tailieunhanh - Shallow Liquid Simulation Using Matlab (2001 Neumann) Episode 2

Tham khảo tài liệu 'shallow liquid simulation using matlab (2001 neumann) episode 2', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | represented as an additional row added to A consisting of 1 0 0 . 0 and a zero added to w. If we want to keep A square we can add the additional row to an existing row instead. For the results that follow this was the method used unless otherwise indicated. It turns out that the non-square additional row method is much slower and also less accurate. Note that these modifications are only needed when solving equation 25 with matrix methods. We do not need the modification when using the FFT method or in setting up equation 37 for time stepping. Comparing Solvers We have various methods of solving equation 15 such as FFT Gaussian Elimination LU decomposition and iterative solvers. To check that they work correctly we examine how well the resulting Ộ satisfies the equation 15 . We use second order central differences on to approximate V2 and compare this to w. Another consideration is that in solving equation 15 we expect Ộ to be symmetric given a symmetric w. An FFT problem When using the Fast Fourier Transform to solve equation 15 we make use of equation 32 . But when both kx and ky are zero the right hand side goes to infinity. To avoid this problem we add a small amount such as 10 4 to the zero entry in the matrix of k numbers. It turns out that if this value is too small on the order of 10-14 then the result is not usable. This can be established using the technique described in section . My guess is that the resulting component is so large that it swamps the accuracy of the other components. 4 Computational Results Results for various initial conditions The following figures show selected frames from the simulations with various initial conditions. Movies of these are viewable at . The spatial dimensions in each case are 10 X 10. 11 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 .