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IMA Journal of Numerical Analysis Advance Access published online on September 7, 2009
IMA Journal of Numerical Analysis, doi:10.1093/imanum/drp022
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© The author 2009. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Optimal stability for trapezoidal–backward difference split-steps

Sohan Dharmaraja, Yinghui Wang and Gilbert Strang{dagger}

Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

{dagger} Corresponding author. Email: gs{at}math.mit.edu

Received on 13 June 2008. Revised on 7 May 2009.


  Abstract

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 ({alpha}{Delta}t trapezoidal, (1 – {alpha}){Delta}t for BDF2) retains second-order accuracy. The ‘magic choice’ Formula 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 Formula also gives the largest region of linearized stability.

Key Words: backward differences; split-step; stability; trapezoidal

Dedicated to Ron Mitchell for a lifetime of leadership by example


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