Hermite Method Research Articles

High order accurate Hermite schemes on curvilinear grids with compatibility boundary conditions

High order accurate Hermite methods for the wave equation on curvilinear domains are presented. Boundaries are treated using centered compatibility conditions rather than more standard one-sided approximations. Both first-order-in-time (FOT) and second-order-in-time (SOT) Hermite schemes are developed. Hermite methods use the solution and multiple derivatives as unknowns and achieve space-time orders of accuracy 2m−1 (FOT) and 2m (SOT) for methods using (m+1)d degree of freedom per node in d dimensions. The compatibility boundary conditions (CBCs) are based on taking time derivatives of the boundary conditions and using the governing equations to replace the time derivatives with spatial derivatives. These resulting constraint equations augment the Hermite scheme on the boundary. The solvability of the equations resulting from the compatibility conditions is analyzed. Numerical examples demonstrate the accuracy and stability of the new schemes in two dimensions.

Non-Gaussian properties and their effects on extreme wind pressure of a 4:1 rectangular cylinder

This paper investigates the wind pressure characteristics of a 4:1 rectangular cylinder under 19 wind angles using a rigid model wind tunnel test, focusing on non-Gaussian properties and extreme wind pressure. The wind angle (α) varies from 0° to 90° and the Reynolds number (Re) ranges from 0.32 × 105 to 2.14 × 105. The spatial and statistical distributions of wind pressure, along with its non-Gaussian properties, are presented. Detailed analyses of these characteristics are provided. The peak factor of wind pressure was calculated using the modified Hermite method. The results indicate that the skewness and kurtosis of the windward region of the 4:1 rectangular cylinder remain relatively unchanged with varying wind angles. In contrast, the skewness and kurtosis in the separation and wake zones are significantly influenced by changes in the wind angle. Most wind pressures on the 4:1 rectangular cylinder exhibit negative bias and a softening process. The non-Gaussian properties are most prominent at the corner of the wake zone. The peak factor of 2.5 (used in Chinese code) or peak factor g,G based on Gaussian processes on the separation and wake zones of interest are much smaller than the peak factors g,NG calculated based on the modified Hermite method. Ignoring non-Gaussian properties underestimates extreme wind pressures by at least 20%–30%. Therefore, non-Gaussian properties must be considered when calculating wind pressure extremes for similar rectangular structures. Additionally, the peak factor value in the Chinese code should be increased appropriately to ensure structural safety.

Utilizing numerical techniques for modeling radiating Casson nanofluid flow with thermophoretic phenomenon on a heated stretching surface

This work aims to use mathematical modeling to investigate the mechanics of heat and mass transport in dissipative Casson nanofluid flows over a linear rough sheet. This study considers various elements such as thermal radiation, magnetic fields, heat generation, the varied properties of porous media, and the thermophoretic impact. Besides that, it looks into the variations in viscosity, diffusivity, and thermal conductivity, as well as the energy dissipation from viscous internal friction and fluid temperature modifications. The method involves coming up with and changing the boundary layer equations into a group of linked nonlinear ordinary differential equations that use variables that do not have any dimensions. The foundation of the solution strategy for these equations is the Hermite collocation method (HCM), which is renowned for its precision and adaptability. It offers an organized approach to solving the complex differential equations, allowing for accurate numerical solutions. The use of graphical representations ensures thorough data analysis and clarity, while also providing insightful information about the computed outcomes. The code validation method uses numerical comparisons with recent research to confirm the algorithm’s accuracy and dependability, as well as its resilience in comparison to the existing literature. Important conclusions from the study show that thermal boundary layers and nanofluid velocity decrease with increases in the porosity parameter, slip parameter, and magnetic field parameter.

Stability Analysis of Hermite Collocation Method for Pulp Washing Models

Pulp washing is concerned with detaching cellulose fibres from black liquor with the use of a minimal amount of wash liquor. An efficient numerical technique of hermite collocation method is used for the solution of mathematical models related to pulp washing. The linear and non-linear models are solved using Quintic Hermite collocation method with Dirichlet’s and mixed Robin’s boundary conditions. Numerical solution of the models are derived using MATLAB ode15s. This study deals with the justification of accuracy of the method with stability analysis. The present method is more convenient, simple and elegant for solving the two point boundary value problems and the results found are very much stable from numerical point of view

Wavelet Collocation Method for HIV-1/HTLV-I Co-Infection Model Using Hermite Polynomial.

