GS534: Solving the 1D Poisson equation using finite differences ...

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The boundary conditions require special care. For x = 0 we hav e a Dirichlet boundary condition which allows us to fix the value u 1 = 0. For x = 1 we hav e a Neumann boundary condition du/dx = 0. This is a symmetry boundary condition, so that in this case we can imagine a ’ghost’ point u N+1 which is always equal to u N−1 . This leads to the expression for point x N : u N = u N−1 + 1 2 (∆x)2 (10) 3. Iterative methods 3.1. Jacobi and Gauss-Seidel A simple approach to solve the differential equation is to start with a ’guess’ and to use the stencil (9) on the internal points and taking care of the boundary conditions to iteratively improve on the estimated solution. The algorithm is as follows: initialize a vector u until converged copy u to backup vector u old update u by applying (9) to the internal points using u old in right hand side update boundary condition values We can modify this ’strict’ Jacobi by not using a duplicate vector u old but rather working on u. This can save memory and has the advantage that that you use updated values whenever possible. Note that it may still be necessary to keep a duplicate vector around for checking on the convergence. A second modification is reached by recognizing that the stencil only uses nearest neighbors. In order to optimally update the values in the center points it would make sense to first do all even points, followed by all odd points. During this last step you would use the updated values. This is called Red-Black Gauss-Seidel, where the ’Red-Black’ term comes from imagining that each odd point is red and each even point is black. This concept is easily extended to 2D and 3D. The algorithm is then modified to: initialize a vector u until converged update u by applying (9) to the odd internal points update u by applying (9) to the even internal points update boundary condition values where you need to make sure that the boundary conditions are treated at the appropriate step in the algorithm. 3.2. Iterative methods: successive overrelaxation The methods above are stable and easy to implement but converge very slowly except for very small problems (N small). In order to speed up the convergence we can use overrelaxation in which you make an overcorrection at each step in the Gauss-Seidel iteration. In our case this reads: u i ′= 1 2 [u i−1 + u i+1 − (∆x) 2 ] (11a) 2
u i : =ωu i ′+(1 −ω)u i (11b) where I’ve used : = to indicate that the left hand side becomes the value of the right hand side (rather than mathematical equality). ω is the relaxation parameter. When 1 < ω < 2 you use overrelaxation but you can also choose 0 < ω < 1 to use underrelaxation (which is sometimes handy when you have strongly nonlinear problems). It can be shown that this algorithm only converges for 0 < ω < 2 and that you can try to get an optimal choice for ω. See Chapter 19.5 of Press et al. (1992) for more details. An evaluation of these algorithms for our equation (1) shows the increase in convergence speed for SOR compared to Gauss-Seidel. The number of iterations are shown for convergence of the rms error to less than 10 −6 compared to the analytical solution. All cases were started with an initial estimate u = 1. N Number of iterations GS ω=1.5 1.9 2.0 11 533 207 62 10 21 2136 776 190 20 41 8552 2983 621 40 Note that we’re lucky that we find convergence (and pretty good convergence at that) for ω=2! This is pretty much guaranteed not to work for any real applications. 3.3. Iterative methods: multigrid The slow convergence of Jacobi can be understood if you consider how information is transferred between nodal points by the stencil. It is clear from the algorithm that it takes the stencil N steps to go from one end of the domain to the other. During this first iteration each nodal point makes itself known to its neighbors, but it takes N iterations to let the point at one end of the domain know about the existence of the point at the other end of the domain. Updates to the values at each point require further iteration and it may be intuitive (and can be formally shown) that it takes N × N iterations to fully converge. The SOR approach allows a speed up but it is logical that it can not be faster than N iterations. This means that the computational cost of these iterative methods is O(N 2 )toO(N 3 ) flops (=floating point operations), since each iteration takes O(N) flops. The main limiting factor is the width of the stencil. The three point stencil is efficient at transfering short wav elength information, but very inefficient for wav elengths that are longer than a few grid points. It would therefore make sense to try to transfer long wav elength information with a stencil that has greater ’reach’. This can be accomplished by subsampling the grid by taking ev ery other gridpoint which halfs the number of iterations that the stencil needs to go from one end of the domain to the other. At this once coarser grid you don’t do the shortest wav elength very accurately, but you could improve on that by going back to the fine grid once you’ve finished on the coarser grid. Of course, you could see the iteration on the coarse grid in a similar light, where it is beneficial to subsample the grid points to improve the convergence of even longer wavelengths. This is the main philosophy behind multigrid, where you use multiple layers of subsampled grids (for an example see Figure 1). The main trick in multigrid is to compute corrections to the solution at fine grids using coarser grids. Let’s write the equation (1a) in more general form to develop an appreciation for the method: 3
Page 1: 1. Example problem GS534: Solving t
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Page 7 and 8: 2 -1 -1 2 -1 -1 2 0 0 2 -1 -1 1 4 -
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GS534: Solving the 1D Poisson equation using finite differences ...

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