We have introduced the Chebyshev collocation method with domain decomposition in the atmospheric modelings. The sub-domain boundary information exchange is by overlapping the sub-domains in one grid spacing interval. By the property of Cheby-shev grids setting and consider the relation between Δ𝑥 and Δ𝑡 with CFL condition, we can enlarge the Δ𝑡 with Chebyshev domain decomposition compared with single domain.
Our domain decomposition Chebyshev collocation method indicates the exponential con-vergence property in 1-D linear advection and diffusion models. In the test of inviscid Burgers equation, we integrate the model up to the shock formation time. We show that the domain decomposition spectral method in general yields a smaller errors when com-pared to the single domain calculations. In a more realistic atmospheric modeling with a 2-D shallow water model, we find our domain decomposition Chebyshev method gives results identical to the single domain spectral method with a 𝐿2 error on the order of 10−4 when 96 degree of freedom is considered. The domain decomposition spectral method is capable of a stable integration of 9 days in our test. It is prominent, considering the fact that the predictability of the typical atmospheric model is about 10 to 12 days. We also argued that the time-splitting method is not well applicable to the 2-D shallow water equation.
Our future work will be evaluate the overhead or the additional cost of the boundary information exchange in domain decomposition. We also may implement the method in the oceanic modeling by incorporating the immerse boundary condition method in the lateral continental shelf and using the Chebyshev domain decomposition method.
Appendix
A Verticle Transform and Shallow Water Equation
In this appendix we will show that hydorstatic atmosphere is equivalent to a set of shallow water equation by the vertical transform.
𝑠 = 𝑠(𝑝) ≡ 𝑐𝑝𝜃0
Linearized equation in 𝑠 coordinate (with 𝐽 represents diabatic heating).
∂−→𝑣
∂ ˜Φ∂𝑡 is the local height change that can be resulted from diabatic heating.
Boundary condition derivation.
𝐽 = ∂
∂𝑠(∂ ˜Φ
∂𝑡) (A.8)
Substitute A.6 and A.8 into A.7 we could obtain:
˙𝑠 = − 1
where
Based on the Sturm-Liouville theorem, the basis function is complete, orthogonal, and the eigenvalue is real.
ℒΨ𝑛= 𝜆𝑛Ψ𝑛 (A.18)
𝑛 > 0, we could obtain the shallow water equation.
∂−→𝑣𝑛
∂𝑡 + 𝑓 ˆ𝑘 × −→𝑣𝑛 = −▽Φ𝑛 (A.21)
∂Φ
∂𝑡 + 𝑐2𝑛▽ ⋅ −→𝑣𝑛 = ∂ ˜Φ
∂𝑡 (A.22)
The above derivation demonstrates that a hydrostatic atmosphere with suitable ver-tical boundary conditions may support free oscillations with several different structurs.
The eigenvalue of each free oscillation is 𝑐2𝑛 = 𝑔ℎ, which is related to the depth of the shallow water equation.
B Amdahl’s Law
In this appendix, we will discuss the Amdahl’s law in parallel computing. Let 𝑊 be the amount of work to be done for a particular job, and let 𝑟 be the rate at which it can be done by one processor. Then the computer time required for one processor to do the job is 𝑇1, given by
𝑇1 = 𝑊
𝑟 (B.1)
Now suppose that 𝑓 fraction of the job, by time, must be done serially and the remaining 1 − 𝑓 fraction can be done perfectly parallelized by 𝑝 processors. Then the time, 𝑇𝑝, for parallel computation is given by
𝑇𝑝 = 𝑓𝑊
𝑟 +(1 − 𝑓)𝑊
𝑝𝑟 (B.2)
Figure B.1: Amdahl speedup as a function of 𝑓.
The above equation indicates that if the entire calculation can be parallelized, that is, 𝑓 = 0, then all the work will be done in 𝑝 fraction of the time. We then claim the speedup SU is p, and
𝑆𝑈 = 𝑇1
𝑇𝑝 = 𝑝 (B.3)
This is the well known linear speedup. But as the equation indicate, the speedup in general will be
𝑆𝑈 = 𝑇1
𝑇𝑝 = 𝑊/𝑟
(𝑊/𝑟)(𝑓 +(1−𝑓)𝑝 )
= 𝑝
𝑓(𝑝 − 1) + 1.
(B.4)
This relation is known in the field of Parallel Computing as the Amdahl’s Law. We are interested from the above equation the speed up 𝑆𝑈 as a function of numbers of processors 𝑝. In particular, we want to know how the 𝑆𝑈 behave as a function of 𝑝 and 𝑓. Figure B.1 shows the Amdahls speed up as a function of 𝑓 for various 𝑝. It is obvious that the steepness near 𝑓 = 0 means that the speedup falls off rapidly for the increase of 𝑓. For example, the 𝑆𝑈 does not change much with 𝑝 processors for 𝑓 = 0.2. Namely, the 𝑆𝑈 becomes insignificant when percentage of code that cannot be parallelized is about 20%. It may appear that the Amdahl’s Law gives a bleak picture as far as the speedup is concerned. However, the fraction 𝑓 is defined by computational time and not by computational code. As a matter of fact, most scientific programs spend the majority of their execution time in a few loops within the program. Thus if these loops parallelize (or vectorize), then Amdahl’s Law predicts that the efficiency will be high. On the other hand, if we employed the domain decomposition method, the theoretical 𝑆𝑈 will be almost proportional to the number of processors 𝑝 with the overhead of information exchange through the decomposed boundaries. The 𝑆𝑈 in the domain decomposition in general is not a function of the 𝑓. More details are on [18].
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