The time-dependent Hermite–Galerkin spectral method (THGSM) of Luo–Yau–Yau (2015) solves nonlinear convection–diffusion equations on the whole line by expanding the solution in generalized Hermite functions whose Gaussian width and center are governed by a time-dependent scaling factor α(t)\alpha(t) and translating factor β(t)\beta(t). The scheme is stable and spectrally convergent for any parameter choice, but its accuracy at a fixed number of modes depends critically on these parameters, and the source paper states that the optimal scaling factor remains an open problem even when the target function is given explicitly — the available guidelines cover only Gaussian-type and super-Gaussian-type shapes, while the exact solution of a PDE is unknown a priori. Later literature has resolved parts of this question: the first author's spectral-viscosity Hermite method (2018) settled the companion problem of extending the method to inviscid conservation laws, and the Hu–Yu error framework (2026) gives systematic optimal-scaling guidance for approximating a known function; adaptive indicator-based, machine-learning, and physics-based strategies supply partial dynamic answers. What remains open is a general theory for choosing α(t)\alpha(t) and β(t)\beta(t) automatically — using only information available during the computation — such that the truncation error is minimized, or provably near-minimized, for arbitrary evolving solutions, including solutions that translate, spread, change their decay and oscillation characteristics, or develop discontinuities.

Background

Spectral methods solve differential equations by expanding the unknown in a basis of orthogonal functions; in a Galerkin scheme the equation is projected onto the finite-dimensional space spanned by the first NN basis functions. On unbounded domains such as the whole real line, the Hermite polynomials — orthogonal with respect to the Gaussian weight e−x2e^{-x^2} — and the associated Hermite functions provide a natural basis, avoiding the artificial boundaries required by domain truncation. Their practical weakness is poor resolution: it has been known since the early days of spectral methods, and is emphasized in the source paper, that resolving MM wavelengths of sin⁡x\sin x with unscaled Hermite functions requires on the order of M2M^2 terms. The standard remedy is to rescale the basis. The generalized Hermite functions

H^nα,β(x)=(α2nn!π)1/2Hn(α(x−β)) e−12α2(x−β)2\hat H_n^{\alpha,\beta}(x)=\Big(\frac{\alpha}{2^n n!\sqrt{\pi}}\Big)^{1/2} H_n\big(\alpha(x-\beta)\big)\,e^{-\frac{1}{2}\alpha^2(x-\beta)^2}

form an orthonormal basis of L2(R)L^2(\mathbb R) for every scaling factor α>0\alpha>0 and translating factor β∈R\beta\in\mathbb R (which recenters the Gaussian envelope), and a well-chosen α\alpha restores fast convergence. Tang (1993) gave an effective scaling rule for Gaussian-type functions, and Luo–Yau (2013) extended such guidelines to super-Gaussian types in the context of nonlinear filtering; both presuppose knowledge of the target function's shape. Ma–Sun–Tang (2005) introduced a time-dependent scaling α(t)\alpha(t) for parabolic equations but gave no rule for choosing it.

