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Shallow Liquid Simulation Using Matlab (2001 Neumann) Episode 3

Tham khảo tài liệu 'shallow liquid simulation using matlab (2001 neumann) episode 3', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 50 100 150 200 250 300 50 100 150 200 250 300 350 50 100 150 200 250 300 50 100 150 200 250 300 350 50 100 150 200 250 300 50 100 150 200 250 300 350 50 100 150 200 250 300 50 100 150 200 250 300 350 Figure 9 Grid mesh pattern at t 0 4.2 8.4 12.6 Using LU Decomposition and V 0.01 At 0.30 n 16. Amplitude ranges from -4 blue thru 0 green to 4 red . because of the central first differences used in estimating the advection term -0 w in equation 14 . The central first difference in y is x y Ay - ệ x y- Ay 2 Ay 44 which depends only on neighboring points. The result is that the grid is that there are effectively two independent simulations. If the grid is colored like a chess board then the white squares are independent of the black squares. Figure 9 shows that the grid mesh is stable and only decays due to diffusion. If the diffusion is set low then the decay takes an extremely long time. The solution recommended in Numerical Recipes is to add some numerical diffusion. And with enough diffusion roughly V 0.01 the artifacts do not appear. 5 Summary and Conclusions We have used a variety of numerical solving techniques to model shallow fluid behavior. Two-dimensional partial differential equations with time behavior 21 are computationally challenging and the efficiency of the technique chosen is crucial. On the computer used for these simulations a mesh size of N 128 was about the maximum for most methods considered. The exception is the Fast Fourier Transform which performed very quickly and could be used for a much larger mesh than N 128. Accuracy is another consideration and here the Fast Fourier Transform did not hold up as well. The accuracy of the FFT in solving the elliptic equation V2 Ộ w was significantly worse than the matrix methods and this showed up in the simulation results. As a result LU Decomposition was the preferred technique having the best speed with very good accuracy. The iterative matrix solvers did not prove to be better than LU Decomposition. .