Time-dependent Hermite–Galerkin spectral method and its applications

Xue Luo Beihang University Shing-Tung Yau Harvard University Stephen S.-T. Yau Tsinghua University

Numerical Analysis and Scientific Computing mathscidoc:1611.25004

Applied Mathematics and Computation, 264, 378-391, 2015
A time-dependent Hermite-Galerkin spectral method (THGSM) is investigated in this paper for the nonlinear convection-diffusion equations in the unbounded domains. The time-dependent scaling factor and translating factor are introduced in the definition of the generalized Hermite functions (GHF). As a consequence, the THGSM based on these GHF has many advantages, not only in theoretical proofs, but also in numerical implementations. The stability and spectral convergence of our proposed method have been established in this paper. The Korteweg-de Vries-Burgers (KdVB) equation and its special cases, including the heat equation and the Burgers' equation, as the examples, have been numerically solved by our method. The numerical results are presented, and it surpasses the existing methods in accuracy. Our theoretical proof of the spectral convergence has been supported by the numerical results.
Hermite–Galerkin spectral method, Time-dependent parameters, Nonlinear convection–diffusion equations
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  title={Time-dependent Hermite–Galerkin spectral method and its applications},
  author={Xue Luo, Shing-Tung Yau, and Stephen S.-T. Yau},
  booktitle={Applied Mathematics and Computation},
Xue Luo, Shing-Tung Yau, and Stephen S.-T. Yau. Time-dependent Hermite–Galerkin spectral method and its applications. 2015. Vol. 264. In Applied Mathematics and Computation. pp.378-391. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20161126011924346744653.
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