$$ \newcommand{\uex}{{u_{\small\mbox{e}}}} \newcommand{\half}{\frac{1}{2}} \newcommand{\tp}{\thinspace .} \newcommand{\Oof}[1]{\mathcal{O}(#1)} $$

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Table of contents

A simple vibration problem
      A centered finite difference scheme; step 1 and 2
      A centered finite difference scheme; step 3
      A centered finite difference scheme; step 4
      Computing the first step
      The computational algorithm
      Operator notation; ODE
      Operator notation; initial condition
      Computing \( u^{\prime} \)
Implementation
      Core algorithm
      Plotting
      Main program
      User interface: command line
      Running the program
Verification
      First steps for testing and debugging
      Checking convergence rates
      Implementational details
      Nose test
Long time simulations
      Effect of the time step on long simulations
      Using a moving plot window
Analysis of the numerical scheme
      Movie of the phase error
      We can derive an exact solution of the discrete equations
      Calculations of an exact solution of the discrete equations
      Solving for the numerical frequency
      Polynomial approximation of the phase error
      Plot of the phase error
      Exact discrete solution
      Convergence of the numerical scheme
      Stability
      The stability criterion
      Summary of the analysis
Alternative schemes based on 1st-order equations
      Rewriting 2nd-order ODE as system of two 1st-order ODEs
      The Forward Euler scheme
      The Backward Euler scheme
      The Crank-Nicolson scheme
      Comparison of schemes via Odespy
      Forward and Backward Euler and Crank-Nicolson
      Phase plane plot of the numerical solutions
      Plain solution curves
      Observations from the figures
      Runge-Kutta methods of order 2 and 4; short time series
      Runge-Kutta methods of order 2 and 4; longer time series
      Crank-Nicolson; longer time series
      Observations of RK and CN methods
      Energy conservation property
      Derivation of the energy conservation property
      Remark about \( E(t) \)
      The Euler-Cromer method; idea
      The Euler-Cromer method; complete formulas
      Euler-Cromer is equivalent to the scheme for \( u^{\prime\prime}+\omega^2u=0 \)
      The schemes are not equivalent wrt the initial conditions
Generalization: damping, nonlinear spring, and external excitation
      A centered scheme for linear damping
      Initial conditions
      Linearization via a geometric mean approximation
      A centered scheme for quadratic damping
      Initial condition for quadratic damping
      Algorithm
      Implementation
      Verification
      Demo program
      Euler-Cromer formulation
      Staggered grid
      Linear damping
      Quadratic damping
      Initial conditions

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