New Multi-implicit Space–Time Spectral Element Methods for Advection–Diffusion–Reaction Problems

  • Chaoxu Pei
  • Mark Sussman
  • M. Yousuff Hussaini


Novel multi-implicit space–time spectral element methods are described for approximating solutions to advection–diffusion–reaction problems characterized by multiple time scales. The new methods are spectrally accurate in space and time and they are designed to be easy to implement and robust. In other words, given an existing stable low order operator split method for approximating solutions to PDEs exhibiting multiple scales, the algorithms described in this article enable one to easily extend a low order method to be a robust space–time spectrally accurate method. In space, two spectrally accurate advective flux reconstructions are proposed: extended element-wise flux reconstruction and non-extended element-wise flux reconstruction. In time, for the hyperbolic term(s), a low-order explicit I-stable building block time integration scheme is introduced in order to obtain a stable and efficient building block for the spectrally accurate space–time scheme. In this article, multiple spectrally accurate space discretization strategies, and multiple spectrally accurate time discretization strategies are compared to one another. It is found that all methods described are spectrally accurate with each method having distinguishing properties.


Space–time Operator splitting Coupling strategy Multiple time scales Spectral accuracy 

Mathematics Subject Classification

65B05 65M70 



This work and the authors were supported in part by the National Science Foundation under Contract DMS 1418983.


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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Chaoxu Pei
    • 1
  • Mark Sussman
    • 1
  • M. Yousuff Hussaini
    • 1
  1. 1.Department of MathematicsFlorida State UniversityTallahasseeUSA

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