Advanced Methods for ODEs and DAEs: Assignment 8: Rosenbrock and DIRK methods

Exercise 1: (10 points) A Rosenbrock method is used to solve the Prothero-Robinson equation. Under which condition the order of Rosenbrock method does not decrease to order 1, if � ≪ 0 ? Deduce this condition (by analogy to the lecture). Exercise 2: (10 points) To show that DIRK methods can have at most stage orderq = 2. Exercise 3: (10 points) Consider the example Prothero and Robinson ú u = �(u − f(t)) + ú f(t), with f(t) = 10 − (10 + t)e t ,� ≪ 0. Solve this problem numerically with the help of DIRK-methods whose Butcher arrays are given by (a)