Journal of Scientific Computing

, Volume 71, Issue 3, pp 959–993 | Cite as

An Asymptotic Preserving Maxwell Solver Resulting in the Darwin Limit of Electrodynamics

  • Yingda Cheng
  • Andrew J. Christlieb
  • Wei Guo
  • Benjamin Ong


In plasma simulations, where the speed of light divided by a characteristic length is at a much higher frequency than other relevant parameters in the underlying system, such as the plasma frequency, implicit methods begin to play an important role in generating efficient solutions in these multi-scale problems. Under conditions of scale separation, one can rescale Maxwell’s equations in such a way as to give a magneto static limit known as the Darwin approximation of electromagnetics. In this work, we present a new approach to solve Maxwell’s equations based on a Method of Lines Transpose (\(\hbox {MOL}^T\)) formulation, combined with a fast summation method with computational complexity \(O(N\log {N})\), where N is the number of grid points (particles). Under appropriate scaling, we show that the proposed schemes result in asymptotic preserving methods that can recover the Darwin limit of electrodynamics.


Implicit method Maxwell’s equations Darwin approximation Method of Lines Transpose Fast summation method Asymptotic preserving method 


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© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of MathematicsMichigan State UniversityEast LansingUSA
  2. 2.Department of Computational Mathematics, Science and EngineeringMichigan State UniversityEast LansingUSA
  3. 3.Department of Mathematical SciencesMichigan Technological UniversityHoughtonUSA

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