1. Chebyshev Spectral Method with Domain Decomposition
Author：Tsai, Yu-Ming Advisor：Prof. Kuo, Hung-Chi
Department of Atmospheric Sciences, National Taiwan University (NTU)
Spectral method seeks the solution to a differential equation in terms of a series of known functions. Chebyshev polynomials from the Sturm-Liouville equation, with the properties of completeness and orthogonality, are suited for the basis function of the spectral method. Chebyshev spectral method is very efficient and accurate with the “exponential convergence” property and the use of the Fast Fourier Transform (FFT). Figure 1 shows the first six Chebyshev polynomials.
Fig. 2: Mean square errors as a function of the number of degree of freedom N in the advection equation. FD1, FD2, and FD4 represent first order, second order and fourth order finite difference methods. COL is the Chebyshev collocation method. The convergent rate of COL is exponential and is much faster than finite difference methods.
Fig. 1: The first six Chebyshev polynomials in the [-1,1] physical domain.
Figure 2 shows the errors in the linear advection equation test. The error in the Chebyshev collocation method decreases exponentially with the increase of N (exponential convergence). Since the convergent rate of Chebyshev polynomials depends only on the smoothness of the function to be represented and not on the boundary conditions, our idea is to develop the Chebyshev spectral method with DD (Domain Decomposition). The DD divides the domain into several pieces of subdomain with each piece can be served with one processor. Ideally, the speed up factor will be M if M pieces of subdomain is used.
The collocation points in the Chebyshev transform are not uniform in the domain with 1/ spacing near the boundary. Thus, the minimum grid spacing in collocation points can increase with DD . The same test problem in Fig. 3 with the Chebyshev spectral method is now studied with the DD. Figure 4 gives the result of DD COL with FD6 and no DD COL for comparison. The absolute accuracy in DD COL is slightly decreased as compare to the no DD COL result, but the exponential convergence property retained with the DD COL. It is interesting to note that the speed up factor from this simple 2-piece DD is equal to 4. This is due to the fact that the grid spacing increases about twice with 2-piece DD as shown in Fig. 3.
In summary, we have demonstrate that the Chebyshev spectral method with DD is very efficient. The speed up factor can be larger than the number of subdomain used due to the increase in the time integration.
Fig. 4: Mean square errors as a function of the number of degree of freedom N in the advection equation with DD. The blue line shows after DD, the accuracy slightly decreases, but it is still much higher than FD6. The speed up factor in this 2-piece DD is 4 due to a larger time step employed.
Name：Tsai, Yu-Ming
Advisor: Prof. Kuo, Hung-Chi
Department of Atmospheric Sciences, National Taiwan University (NTU)
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Fig. 3: The domain is -1 ~ 1. Original domain (upper) and divided into two domains (lower) are shown. Max means maximum grid space (dx), and Min means minimum dx. After divided, the minimum dx increases about twice.