hw2_228af10 - problem gives an O ( h 2 ) LTE provided the...

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Math 228A Homework 2 Due Friday, 10/22/08, 4:00 1. Use the standard 3-point discretization of the Laplacian on a regular mesh to Fnd a numerical solution to the PDEs below. Perform a reFnement study using the exact solution to compute the error that shows the rate of convergence for both the 1-norm and the max norm. (a) u xx = exp( x ) , u (0) = 0 , u (1) = 1 (b) u xx = 2 cos 2 ( πx ) , u x (0) = 0 , u x (1) = 1 2. Propose a discretization scheme for u xx = f, u x (0) - αu (0) = g, u (1) = b. What is the form of the matrix and right hand side in your discrete equations? What order of accuracy do you expect? 3. As a general rule, we usually think that an O ( h p ) local truncation error (LTE) leads to an O ( h p ) error. However, in some cases the LTE can be lower order at some points without low- ering the order of the error. Consider the standard second-order discretization of the Poisson equation on [0 , 1] with homogeneous boundary conditions. The standard discretization of this
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Unformatted text preview: problem gives an O ( h 2 ) LTE provided the the solution is at least C 4 . The LTE may be lower order because the solution is not C 4 or because we use a lower order discretization at some points. (a) Suppose that the LTE is O ( h p ) at the Frst grid point ( x 1 = h ). What eect does this have on the error? What is the smallest value of p that gives a second order accurate error? Hint: Use equation (2.46) from LeVeque to aid in your argument. (b) Suppose that the LTE is O ( h p ) at an interior point (i.e. a point that does not limit to the boundary as h 0). What eect does this have on the error? What is the smallest value of p that gives a second order accurate error? (c) Verify the results of your analysis from parts (a) and (b) using numerical tests. 1...
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