Time-dependent Hermite–Galerkin spectral method and its applications

Abstract

Abstract. A time-dependent Hermite-Galerkin spectral method (THGSM) is investigated in this paper for the nonlinear convection-diffusion equations in the unbounded domains. The timedependent 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 theorethical 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.

Spectral viscosity method with generalized Hermite functions for nonlinear conservation laws

“…The author and her co-worker provide a practical guideline to choose the suitable scaling factor [14] for Gaussian and super-Gaussian functions. The time-dependent scaling factor of the Hermite spectral method in solving evolution equations has also been investigated in [15]. However, all the guidelines can not be applied in our case, due to the discontinuity.…”

New unconditionally stable scheme for solving the convection–diffusion equation based on the Associated Hermite orthogonal functions

“…Funaro [21] first considered the use of Hermite spectral method (HSM) to approximate the numerical solutions of diffusion evolution equation in unbounded domains, Guo [22,23] developed the HSM and the Hermite pseudospectral methods. To overcome the drawback of unstable for HSM, Ma [24,25] and Luo [26] introduced a timedependent scaling factor in the Hermite expansions to obtain a conditionally stable scheme. However, the AH basis functions have not been combined with FDS to form an unconditionally stable scheme in the convection-diffusion equation.…”

A decision-making machine learning approach in Hermite spectral approximations of partial differential equations

“…More recently, generalized Hermite functions have been applied to solving time dependent problems, assuming that α = α(t) and β = β(t) may also change in time. Noteworthy developments in this direction are found for instance in [24,23,48]. Those papers cover a wide range of applications ranging from the discretization of linear and nonlinear diffusion and convection-diffusion equations, to the generalized Ginzburg-Landau equation and the Fokker-Planck equation.…”