In this study, the dynamic behavior of fractional order co-infection model with human immunodeficiency virus type 1 (HIV-1) and human T-lymphotropic virus type I (HTLV-I) is analyzed using operational matrix of Hermite wavelet collocation method. Also, the uniqueness and existence of solutions are calculated based on the fixed point hypothesis. For the fractional order co-infection model, its positivity and boundedness are demonstrated. Furthermore, different types of Ulam-Hyres stability are also discussed. The numerical solution of the model are obtained by using the operational matrix of the Hermite wavelet approach. This scheme is used to solve the system of nonlinear equationsthat are very fruitful and easy to implement. Additionally, the stability analysis of the numerical scheme is explained. The mathematical model taken in this work incorporates the biological characteristics of both HIV-1 and HTLV-I. After that all the equilibrium points of the fractional order co-infection model are found and their existence conditions are explored with the help of the Caputo derivative. The global stability of all equilibrium points of this model are determined with the help of Lyapunov functions and the LaSalle invariance principle. Convergence analysis is also discussed. Hermite wavelet operational matrix methods are more accurate and convergent than other numerical methods. Lastly, variations in model dynamics are found when examining different fractional order values. These findings will be valuable to biologists in the treatment of HIV-1/HTLV-I.

Modeling heat conduction with dual-dissipative variables: A mechanism-data fusion method.

Many macroscopic non-Fourier heat conduction models have been developed in the past decades based on Chapman-Enskog, Hermite, or other small perturbation expansion methods. These macroscopic models have achieved great success in capturing non-Fourier thermal behaviors in solid materials, but most of them are limited by small Knudsen numbers and incapable of capturing highly nonequilibrium or ballistic thermal transport. In this paper, we provide a different strategy for constructing macroscopic non-Fourier heat conduction modeling, that is, using data-driven deep-learning methods combined with nonequilibrium thermodynamics instead of small perturbation expansion. We present the mechanism-data fusion method, an approach that seamlessly integrates the rigorous framework of conservation-dissipation formalism (CDF) with the flexibility of machine learning to model non-Fourier heat conduction. Leveraging the conservation-dissipation principle with dual-dissipative variables, we derive an interpretable series of partial differential equations, fine tuned through a training strategy informed by data from the phonon Boltzmann transport equation. Moreover, we also present the inner-step operation to narrow the gap from the discrete form to the continuous system. Through numerical tests, our model demonstrates excellent predictive capabilities across various heat conduction regimes, including diffusive, hydrodynamic, and ballistic regimes, and displays its robustness and precision even with discontinuous initial conditions.

From low-rank retractions to dynamical low-rank approximation and back

In algorithms for solving optimization problems constrained to a smooth manifold, retractions are a well-established tool to ensure that the iterates stay on the manifold. More recently, it has been demonstrated that retractions are a useful concept for other computational tasks on manifold as well, including interpolation tasks. In this work, we consider the application of retractions to the numerical integration of differential equations on fixed-rank matrix manifolds. This is closely related to dynamical low-rank approximation (DLRA) techniques. In fact, any retraction leads to a numerical integrator and, vice versa, certain DLRA techniques bear a direct relation with retractions. As an example for the latter, we introduce a new retraction, called KLS retraction, that is derived from the so-called unconventional integrator for DLRA. We also illustrate how retractions can be used to recover known DLRA techniques and to design new ones. In particular, this work introduces two novel numerical integration schemes that apply to differential equations on general manifolds: the accelerated forward Euler (AFE) method and the Projected Ralston–Hermite (PRH) method. Both methods build on retractions by using them as a tool for approximating curves on manifolds. The two methods are proven to have local truncation error of order three. Numerical experiments on classical DLRA examples highlight the advantages and shortcomings of these new methods.

A Type of Interpolation between Those of Lagrange and Hermite That Uses Nodal Systems Satisfying Some Separation Properties

In this paper, we study a method of polynomial interpolation that lies in-between Lagrange and Hermite methods. The novelty is that we use very general nodal systems on the unit circle as well as on the bounded interval only characterized by a separation property. The way in which we interpolate consists in considering all the nodes for the prescribed values and only half for the derivatives. Firstly, we develop the theory on the unit circle, obtaining the main properties of the nodal polynomials and studying the convergence of the interpolation polynomials corresponding to continuous functions with some kind of modulus of continuity and with general conditions on the prescribed values for half of the derivatives. We complete this first part of the paper with the study of the convergence for smooth functions obtaining the rate of convergence, which is slightly slower than that when equidistributed nodal points are considered. The second part of the paper is devoted to solving a similar problem on the bounded interval by using nodal systems having good properties of separation, generalizing the Chebyshev–Lobatto system, and well related to the nodal systems on the unit circle studied before. We obtain an expression of the interpolation polynomials as well as results about their convergence in the case of continuous functions with a convenient modulus of continuity and, particularly, for differentiable functions. Finally, we present some numerical experiments related to the application of the method with the nodal systems dealt with.

