Lecture 25: 8 November 2010Lecture 25: 8 November 2010 Discretization of the (1d scalar) wave equation, simplifying for now to an infinite domain (no boundaries) and constant coefficients (c=1). This corresponds to the equations ∂u/∂t=∂v/∂x and ∂v/∂t=∂u/∂x. The obvious strategy is to make everything a center difference. First concentrating on the spatial discretization, showed that this means that u and v should be discretized on different grids: for integers m, we should discretize u(mΔx)≈um and v([m+0.5]Δx)≈vm+0.5. That is, the u and v spatial grids are offset, or staggered, by Δx/2. For discretizing in time, one strategy is to discretize u and v at the same timesteps nΔt. Center-differencing then leads to a Crank-Nicolson scheme, which you analyze in homework and show to be unconditionally stable (albeit implicit) for anti-Hermitian spatial discretizations. Alternatively, we can ues an explicit leap-frog scheme in which u is discretized at times nΔt and v is discretized at times [n-0.5]Δt. Discretization of the (1d scalar) wave equation: staggered grids and leap-frog schemes. Von Neumann and CFL analysis. Dispersion relation. Further reading: Strang book, section 6.4 on the leapfrog scheme for the wave equation.MIT OpenCourseWarehttp://ocw.mit.edu 18.303 Linear Partial Differential Equations: Analysis and NumericsFall 2010 For information about citing these materials or our Terms of Use, visit:
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