A leapfrog finite-difference scheme models P-waves in granite

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Oscillations, Waves & Acoustics · Wave motion, the wave equation and wave energy

Error, Chance and the Machine, Ch Partial differential equations: stability, convergence and solvers

Problem

A leapfrog finite-difference scheme models P-waves in granite, with wave speed c=6000 m s−1c = 6000\,\mathrm{m\,s^{-1}}, on a uniform grid with spacing Δx=25 m\Delta x = 25\,\mathrm{m}. ((a)) Find the largest time step Δt\Delta t allowed by the CFL condition. ((b)) The grid is then refined to Δx=12.5 m\Delta x = 12.5\,\mathrm{m} (spacing halved) to resolve shorter wavelengths. Find the new largest stable step, and state how it scales with Δx\Delta x compared with an explicit diffusion scheme.

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