Two-grid finite element method with an H2N2 interpolation for two-dimensional nonlinear fractional multi-term mixed sub-diffusion and diffusion wave equation

<abstract><p>In this paper, we studied the two-grid method (TGM) for two-dimensional nonlinear time fractional multi-term mixed sub-diffusion and diffusion wave equation. A fully discrete scheme with the quadratic Hermite and Newton interpolation (H2N2) method was considered in the temporal direction and the expanded finite element method is used to approximate the spatial direction. In order to reduce computational time, a dual grid method based on Newton iteration was constructed with order $ \alpha\in(0, 1) $ and $ \beta\in(1, 2) $. The global convergence order of the two-grid scheme reaches $ O(\tau^{3-\beta}+h^{r+1}+H^{2r+2}) $, where $ \tau $, $ H $ and $ h $ are the time step size, coarse grid mesh size and fine grid mesh size, respectively. The error estimation and stability of the fully discrete scheme were derived. Theoretical analysis shows that the two grid algorithms maintain asymptotic optimal accuracy while saving computational costs. In addition, numerical experiments further confirmed the theoretical results.</p></abstract>

Estimatig the amount of potassium radiation effect on soil using Neville and Hermite numerical methods

The aim of this work is to estimate the effect of potassium radiation on soil in Nineveh governorate using some methods of numerical interpolation. We used (Neville, and Hermite) methods and obtained mathematical models by which we quantified the effect of radiation on soil in Nineveh governorate. The deductible results were very close to the real results. We then estimated the effect of potassium radiation on soil in Nineveh governorate outside the field measured areas.

Efficient SAV–Hermite methods for the nonlinear Dirac equation

We construct two SAV/CN–Hermite schemes for one and two dimensional nonlinear Dirac equation in this paper. Both SAV/CN methods preserve the energy and mass and an efficient and accurate scheme is proposed by Hermite approximation. The numerical analysis of error estimates is established on SAV/CN scheme. Numerical experiments confirm the expected convergence order. In addition, they are given to show the simulations of binary and ternary collisions.

Numerical scheme for Erdélyi–Kober fractional diffusion equation using Galerkin–Hermite method

The aim of this work is to devise and analyse an accurate numerical scheme to solve Erdélyi–Kober fractional diffusion equation. This solution can be thought as the marginal pdf of the stochastic process called the generalized grey Brownian motion (ggBm). The ggBm includes some well-known stochastic processes: Brownian motion, fractional Brownian motion, and grey Brownian motion. To obtain a convergent numerical scheme we transform the fractional diffusion equation into its weak form and apply the discretization of the Erdélyi–Kober fractional derivative. We prove the stability of the solution of the semi-discrete problem and its convergence to the exact solution. Due to the singularity in the time term appearing in the main equation, the proposed method converges slower than first order. Finally, we provide a numerical analysis of the full-discrete problem using orthogonal expansion in terms of Hermite functions.

Radial Point Interpolation Method for Isotropic Nanoplates in Bending Using Strain Gradient Theory

This paper presents the static bending of isotropic Kirchhoff’s nanoplates modelled using the second-order strain gradient theory. The numerical analysis is conducted using mesh free methods instead of traditional finite elements. To the best of the authors’ knowledge, no such meshless methods have been employed in the analysis of strain gradient nanoplates. Hermite radial point interpolation method is used to approximate the bending degrees of freedom. Plates with different geometries and simply supported boundary conditions are studied. The results are then compared with the analytical solution available in the literature.

Feasibility and method of multi-step Hermitization of crypto-Hermitian quantum Hamiltonians

In the popular ${\cal PT}-$symmetry-based formulation of quantum mechanics of closed systems one can build unitary models using non-Hermitian Hamiltonians (i.e., $H \neq H^\dagger$) which are Hermitizable (so that one can write, simultaneously, $H = H^\ddagger$). The essence of the trick is that the reference Hilbert space $\cal R$ (in which we use the conventional inner product $\langle \psi_a|\psi_b\rangle$ and write $H \neq H^\dagger$) is declared unphysical. The necessary Hermiticity of the Hamiltonian $H = H^\ddagger$ can be then achieved by the mere metric-mediated amendment $\langle \psi_a|\Theta|\psi_b\rangle$ to the inner product. This converts $\cal R$ into a correct physical Hilbert space $\cal H$. The feasibility of the construction is based on a factorization postulate $\Theta={\cal PC}$ where, usually, ${\cal P}$ is parity and ${\cal C}$ is charge. In our paper we propose a more general factorization recipe in which one constructs $\Theta=Z_NZ_{N-1}\ldots Z_1$, at any $N$, in terms of suitable auxiliary pre-metric operators $Z_k$.

Spectral preprocessing to improve accuracy of quantitative detection of elemental Cr in austenitic stainless steel by laser-induced breakdown spectroscopy.

