The result is x y 1 h2 t y 1x 1 2 k k i k i k j k k k

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Unformatted text preview: interpolate (x f (x y(0) )), j = i ; 2 i ; 1 i i + 1. If the points are equally spaced then j j j ^ ^ (1) (y(0) ) = h2 T(1) = 1 (;f 2 + f 1 + f ; f +1): 24 13 i i i; i; i i This formula would have to be modi ed near the boundaries. Problems 1. Consider a logarithmically graded mesh ln(1 + ) x = ln 2 i i = 0 1 : : : N: N i This grading is less severe than the quadratic grading of Example 8.3.1. Is this mesh quasi uniform? 8.4 Adaptive Mesh Selection We seek to solve the nonlinear BVP (8.2.1) to a speci ed degree of accuracy in an adaptive manner by (i) computing a preliminary solution on a coarse mesh, (ii) estimating the discretization error of this solution, and (iii) recursively re ning the mesh where the accuracy is not su cient. Thus, in the end, we would only use a ne mesh where the solution varied rapidly. A simple example serves to illustrate the principle. Example 8.4.1. Consider the linear two-point BVP y + y =0 00 0<x<1 0 y(0) = 1 y(1) = 1=2 which has the exact solution e ;e y(x) = 1 + (1 ; e 2) : 2 ; x ; ; Assuming that 0 < 1= 1, this is approximately y(x) 1 (1 + e 2 ; x ) As shown in Figure 8.4.1, the solution has a boundary layer near x = 0 and we might be able to improve both accuracy and e ciency by using a mesh that is concentrated in the boundary layer. Once again, we'll focus on the trapezoidal rule solution of (8.2.1) which is (y) = y ; y 1 ; f (x y ) + f (x 1 y 1) h 2 i i i; i i i; i 14 i; i = 1 2 ::: N (8.4.1a) 1 0.9 0.8 0.7 y 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 x 0.6 0.7 0.8 0.9 1 Figure 8.4.1: The solution of Example 8.4.1 with = 100. g (y0) = g (y ) = 0: L R (8.4.1b) N Using (8.3.2a) h2 = ; 12 f (x 1 2 y(x 1 2 )) + O(h4): i i 00 i; = i; = (8.4.1c) While concentrating on the trapezoidal rule, we'll consider a somewhat more abstract setting that will be useful when working with, e.g., high-order solutions via Richardson's extrapolation or deferred corrections. Thus, let e be an \error indicator" on the subinterval x 1 x ). Some possibilities for e are: i i; i i 1. An approximation of the local discretization error. For the trapezoidal rule this would be h2 e = 12 jf (x 1 2 y 1 2 )j : i 00 i i; = 15 i; = 1 2. An approximation of the global discretization error e = C h ky( )( )k s i s s i1 i where C is a known constant that depends upon the method and k k maximum norm restricted to x 1 x ). s is the i1 i; i 3. An ad hoc metric. One could use a scaled approximation of the gradient or curvature, e.g., e = h k y 1 2k i i i; = where is the central di erence operator (Table 6.3.3). The key ingredient is that the error indicator be small where the solution is well-resolved and large where additional resolution is necessary. Given a mesh of N + 1 points, we try to determine the coordinates fx0 < x1 < : : : < x g such that some global metric of solution quality is minimized. We'll illustrate a solution of this problem by determining the mesh such that N jej = 1max je j 1 i N i is minimized. Although still in a rather abstract setting, de Boor 3], Chapter XIV, found a rather simple solution to this extremal p...
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This document was uploaded on 03/16/2014 for the course CSCI 6820 at Rensselaer Polytechnic Institute.

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