Optimal stability for trapezoidal-backward difference split-steps

The marginal stability of the trapezoidal method makes it dangerous to use for highly non-linear oscillations. Damping is provided by backward differences. The split-step combination (t trapezoidal, (1 – )t for BDF2) retains second-order accuracy. The ‘magic choice’
allows the same Jacobian for both steps, when Newton's method solves these implicit difference equations. That choice is known to give the smallest error constant, and we prove that
also gives the largest region of linearized stability.

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Original source : http://imajna.oxfordjournals.org/cgi/content/short...