AM 255 : Problem Set 2

My implementation of the discrete difference approximation, as defined by Equation 1, was completed using Matlab and is included as prob1.m. Note that prob1.m accepts a single input argument kMode which is used to toggle λ = 1/2 or λ = h. Before presenting the results of my program, I will briefly outline the architecture of the source code. On lines 21-55 I select the values of {N, h, k} and determine the resulting grid points {x, t}. (Note that on lines 44-47 I ensure that the last time is given by T = 2π.) Lines 56-66 implement Equation 1. Note that I have implemented the central difference operator D0 as a stand-alone program D0.m. Finally, lines 67-104 create the tables and plots shown in this write-up. Recall from Equation 2.1.3 in [1] that the general solution to the first-order wave equation is given by u(x, t) = f(x+ t), where f(x) is the initial condition along t = 0. As a result, the solution at T = 2π is given by u(x, 2π) = sin(x + 2π) = sin(x). As specified in the problem statement, I have plotted the numerical approximation along with the analytic solution in Figure 1. (Note that the cases λ = {1/2, h} are shown in Figure 1(a) and 1(b), respectively. Also recall that λ = h ⇒ k = h from page 44 in [1].) The approximation results for both λ = 1/2 and λ = h are tabulated below.