Hermite Spectral Method Research Articles

The Generalized Hermite Spectral Method Combined with Laplace Transform for Solution of the Time Fractional Fokker–Planck Equation

The paper proposes a method to solve the time fractional Fokker–Planck equation using the generalized Hermite spectral method and Laplace transform. By the Laplace transform, the ordinary differential equation about the orthogonal expansion coefficient is transformed into a linear algebraic equation system, with the coefficient matrix being a tridiagonal lower triangular matrix. The orthogonal expansion coefficients of frequency domain components are obtained through this transformation. The approximate solution at any time is then obtained by using the numerical inverse Laplace transform with the Talbot algorithm. Numerical experiments have been carried out to demonstrate the high accuracy and efficiency of the proposed method.

Numerical simulation of the time-fractional Fokker–Planck equation and applications to polymeric fluids

We introduce a new approach to the numerical approximation of the time-fractional Navier–Stokes–Fokker–Planck (TFNSFP) system, which involves the coupling of the incompressible Navier–Stokes equations to a time-fractional Fokker–Planck equation. The model arises in the context of dilute polymeric fluids, and it enhances the standard integer-derivative version of the model by including memory effects. The key challenge associated with the numerical solution of the TFNSFP system is that, in addition to it being nonlocal in time, it is, even in its simplest form, posed on a spatial domain that is the Cartesian product of two d-dimensional domains, for d∈{2,3}, the d-dimensional flow domain and the d-dimensional configuration space domain. We combine a kernel compression technique based on rational approximation of the integral kernel of the time-fractional derivative with a space-splitting method. By doing so, we transform the time-fractional partial differential equation into a fixed number of integer-order in time partial differential equations. The Fokker–Planck equation posed on the 2d-dimensional domain is decoupled into two d-dimensional problems, and a standard combination of the Hermite spectral method on the configuration space domain and a finite element method on the flow domain is then applied. Finally, we combine our numerical scheme for the time-fractional Fokker–Planck equation (TFFPE) with a standard Navier–Stokes solver. We propose an efficient implementation strategy based on the algebraic structure of the discretized time-fractional Fokker–Planck equation, which significantly reduces the computational cost.

Hermite spectral method for the inelastic Boltzmann equation

We propose a Hermite spectral method for the inelastic Boltzmann equation, which makes two-dimensional periodic problem computation affordable by the hardware nowadays. This new approach involves utilizing a Hermite expansion, whereby the expansion coefficients for the variable hard sphere model are simplified into a series of summations that can be precisely derived. Additionally, a new collision model is built with a combination of the quadratic collision operator and a simplified collision operator, which helps us to balance the computational cost and the accuracy. Various numerical experiments, including spatially two-dimensional simulations, demonstrate the accuracy and efficiency of this numerical scheme.

Analysis of the Buried Pipeline Response Induced by Twin Tunneling Using the Generalized Hermite Spectral Method

For the sustainability of economic, ecological and social development, the safety of infrastructure, including buried pipelines, is extremely important. Undercrossing tunneling can compromise the safety of buried pipelines because of deformations, cracks and dislocations, which can result in wasted resources, environmental pollution and economic losses. Therefore, it is important to assess the pipeline response accurately during tunnel excavation. This paper proposes a generalized Hermite spectral solution to estimate the pipeline response induced by twin tunneling. The proposed solution is formulated by a truncated series of Hermite functions and it is available in an unbounded domain. On the basis of the two-stage analysis method, a general formula for calculating the soil greenfield displacement induced by twin tunneling is first derived using the superposition principle. To obtain the final solution, the soil greenfield displacement and pipeline displacement are expanded using two truncated series of Hermite functions, and the governing differential equation of pipeline displacement is subsequently simplified into a linear algebraic system. After solving this system, a general solution for calculating pipeline displacement is formulated. Then, the convergence of the developed solution is proven, and its validity is verified against existing theoretical solutions and centrifuge test results. The effects of the truncated series number and its scaling factor are investigated. Finally, parametric studies are conducted to discuss pipeline responses induced by twin tunneling.

Hermite spectral method for multi-species Boltzmann equation

We introduce a numerical scheme for the full multi-species Boltzmann equation based on Hermite spectral method. With the proper choice of expansion centers for different species, a practical algorithm is derived to evaluate the complicated multi-species binary collision operator. New collision models are built by combining the quadratic collision model and the simple BGK collision model under the framework of the Hermite spectral method, which enables us to balance the computational cost and accuracy. Several numerical experiments are implemented to validate the dramatic efficiency of this new Hermite spectral method. Moreover, we can handle the problems with as many as 100 species, which is far beyond the capability of the state-of-art algorithms.

