運用在分解定義域之淺水方程模型的切比雪夫排列法

Abstract

本篇論文介紹使切比雪夫排列法加速的分解定義域方法。我們將定義域分割成許多子定義域，使其可以進行平行運算，並且在完成每一時步的積分後在子定義域之間的邊界進行資訊的交換。我們先使用此方法運作在三個一維的方程做為例子，分別是傳導方程、擴散方程以及inviscid Burgers方程，並且提供了指數收斂的計算結果。另外再延伸至更實際的二維淺水方程做為例子，並且獲得與未分割定義域之前十分相近的計算結果。因此我們也得到結論，對於大氣或海洋的模式，分解定義域的切比雪夫排列法是一個十分有效率的數值方法。

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 sub-domains 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.