The conservation of the quadratic quantities are important to model the turbulence under strong rotation and stratification. In this paper, we introduce the domain decomposition method to speed up the Chebyshev collocation method. The domain decomposition is to divide the domain into many sub-domains to run the computation in parallel and to exchange the information through the sub-domain boundaries during the time integration. We implement the domain decomposition Chebyshev collocation method with overlapping the subdomains in one grid spacing interval for 1-D tests such as advection, diffusion and inviscid Burgers equations. We show the exponential convergence property and error characteristics in these tests. In a more realistic atmospheric modeling, we study the spectral method with 2-D shallow water equations. The domain decomposition results compared favorably with that of the single domain calculations. Thus, Chebyshev domain decomposition method may be an efficient alternative method for the atmospheric/oceanic limited area modeling. ∗Department of Atmospheric Sciences, National Taiwan University, No. 1, Sec. 4, Roosevelt Road, Taipei, 10617, Taiwan 0