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Two Point Boundary Value Problems part 6

In relaxation problems, you have to choose values for the independent variable at the mesh points. This is called allocating the grid or mesh. The usual procedure is to pick a plausible set of values and, if it works, to be content. If it doesn’t work | 17.5AutomatedAllocation ofMesh Points 783 17.5AutomatedAllocation ofMesh Points In relaxation problems you have to choose values for the independent variable at the mesh points. This is called allocating the grid or mesh. The usual procedure is to pick a plausible set of values and if it works to be content. If it doesn t work increasing the number of points usually cures the problem. If we know ahead of time where our solutions will be rapidly varying we can put more grid points there and less elsewhere. Alternatively we can solve the problem first on a uniform mesh and then examine the solution to see where we should add more points. We then repeat the solution with the improved grid. The object of the exercise is to allocate points in such a way as to represent the solution accurately. It is also possible to automate the allocation of mesh points so that it is done dynamically during the relaxation process. This powerful technique not only improves the accuracy of the relaxation method but also as we will see in the next section allows internal singularities to be handled in quite a neat way. Here we learn how to accomplish the automatic allocation. We want to focus attention on the independent variable x and consider two alternative reparametrizations of it. The first we term q this is just the coordinate corresponding to the mesh points themselves so that q 1 at k 1 q 2 at k 2 and so on. Between any two mesh points we have Aq 1. In the change of independent variable in the ODEs from x to q becomes g dx g dy dx dq g dq In terms of q equation 17.5.2 as an FDE might be written yk - yk-i - 2 0 17.5.1 17.5.2 17.5.3 or some related version. Note that dx dq should accompany g. The transformation between x and q depends only on the Jacobian dx dq. Its reciprocal dq dx is proportional to the density of mesh points. Now given the function y x or its approximation at the current stage of relaxation we are supposed to have some idea of how we want to specify the density .