The production and consumption of austenitic stainless steel account for about 70% of stainless steel worldwide. The content of chromium (Cr) must be accurately detected and controlled to form a stable austenite structure and obtain strong properties in production. Laser-induced breakdown spectroscopy (LIBS) can be used to detect the Cr content of austenitic stainless steel in a complex production process. However, LIBS signals may be weak and unstable because the experimental signals are seriously affected by noise, self-absorption, the matrix effect, and the instability of the shot-to-shot signal, rendering the quantitative detection results inaccurate and unstable. The spectral-preprocessing methods of baseline correction and denoising can improve the accuracy of quantitative detection of LIBS. An improved segmented Hermite cubic-interpolation method is proposed herein to correct the baseline offset and produce baseline signals that are smooth and convergent (to overcome the Runge phenomenon). Empirical mode decomposition (EMD) based on the wavelet method is proposed to remove LIBS noise; this is done by exploiting the adaptivity of EMD to refine the wavelet-scaling coefficients. Compared with other denoising methods, the proposed method has good denoising evaluation indices and stability and, thus, effectively removes the noise. To verify detection accuracy, the internal standard quantitative method is used to detect the Cr content, and a cyclic-inversion prediction method is designed to verify detection stability. The results show that the correlation coefficient of the calibration curve is improved, the root-mean-square error is reduced, and the average relative error of the predicted Cr content decreases from 10.46% to 3.858%.

A Hermite Method with a Discontinuity Sensor for Hamilton–Jacobi Equations

We present a Hermite interpolation based partial differential equation solver for Hamilton-Jacobi equations. Many Hamilton-Jacobi equations have a nonlinear dependency on the gradient, which gives rise to discontinuities in the derivatives of the solution, resulting in kinks. We built our solver with two goals in mind: 1) high order accuracy in smooth regions and 2) sharp resolution of kinks. To achieve this, we use Hermite interpolation with a smoothness sensor. The degrees-of freedom of Hermite methods are tensor-product Taylor polynomials of degree $m$ in each coordinate direction. The method uses $(m+1)^d$ degrees of freedom per node in $d$-dimensions and achieves an order of accuracy $(2m+1)$ when the solution is smooth. To obtain sharp resolution of kinks, we sense the smoothness of the solution on each cell at each timestep. If the solution is smooth, we march the interpolant forward in time with no modifications. When our method encounters a cell over which the solution is not smooth, it introduces artificial viscosity locally while proceeding normally in smooth regions. We show through numerical experiments that the solver sharply captures kinks once the solution losses continuity in the derivative while achieving $2m+1$ order accuracy in smooth regions.

Hermite Method of Approximate Particular Solutions for Solving Time-Dependent Convection-Diffusion-Reaction Problems

This article describes the development of the Hermite method of approximate particular solutions (MAPS) to solve time-dependent convection-diffusion-reaction problems. Using the Crank-Nicholson or the Adams-Moulton method, the time-dependent convection-diffusion-reaction problem is converted into time-independent convection-diffusion-reaction problems for consequent time steps. At each time step, the source term of the time-independent convection-diffusion-reaction problem is approximated by the multiquadric (MQ) particular solution of the biharmonic operator. This is inspired by the Hermite radial basis function collocation method (RBFCM) and traditional MAPS. Therefore, the resultant system matrix is symmetric. Comparisons are made for the solutions of the traditional/Hermite MAPS and RBFCM. The results demonstrate that the Hermite MAPS is the most accurate and stable one for the shape parameter. Finally, the proposed method is applied for solving a nonlinear time-dependent convection-diffusion-reaction problem.

Hermite wavelet method for solving nonlinear Rosenau–Hyman equation

In this paper, we present an approximate solution for solving the nonlinear Rosenau–Hyman equation. The method is based on adapting the wavelet technique accompanied with the Hermit polynomials. Convergence analysis for the proposed method is being investigated, proving that the Hermite wavelet expansion is uniformly convergent. A test example is presented for different values of the parameters, and the obtained results are compared to other relevant methods from the literature. The process proves to have the ability to produce accurate results than the other compared methods. Some graphical representations for the problem are drawn to illustrate the behavior of the solution.

On a high-order Gaussian radial basis function generated Hermite finite difference method and its application

On a high-order Gaussian radial basis function generated Hermite finite difference method and its application

Hermite wavelet method for solving oscillatory electrical circuit equations

In this research, numerical solutions based on Hermite wavelets have been presented for solving oscillatory electrical circuit equations. The proposed numerical technique is based on Hermite wavelet basis functions, operational matrices of integration and collocation points. Numerical experiments have been performed to illustrate the accuracy and efficiency of the proposed scheme.