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# NUMERICAL ANALYSIS MATH 609

Texas A&M

GPA 3.6

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This 2 page Class Notes was uploaded by Vivien Bradtke V on Wednesday October 21, 2015. The Class Notes belongs to MATH 609 at Texas A&M University taught by Joseph Pasciak in Fall. Since its upload, it has received 14 views. For similar materials see /class/226031/math-609-texas-a-m-university in Mathematics (M) at Texas A&M University.

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Date Created: 10/21/15

Programming assignment3 Due Nov 21 Math 609 We consider applying explicit and implicit methods for solving a mildly stiff system of ODE s in this assignment Speci cally7 we consider the system Y Yt 01 a R where Y is the solution to n 12 Y iAY 0 01 t 5 Y0 Y0 E 111 Here A is the n x 71 matrix given by 2 ifz j A 71 iflz ijl17 0 otherwise Problem 1 Show that the set of vectors 117 7 1 C R with compo nents given by i ij gtgtjsmn1 241m are eigenvectors for A and compute the corresponding eigenvalues Write down the solution to 01 given the expansion of the initial data7 V L Y0 Zimm i1 You need not compute the coef cients above and they will appear in your expansion of the solution Note that every component of the solution decays exponentially as 25 increases Problem 2 Given a time step size k let Y denote the sequence obtained by applying the forward Euler method to 01 Give an expression for the coef cients of Y expanded in the basis of eigenvectors The forward Euler method is considered stable for this problem provided that every component of the solution decays Find a resonably sharp see the next problem77 condition on the time step size as a function of n which guarantees stability Problem 3 Code the forward Euler method applied to the ODE 01 You can use CRS to store the matrix or simply store the diagonals in three vectors The forward Euler method only requires matrix evaluation and is simple to code using either storage mode note that the generation of a full 71 x 71 matrix is unacceptable Apply your method to the cases of 1 2 n 16 32 64 128 First use time step sizes which are twice as large as your condition in the previous problem If your condition above is sharp enough you will see blowup of the ODE approximation Next use time step sizes which are smaller than you condition by a factor of two If your condition above is sharp enough you will now see convergence at t 1 Problem 4 Code the backward Euler method applied to the ODE 01 As the matrix A is tri diagonal the matrix systems appearing are also tri diagonal These can be solved using a tri diagonal solver as illustrated in the c code httpwwwmathtamuedu pasciakclasses609trisolc This code is simply Gaussian elimination in a storage mode where calculations involving the zeroes of the matrix are avoided As above the construction of the full 71 x 71 matrix is not allowed This method is stable for any time step size Run the method for n 163264128 and in each case try to nd a time step size where you observe two decimal places of accuracy in each component This you can do by comparing the results of different decreasing time step sizes while observing the resulting changes in the solution at t 1 You should be able to get the desired accuracy with fewer time steps than necessary for stability of the forward Euler method above

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