Math 128A, Spring 2016 Problem Set 10 Question 1 Suppose y(t) is the exact solution of the initial value problem y (t) = f (t, y(t)), y(0) = y0 , and u(t) is any approximation to y(t) with u(0) = y(0). Dene the error e(t) = y(t) u(t). (a) Show that e(t) satises the initial value problem e (t) = f (t, u(t) + e(t)) u (t) e(0) = 0 (b) Suppose f (t, y) = y for some constant . Solve the initial value problem from (a) exactly to show that u(t) + e(t) = y(t). Question 2 Dene a family of explicit Runge-Kutta methods parametrized by order q, by applying up to p 1 passes of deferred correction to p steps of Euler's method. I.e. starting from un dene the uncorrected solution by 1 1 u1 n+j+1 = un+j + hf (tn+j , un+j ) for 0 j p 1. Let u(t) = U1 (t) be the degree-p polynomial that interpolates the p + 1 values u1 at the p points t = tn+j for 0 j p. Solve the error equation from question 1 n+j by Euler's method, yielding approximate errors e1 , e1 , . . . , e1 . Produce a second-order n+1 n+2 n+p accurate corrected solution u2 = u1 + e 1 n+j n+j n+j for 1 j p. Repeat the procedure to produce u2 , . . . , up . n+j n+j (a) Verify that p = 1 gives Euler's method. (b) For p = 2 express your method as a Runge-Kutta method. (c) For arbitrary p, verify that your method gives the correct Taylor expansion up to order p when you solve y = y. Question 3 (a) Write, test and debug a matlab function function yb = idec(a, b, ya, f, p, tau) % a,b: interval endpoints with a < b % ya: vector y(a) of initial conditions % f: function handle f(t, y) to integrate (y is a vector) % p: number of euler substeps / correction passes % tau: user-specified local error tolerance % yb: output approximation to the final solution vector y(b) which approximates the nal solution vector y(b) of the vector initial value problem y = f (t, y) y(a) = ya 1 Math 128A, Spring 2016 Problem Set 10 by the method you derived in problem 2, with u0 = ya . p1 Use variable step sizes determined by the order-p error estimate en+j , to keep local error below the user-specied tolerance . (b) Use idec.m with a selection of orders p and tolerances to approximate the nal solution vector u(T ) of the orbit computation from problem 1 of problem set 09. Tabulate the errors Ep = max |uj (T ) uj (0)|. 1j4 Measure the CPU time for each run and estimate the total CPU time necessary to obtain an orbit which is periodic to three-digit, six-digit and twelve-digit accuracy. Plot some inaccurate solutions and some accurate solutions and draw conclusions about values of the order p which give three, six or twelve digits of accuracy at minimal cost( i.e. CPU time). 2