Hermite Spectral Methods with a Time-Dependent Scaling for Parabolic Equations in Unbounded Domains

Heping Ma*, Weiwei Sun, Tao Tang

*Corresponding author for this work

Research output: Contribution to journalJournal articlepeer-review

61 Citations (Scopus)
26 Downloads (Pure)


Hermite spectral methods are investigated for linear diffusion equations and nonlinear convection-diffusion equations in unbounded domains. When the solution domain is unbounded, the diffusion operator no longer has a compact resolvent, which makes the Hermite spectral methods unstable. To overcome this difficulty, a time-dependent scaling factor is employed in the Hermite expansions, which yields a positive bilinear form. As a consequence, stability and spectral convergence can be established for this approach. The present method plays a similar role in the stability of the similarity transformation technique proposed by Punaro and Kavian [Math. Comp., 57 (1991), pp. 597-619]. However, since coordinate transformations are not required, the present approach is more efficient and is easier to implement. In fact, with the time-dependent scaling the resulting discretization system is of the same form as that associated with the classical (straightforward but unstable) Hermite spectral method. Numerical experiments are carried out to support the theoretical stability and convergence results.

Original languageEnglish
Pages (from-to)58-75
Number of pages18
JournalSIAM Journal on Numerical Analysis
Issue number1
Publication statusPublished - 26 Apr 2005

Scopus Subject Areas

  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics

User-Defined Keywords

  • Convergence
  • Hermite spectral method
  • Stability
  • Time-dependent scaling


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