Fokker-Planck multi-species equations in the adiabatic asymptotics

The main concern of the present paper is the study of the multi-scale dynamics of thermonuclear fusion plasmas via a multi-species Fokker-Planck kinetic model. One of the goals is the generalization of the standard Fokker-Planck collision operator to a multi-species one, conserving mass, total momentum and energy, as well as satisfying Boltzmann's H-theorem. Secondly, we perform on one hand a mathematical asymptotic limit, letting the electron/ion mass ratio converging towards zero, to obtain a thermodynamic equilibrium state for the electrons (adiabatic regime), whereas the ions are kept kinetic. On the other hand, we develop a first numerical scheme, based on a Hermite spectral method, and perform numerical simulations to investigate in more details this asymptotic limit.

Adaptive Hermite spectral methods in unbounded domains

A novel adaptive spectral method has been recently developed to numerically solve partial differential equations (PDEs) in unbounded domains. To achieve accuracy and improve efficiency, the method relies on the dynamic adjustment of three key tunable parameters: the scaling factor, a displacement of the basis functions, and the spectral expansion order. In this paper, we perform the first numerical analysis of the adaptive spectral method using generalized Hermite functions in both one- and multi-dimensional problems. Our analysis reveals why adaptive spectral methods work well when a “frequency indicator” of the numerical solution is controlled. We then investigate how the implementation of the adaptive spectral methods affects numerical results, thereby providing guidelines for the proper tuning of parameters. Finally, we further improve performance by extending the adaptive methods to allow bidirectional basis function translation, and the prospect of carrying out similar numerical analysis to solving PDEs arising from realistic difficult-to-solve unbounded models with adaptive spectral methods is also briefly discussed.

A Decision-Making Machine Learning Approach in Hermite Spectral Approximations of Partial Differential Equations

The accuracy and effectiveness of Hermite spectral methods for the numerical discretization of partial differential equations on unbounded domains are strongly affected by the amplitude of the Gaussian weight function employed to describe the approximation space. This is particularly true if the problem is under-resolved, i.e., there are no enough degrees of freedom. The issue becomes even more crucial when the equation under study is time-dependent, forcing in this way the choice of Hermite functions where the corresponding weight depends on time. In order to adapt dynamically the approximation space, it is here proposed an automatic decision-making process that relies on machine learning techniques, such as deep neural networks and support vector machines. The algorithm is numerically tested with success on a simple 1D problem, but the main goal is its exportability in the context of more serious applications. As a matter of fact we also show at the end an application in the framework of plasma physics.

Nontensorial generalised hermite spectral methods for PDEs with fractional Laplacian and Schrödinger operators

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 ℝd. 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|2μ) in ℝd. 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 [u,v]Hs(ℝd)= ((-Δ)s/2u, (-Δ)s/2v)ℝdassociated with the IFL of orders> 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: (-Δ)s+ |x|2μwiths∈ (0,1] andμ> −1/2 in ℝdWe 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.

Finite difference/generalized Hermite spectral method for the distributed-order time-fractional reaction-diffusion equation on multi-dimensional unbounded domains

Distributed-order fractional differential equations, where the differential order is distributed over a range of values rather than being just a fixed value as it is in the classical differential equations, offer a powerful tool to describe multi-physics phenomena. In this article, we develop and analyze an efficient finite difference/generalized Hermite spectral method for the distributed-order time-fractional reaction-diffusion equation on one-, two-, and three-dimensional unbounded domains. Considering the Gauss-Legendre quadrature rule for the distributed integral term in temporal direction, we first approximate the original distributed-order time-fractional problem by the multi-term time-fractional differential equation. Then, we apply the L2-1σ formula for the discretization of the multi-term Caputo fractional derivatives. Moreover, we employ the generalized Hermite functions with scaling factor for the spectral approximation in space. The detailed implementations of the method are presented for one-, two-, and three-dimensional cases of the fractional problem. The stability and convergence of the method are strictly established, which shows that the proposed method is unconditionally stable and convergent with second-order accuracy in time. In addition, the optimal error estimate is derived for the space approximation. Finally, we perform numerical examples to support the theoretical claims.

Hermite spectral method for Fokker-Planck-Landau equation modeling collisional plasma

We propose an Hermite spectral method for the Fokker-Planck-Landau (FPL) equation. Both the distribution functions and the collision terms are approximated by series expansions of the Hermite functions. To handle the complexity of the quadratic FPL collision operator, a reduced collision model is built by adopting the quadratic collision operator for the lower-order terms and the diffusive Fokker-Planck operator for the higher-order terms in the Hermite expansion of the reduced collision operator. The numerical scheme is split into three steps according to the Strang splitting, where different expansion centers are employed for different numerical steps to take advantage of the Hermite functions. The standard normalized Hermite basis [36] is adopted during the convection and collision steps to utilize the precalculated coefficients of the quadratic collision terms, while the one constituted by the local macroscopic velocity and temperature is utilized for the acceleration step, by which the effect of the external force can be simplified to an ODE. Projections between different expansion centers are achieved by an algorithm proposed in [29]. Several numerical examples are studied to test and validate our new method.