Luo–Yau–Yau (2015) proposed the time-dependent Hermite–Galerkin spectral method (THGSM) for nonlinear convection–diffusion equations on the whole line, allowing both α(t)\alpha(t) and β(t)\beta(t) to depend on time. Stability and spectral convergence were proved for arbitrary parameter choices, and numerical experiments on the heat equation, the viscous Burgers equation, and the Korteweg–de Vries–Burgers (KdVB) equation — a model combining Burgers-type diffusion with KdV-type dispersion — showed improved accuracy over fixed-parameter schemes. Yet the truncation error at a fixed mode count NN depends strongly on α(t)\alpha(t) and β(t)\beta(t), and the paper states plainly that the optimal scaling factor remains an open problem even when the function to be approximated is given explicitly, since the existing guidelines cover only Gaussian-type targets and the PDE solution is unknown a priori. The paper also noted that the method was demonstrated only for problems with sufficient diffusion: for inviscid equations such as the inviscid Burgers equation, global spectral approximations exhibit Gibbs oscillations (spurious high-frequency oscillations near a discontinuity), and a combination with vanishing viscosity and post-processing was left as an open direction. That inviscid strand was subsequently addressed by the first author's own spectral viscosity method with generalized Hermite functions (Luo 2018) — scale-dependent vanishing viscosity added to the Hermite discretization, with convergence to the entropy solution (the physically admissible weak solution selected by an entropy condition) proved via compensated compactness, a technique for passing to limits in nonlinear functions of approximate solutions. The scaling-factor question, by contrast, was reiterated as open there, because the Gaussian-type guidelines fail for discontinuous solutions. Activity on the selection question continues through 2025–2026: frequency-indicator scaling adaptivity has been extended to the Boltzmann equation and the Vlasov–Poisson system of plasma kinetics (Shao–Wang–Wu), the first error estimates for Hermite approximation under adaptive coordinate transformations have appeared (Saleh 2026), and optimal scaling factors have been derived for static approximation composed with exponential transforms (Wang 2026) — none of them an automatic, provably near-optimal rule for time-dependent parameters. The surviving open problem is therefore the selection theory for α(t)\alpha(t) and β(t)\beta(t), in both the smooth diffusion-dominated setting of the THGSM and the discontinuous regime of nonlinear conservation laws.

Problem Statement

Develop a general theory for the automatic selection of the time-dependent scaling factor α(t)\alpha(t) and translating factor β(t)\beta(t) in the time-dependent Hermite–Galerkin spectral method (THGSM) of Luo–Yau–Yau (2015), under which the truncation error of the NN-term generalized Hermite approximation

uN(x,t)=∑n=0Nu^n(t) H^nα(t),β(t)(x)u_N(x,t)=\sum_{n=0}^{N}\hat u_n(t)\,\hat H_n^{\alpha(t),\beta(t)}(x)

of the solution of an evolution equation on the whole line — in the source setting, nonlinear convection–diffusion and dispersive model equations such as the viscous Burgers and Korteweg–de Vries–Burgers equations, and in the non-diffusive limit, scalar conservation laws — is minimized, or provably near-minimized in a sense the theory must make precise, using only information available during the computation and no a priori knowledge of the exact solution.

The theory should cover arbitrary evolving solutions: solutions whose spatial extent, center, decay rate, and oscillation characteristics change over time (translating, spreading, and non-Gaussian profiles, where the Gaussian-type and super-Gaussian-type guidelines of Tang (1993) and Luo–Yau (2013) do not apply), and the weakly or non-diffusive regime, including solutions that develop discontinuities, for which Luo (2018) explicitly reiterates that the optimal choice of the scaling factor is open. It should specify (i) the optimality criterion and the class of solutions covered; (ii) a concrete, implementable rule for updating α(t)\alpha(t) and β(t)\beta(t) during the computation; (iii) error bounds showing that the rule attains the criterion; and (iv) how the parameter updates interact with the stability of the resulting time-stepping scheme. This is the surviving core of the 2015 open problem: the companion question of extending Hermite spectral methods to inviscid conservation laws has been addressed by the spectral-viscosity method of Luo (2018), and the static question of optimal scaling for a known function has been substantially addressed by the Hu–Yu framework (2026); the general time-dependent, solution-adaptive selection theory remains unavailable. Whether such a rule exists is part of the open question: a proof that no admissible, computable rule attains the stated criterion on the stated class of solutions settles it negatively.

