SEMT20001-R: Principles of Computational Modelling
COURSEWORK REASSESSMENT: QUESTION 3
Question 3: Discrete vs continuous
(a) Consider the map
xk+1 = r sin(πxk) , k = 1, 2, . . . , (1)
where r is a parameter from the interval r ∈ [0, 1].
(i) For parameter value r = 0.5 find all fixed points, for parameter value r = 0.8 find all period-two orbits, and for parameter value r = 0.96 find all period-three orbits of map (1) within the interval [-1, 1]. Use a combination of interval bisection and interval Newton’s method. (3 marks)
(ii) Using indicator functions, calculate the transition matrix K for map (1) and for parameter values r = 0.5, r = 0.8 and r = 0.96. Find the stationary distribution for the three parameter values. Are there more than one ways to calculate stationary distributions from a given transition matrix? (3 marks)
(iii) Explain what would happen if instead of indicator functions, one used Chebyshev collocation to approximate the transition matrix and then calculate the stationary distribution. (2 marks)
(b) Consider the function
f(t) = exp(2cos(2t)) , t ∈ [-π,π]. (2)
(i) Approximate function f, using orthogonal collocation with a library of functions ϕn,j(t), con- structed by appropriately scaling and shifting the order-n Dirichlet kernel
For each n there are 2n + 1 collocation points, defined by
The library of functions is
Illustrate the convergence of your approximation as a function of n on a diagram. (2 marks)
(ii) Now use the constant, sine, and cosine functions as your library functions to carry out (non- orthogonal) collocation on the grid given by equation (3). (2 marks)
(iii) The truncated Fourier series of order n of a function f can be written as
Show that the collocation scheme using the Dirchlet kernel Dn is orthogonal and that it is equivalent to the non-orthogonal scheme using constant, sine, and cosine functions. (3 marks)
(iv) Now use Chebyshev approximation to approximate function f and illustrate the covergence in a diagram. (2 marks)
(c) Consider the van der Pol oscillator
and its linearised version
(5)
about the equilibrium at x = 0.
(i) A solutions of equation (5) is
Set the boundary conditions to x(-1) = f(-1) and x(1) = f(1) and compute a numerical so- lution of equation (5) using Chebyshev approximation with the optimal Chebyshev polynomial coefficients. (2 marks)
(ii) Now use the grid
to compute a numerical solution of equation (5) and compare the accuracy of the two results. (3 marks)
(iii) Rewrite the nonlinear equation (4) into first order form and compute its numerical solution as an initial value problem on the interval [-1, 1] using Chebyshev collocation. Use initial conditions x(-1) = 1, x.(-1) = 0. Illustrate and explain the convergence properties of the numerical solution as the number of grid points increases. Use the numerical solution with n = 256 grid points as the reference. (3 marks)