Nontensorial generalised Hermite spectral methods for PDEs with fractional Laplacian and Schrodinger operators
Nontensorial generalised Hermite spectral methods for PDEs with fractional Laplacian and Schrodinger operators
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DOI：
10.1051/m2an/2021049
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发表时间：
2021-09
期刊：
影响因子：
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通讯作者：
Changtao Sheng;Suna Ma;Hui-yuan Li;Lilian Wang;Lueling Jia
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文献类型：
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作者：
Changtao Sheng;Suna Ma;Hui-yuan Li;Lilian Wang;Lueling Jia
抽象的。在本文中，我们介绍了任意维度的两个非张量广义 Hermite 多项式/函数 (GHP/GHF) 族，并开发了高效且准确的谱方法，用于在 R 中使用积分分数拉普拉斯 (IFL) 和/或薛定谔算子求解偏微分方程。作为 G. Szegö 族在一维 (1939) 中的推广，第一个多元多元多项式族 GHP (resp.GHF) 与权重函数 |x|2μe ́|x| 正交R 中的 2 (resp. |x|)。我们进一步构造了伴随广义 Hermite 函数 (A-GHF)，它通过傅立叶变换与相应的 GHF 相互交织，并且与与 s 阶 IFL 相关的内积 ru, vsHspRdq 正交 pp ́Δqs{2u, p ́ΔqvqRd 与 s ± 0 阶的 IFL 相关。 使用 A-GHF 作为基函数的谱伽辽金方法会产生 IFL 的对角刚度矩阵（众所周知，离散化非常困难且昂贵）。新的基础还发现使用分数薛定谔算子求解偏微分方程非常有效：p ́Δq ` |x|与 R 中的 s P p0, 1s 和 μ ą ́1{2 。我们构建了第二个族 多元非张量 Müntz 型 GHF，与与基础薛定谔算子相关的内积正交，并且针对原点处解的奇异性进行定制。我们证明了 Müntz 型 GHF 谱方法可以产生稀疏矩阵，并能对一些薛定谔特征值问题提供谱上精确的解。
Abstract. In this paper, we introduce two families of nontensorial generalised Hermite polynomials/functions (GHPs/GHFs) in arbitrary dimensions, and develop efficient and accurate spectral methods for solving PDEs with integral fractional Laplacian (IFL) and/or Schrödinger operators in R. As a generalisation of the G. Szegö’s family in 1D (1939), the first family of multivariate GHPs (resp.GHFs) are orthogonal with respect to the weight function |x|2μe ́|x| 2 (resp. |x|) in R. We further construct the adjoint generalised Hermite functions (A-GHFs), which have an interwoven connection with the corresponding GHFs through the Fourier transform, and are orthogonal with respect to the inner product ru, vsHspRdq “ pp ́Δqs{2u, p ́ΔqvqRd associated with the IFL of order s ą 0. As an immediate consequence, the spectral-Galerkin method using A-GHFs as basis functions leads to a diagonal stiffness matrix for the IFL (which is known to be notoriously difficult and expensive to discretise). The new basis also finds remarkably efficient in solving PDEs with the fractional Schrödinger operator: p ́Δq ` |x| with s P p0, 1s and μ ą ́1{2 in R. We construct the second family of multivariate nontensorial Müntz-type GHFs, which are orthogonal with respect to an inner product associated with the underlying Schrödinger operator, and are tailored to the singularity of the solution at the origin. We demonstrate that the Müntz-type GHF spectral method leads to sparse matrices and spectrally accurate solution to some Schrödinger eigenvalue problems.