18.303 Problem Set 9 Due Wednesday, 17 November 2010. Problem 1: Discretizing a 2d wave equation Consider the scalar wave equation with constant coefficients, which can be written as in the notes as two coupled equations ∂u = � · v, ∂∂t v = �u. Now, let us consider this in two dimensions (for an infinite spatial domain Ω = R2 ,∂t don’t worry about boundary conditions). n(a) Suppose that we discretize u(x, y, t) as umx,my ≈ u(mxΔx, my Δy, nΔt) for integers mx, my, n. Explain how (i.e. where/when) we should discretize vx(x, y, t) and vy(x, y, t) so that our equations ∂u = �u can∂t = � · v and ∂∂t v all be implemented as explicit center differences, similar to the staggered-grid leap-frog scheme in class for the 1d case. Give the discretized equations. (Hint: vx and vy don’t have to be discretized on the same spatial grid.) u 2(b) Combine your discretized equations back into a discretization of ∂∂t22 = � u, as in class, by writing: n+1 n n n−1 ∂2n+1 n n−1 mx,my −u mx,my mx,my −u mx,myuumx,my − 2umx,my + umx,my u u ∂t2 ≈ Δt2 = Δt Δ− t Δt = · · · , u 2where the right-hand side is some spatial derivatives (finite differences) of u. (Do not re-discretize ∂∂t22 = � u from the beginning; make sure you plug in your discretizations from the previous part.) (c) Using the previous part (which helpfully is only in terms of u), perform a Von Neumann analysis to relate Δt to Δx and Δy: plug in umn x,my = ei(kxmxΔx+ky my Δy−ωnΔt) and solve for ω (kx, ky ), and find out under what conditions (on Δt) ω is real for all possible values of kx and ky. Problem 2: Bouncing waves u 2 ∂2 uIn class, I gave an animwave.m file (available on the web site) that animated the 1d wave equation ∂∂t22 = c∂x2 with periodic boundary conditions. For example, to animate a Gaussian-shaped pulse propagating to the right, you can use the command: animwave(@(x) exp(-(x-0.5).^2/0.1^2), 100, 1, 5, 0.9); (a) Modify the animwave.m file implement Dirichlet boundary conditions u(0) = u(L) = 0 instead of periodic boundary conditions. Hint: define um for m = 1 . . . M , and vm+0.5 for m = 0 . . . M (that is, v will have one more point than u), with u0 = uM+1 = 0. You shouldn’t need to explicitly implement any boundary condition for v. Run it for the gaussian pulse as above, and show the graph at a few timesteps to illustrate what happens when your pulse hits a boundary. u 2 ∂2 u(b) In an infinite 1d problem (Ω = R) for the wave equation ∂∂t22 = c∂x2 with constant c, there is a solution u(x, t) = f(x − ct) for a right-going wave. Suppose instead that the domain is x < 0 (i.e. semi-infinite), with Dirichlet boundary conditions u(0) = 0. Construct a solution u(x, t) using f(x) and f (−x) that satisfies both the wave equation and the boundary condition, and relate it to what you observed in the previous part. 1MIT OpenCourseWare http://ocw.mit.edu 18.303 Linear Partial Differential Equations: Analysis and Numerics Fall 2010 For information about citing these materials or our Terms of Use, visit:
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