Prandtl’s extended mixing length model applied to the two-dimensional turbulent classical far wake

Despite its limitations, Prandtl’s mixing length model is widely applied in modelling turbulent free shear flows. Prandtl’s extended model addresses many of the shortfalls of the original model, but is not so widely used, in part due to additional mathematical complexities that arise in its derivation and implementation. Furthermore, in both models, Prandtl neglects the kinematic viscosity on the basis that it is much smaller in magnitude than the turbulent viscosity. Recent work has shown that including the kinematic viscosity in the original model has both mathematical and physical advantages. In the present work, a novel derivation of the extended model is provided, and it is demonstrated that similar advantages are again obtained when the kinematic viscosity is included. Additionally, through the use of scaling techniques, similarity mean velocity profiles of the extended model are derived, resulting in a single nonlinear ordinary differential equation that is solved numerically with a Hermite spectral method. The computed profiles for the normalized similarity mean velocity and shear stress are compared with experimental observations and shown to be in excellent agreement.

Efficient Scaling and Moving Techniques for Spectral Methods in Unbounded Domains

When using Laguerre and Hermite spectral methods to numerically solve PDEs in unbounded domains, the number of collocation points assigned inside the region of interest is often insufficient, parti...

Solving Nonlinear Filtering Problems in Real Time by Legendre Galerkin Spectral Method

It is well known that the nonlinear filtering (NLF) problem has important applications in both military and civil industries. The central question is to solve the posterior conditional density function of the states, which satisfies the Kushner or the Duncan–Mortensen–Zakai (DMZ) equation after suitable change of probability measure. In this article, we shall follow the so-called Yau–Yau's algorithm to split the solution of the DMZ equation into on- and off-line part, where the off-line part is to solve the forward Kolmogorov equation (FKE) with the initial conditions to be the orthonormal bases in some suitable function space. Instead of the generalized Hermite function investigated by the second and the third author of this article, we shall explore the generalized Legendre polynomials. The Legendre spectral method (LSM) is used to numerically solve the FKE. Under certain conditions, the convergence rate of LSM is twice faster than that of the Hermite spectral method. Two two-dimensional numerical experiments of NLF problems (time-invariant and time-varying cases) have been numerically solved to illustrate the feasibility of our algorithm. Our algorithm outperforms the extended Kalman Filter and particle filter in both real-time manner and accuracy.

Efficient Laguerre and Hermite Spectral Methods for Odd-order Differential Equations in Unbounded Domains

Efficient Laguerre and Hermite Spectral Methods for Odd-order Differential Equations in Unbounded Domains

Numerical Simulation of Microflows Using Hermite Spectral Methods

We propose a Hermite spectral method for the spatially inhomogeneous Boltzmann equation. For the inverse-power-law model, we generalize an approximate quadratic collision operator defined in the normalized and dimensionless setting to an operator for arbitrary distribution functions. An efficient algorithm with a fast transform is introduced to discretize this new collision operator. The method is tested for one-dimensional benchmark microflow problems.

A Hermite spectral method for fractional convection diffusion equations on unbounded domains

In this paper, a Hermite spectral method is used for solving the fractional convection diffusion equations on unbounded domains. The scaled Hermite functions are used as basis functions, and the problems are solved in Fourier space. Multi-dimensional problems are considered in this paper, and the errors are estimated in Fourier space as well. At the end of the paper, the method is introduced to solve the fractional convection diffusion equations. Numerical examples are presented to verify our results.

Approximation of the Boltzmann collision operator based on hermite spectral method

Based on the Hermite expansion of the distribution function, we introduce a Galerkin spectral method for the spatially homogeneous Boltzmann equation with the realistic inverse-power-law models. A practical algorithm is proposed to evaluate the coefficients in the spectral method with high accuracy, and these coefficients are also used to construct new computationally affordable collision models. Numerical experiments show that our method captures the low-order moments very efficiently.

Hermite spectral method for Long-Short wave equations

We are concerned with the initial boundary value problem of the Long-Short wave equations on the whole line. A fully discrete spectral approximation scheme is structured by means of Hermite functions in space and central difference in time. A priori estimates are established which are crucial to study the numerical stability and convergence of the fully discrete scheme. Then, unconditionally numerical stability is proved in a space of $H^1({\Bbb R})$ for the envelope of the short wave and in a space of $L^2({\Bbb R})$ for the amplitude of the long wave. Convergence of the fully discrete scheme is shown by the method of error estimates. Finally, numerical experiments are presented and numerical results are illustrated to agree well with the convergence order of the discrete scheme.

A conservative scheme for the Fokker–Planck equation with applications to viscoelastic polymeric fluids

We propose a conservative scheme for a high-dimensional Fokker–Planck equation that arises in the dynamics of infinitely extensible polymer molecules. This leads to a challenging problem of unbounded domain. Our scheme combines the Lagrange–Galerkin method and the Hermite spectral method together with a space splitting approach. We prove that the scheme preserves the discrete mass. Combining it with a stabilized Lagrange–Galerkin method for the Navier–Stokes equations, we further propose a multiscale scheme for the simulation of some viscoelastic polymeric fluids. Several numerical experiments are presented to illustrate the performance of the schemes, and to confirm the conservation of mass at the discrete level.