Even the simplest advection-diffusion problems can exhibit multiple time scales. This means that robust variable step time integrators are a prerequisite if such problems are to be efficiently solved computationally. The performance of the second order trapezoid rule using an explicit Adams–Bashforth method for error control is assessed in this work. This combination is particularly well suited to long time integration of advection-dominated problems. Herein it is shown that a stabilized implementation of the trapezoid rule leads to a very effective integrator in other situations: specifically diffusion problems with rough initial data; and general advection-diffusion problems with different physical time scales governing the system evolution.

MSC codes

  1. 65M12
  2. 65M15
  3. 65M20


  1. time-stepping
  2. adaptivity
  3. convection-diffusion

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Information & Authors


Published In

cover image SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Pages: 2018 - 2054
ISSN (online): 1095-7197


Submitted: 11 April 2007
Accepted: 7 September 2007
Published online: 14 May 2008

MSC codes

  1. 65M12
  2. 65M15
  3. 65M20


  1. time-stepping
  2. adaptivity
  3. convection-diffusion



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