- Question 2 The Wave Equation Utt C Uxx Can Be Discretised As Follows U 2u U 24 1 2u 4 1 0 1 At 4x Whe 1 (97.4 KiB) Viewed 54 times
Question 2. The wave equation Utt = c²uxx can be discretised as follows: u+¹-2u; + u-¹ 24+1-2u² +4²1 = 0 (1) At² 4x² where Ax is the grid spacing and At the time step size. (a) Using a Taylor Series expansion, determine the order of accuracy of the approximation Uxx = +1-20+1 [Ans: Expand each term as a function of u in space, simplify. The leading order 4x² term is 2nd order in space.] (6 marks) (b) via a von Neumann stability analysis, show that the maximum stable time step size is At = Ax/c. [Ans: See notes] (6 marks) (c) Given the initial conditions: uf-2 = 1.0 = 1.0 uf = 0.3 = 0.0 uitl U₁+2 = 0.0 n+1 = and u?-¹ = 0.25, calculate u?+¹ using the above scheme given At = Ax/c. [Ans: u?+¹: 0.75] (4 marks) (d) Discuss the physical behaviour of the solution this equation where the initial conditions are now u(x,0) = sin and with boundary conditions u(0, t) = u(L, t) = 0.. Sketch the behaviour of the solution for three representative times. [Ans: The solution is an oscillation in space and time. In this case it is a mode 1 standing wave between 0≤x≤ 1. This will not dissipate in time. The sketch should show a half period sin wave with a positive amplitude (t=0), then with a negative amplitude after half a full period of oscillation, and then a flat dispacement but with a velocity.] (4 marks)