References

  1. Xue Luo, Shing-Tung Yau, Stephen S.-T. Yau, "Time-dependent Hermite–Galerkin spectral method and its applications", Applied Mathematics and Computation 264 (2015) 378–391. DOI: 10.1016/j.amc.2015.04.088, https://doi.org/10.1016/j.amc.2015.04.088 (arXiv:1412.0427, https://arxiv.org/abs/1412.0427)
  2. Xue Luo, "Spectral viscosity method with generalized Hermite functions for nonlinear conservation laws", Applied Numerical Mathematics 123 (2018) 256–274. DOI: 10.1016/j.apnum.2017.09.014, https://doi.org/10.1016/j.apnum.2017.09.014 (arXiv:1402.3775, https://arxiv.org/abs/1402.3775)
  3. Lorella Fatone, Daniele Funaro, Gianmarco Manzini, "A Decision-Making Machine Learning Approach in Hermite Spectral Approximations of Partial Differential Equations", Journal of Scientific Computing 92 (2022), article 3. DOI: 10.1007/s10915-022-01853-4, https://doi.org/10.1007/s10915-022-01853-4 (arXiv:2104.02181, https://arxiv.org/abs/2104.02181)
  4. Mingtao Xia, Sihong Shao, Tom Chou, "Efficient Scaling and Moving Techniques for Spectral Methods in Unbounded Domains", SIAM Journal on Scientific Computing 43 (2021) A3244–A3268. DOI: 10.1137/20M1347711, https://doi.org/10.1137/20M1347711
  5. Tom Chou, Sihong Shao, Mingtao Xia, "Adaptive Hermite spectral methods in unbounded domains", Applied Numerical Mathematics 183 (2023) 201–220. DOI: 10.1016/j.apnum.2022.09.003, https://doi.org/10.1016/j.apnum.2022.09.003 (arXiv:2203.15630, https://arxiv.org/abs/2203.15630)
  6. Hao Hu, Haijun Yu, "Scaling Optimized Hermite Approximation Methods", SIAM Journal on Numerical Analysis 64 (2026) 125–147. DOI: 10.1137/25M1737146, https://doi.org/10.1137/25M1737146 (arXiv:2412.08044, https://arxiv.org/abs/2412.08044)
  7. Hao Hu, Haijun Yu, "Scaling Optimized Spectral Approximations on Unbounded Domains: The Generalized Hermite and Laguerre Methods", arXiv:2602.03083 (2026), https://arxiv.org/abs/2602.03083
  8. Heping Ma, Weiwei Sun, Tao Tang, "Hermite Spectral Methods with a Time-Dependent Scaling for Parabolic Equations in Unbounded Domains", SIAM Journal on Numerical Analysis 43 (2005) 58–75. DOI: 10.1137/S0036142903421278, https://doi.org/10.1137/S0036142903421278
  9. Tao Tang, "The Hermite Spectral Method for Gaussian-Type Functions", SIAM Journal on Scientific Computing 14 (1993) 594–606. DOI: 10.1137/0914038, https://doi.org/10.1137/0914038
  10. Xue Luo, Stephen S.-T. Yau, "Hermite Spectral Method to 1-D Forward Kolmogorov Equation and Its Application to Nonlinear Filtering Problems", IEEE Transactions on Automatic Control 58 (2013) 2495–2507. DOI: 10.1109/TAC.2013.2259975, https://doi.org/10.1109/TAC.2013.2259975
  11. Cecilia Pagliantini, Gian Luca Delzanno, Stefano Markidis, "Physics-based adaptivity of a spectral method for the Vlasov–Poisson equations based on the asymmetrically-weighted Hermite expansion in velocity space", Journal of Computational Physics 488 (2023) 112252. DOI: 10.1016/j.jcp.2023.112252, https://doi.org/10.1016/j.jcp.2023.112252
  12. Sihong Shao, Yanli Wang, Jie-Zhi Wu, "An adaptive Hermite spectral method for the Boltzmann equation", arXiv:2509.17981 (2025), https://arxiv.org/abs/2509.17981
  13. Sihong Shao, Yanli Wang, Jie Wu, "Solving Vlasov-Poisson system with an adaptive Hermite spectral method", arXiv:2605.17820 (2026), https://arxiv.org/abs/2605.17820
  14. Yahya Saleh, "Convergence theory for Hermite approximations under adaptive coordinate transformations", arXiv:2604.16975 (2026), https://arxiv.org/abs/2604.16975
  15. Haiyong Wang, "Hermite spectral approximation for functions with endpoint singularities using exponential transforms", arXiv:2607.12648 (2026), https://arxiv.org/abs/2607.12648