High-order Method Research Articles

Stability and convection in compressible partially ionized plasma layers: nonlinear and linear analysis

Abstract The thermal convection of compressible, partially ionized plasma has been investigated using nonlinear and linear analyses across three boundary configurations. Nonlinear analysis was carried out via the energy method, while linear analysis was assessed using the normal mode method. For free-free boundaries, exact solutions were obtained, whereas, for rigid-rigid and rigid-free boundaries, the higher-order Galerkin-weighted residual method was employed for numerical results. The critical Rayleigh numbers for both analyses coincide, indicating global stability and confirming the absence of subcritical regions. The impact of collisional frequency on energy decay was quantified, revealing a significant effect on the decay rate, although it does not affect the Rayleigh number. The principle of exchange of stabilities was confirmed in the linear analysis. Compressibility delays the onset of convection. The critical Rayleigh numbers were computed as 986.267, 2,561.64, and 1,650.97 for the free–free, rigid–rigid, and rigid–free cases, respectively, demonstrating that plasma confined between rigid–rigid surfaces exhibits the highest thermal stability.

A high-order multiscale finite element method for 3D direct current resistivity log modeling

When solving the 3D direct current resistivity logging problem with complex heterogeneous structure, the finite element method requires the construction of a fine discretized mesh to accurately reflect the significant heterogeneity of the steel-casing and the complexity of the underground structure. This fine mesh division will lead to an excessively large calculation, thus occupying a large amount of computing resources and making it difficult to solve. The multiscale finite element method can significantly reduce the size of the coefficient matrix during the solving process, which greatly reduces the computational cost. Nevertheless, the low-order multiscale finite element cannot accurately transfer the heterogeneous information of the fine mesh to the coarse mesh, it leads to numerical errors. And the imposition of linear boundary conditions to construct multiscale basis functions may introduce resonance error. Consequently, this article implements a 3D direct current resistivity logging forward algorithm based on high-order multiscale finite element. On the one hand, we construct boundary and internal high-order multiscale basis functions by utilizing arbitrary order orthogonal polynomials within local cells, and then linearly combine them to form complete high-order multiscale basis functions, this ensures the approximation ability and convergence accuracy of the numerical solution. On the other hand, we construct temporary high-order multiscale basis functions on extended local cells, combined with an oversampling technique, which can mitigate the error problem caused by boundary resonance effects. Simulate in the scenarios with three different electrode configurations: pole-pole, pole-dipole, and dipole-dipole. The results show that our new method can approximate the fine-scale reference solution on the coarse mesh with high accuracy and significantly reduced computational time at the linear system solving stage.

High-Order Compact Difference Method for Solving Population Balance Equations in Batch Crystallization

High-Order Compact Difference Method for Solving Population Balance Equations in Batch Crystallization

A High-Order Shifted Interface Method for Lagrangian Shock Hydrodynamics

We present a new method for two-material Lagrangian hydrodynamics, which combines the Shifted Interface Method (SIM) with a high-order Finite Element Method. Our approach relies on an exact (or sharp) material interface representation, that is, it uses the precise location of the material interface. The interface is represented by the zero level-set of a continuous high-order finite element function that moves with the material velocity. This strategy allows to evolve curved material interfaces inside curved elements. By reformulating the original interface problem over a surrogate (approximate) interface, located in proximity of the true interface, the SIM avoids cut cells and the associated problematic issues regarding implementation, numerical stability, and matrix conditioning. Accuracy is maintained by modifying the original interface conditions using Taylor expansions. We demonstrate the performance of the proposed algorithms on established numerical benchmarks in one, two and three dimensions.

Extrapolated splitting methods for multilinear PageRank computations

Multilinear PageRank is a variant of the well-known PageRank model. With this model, web page ranking can be more accurate and efficient by taking into account higher-order connections between pages. The higher-order power method is commonly employed for computing the multilinear PageRank vector, since it is a natural extension of the traditional power method used in the PageRank algorithm. However, this method may not be efficient when the hyperlink tensor becomes large or the damping factor value fails to meet the necessary conditions for convergence. In this work, we propose a novel approach to efficiently computing the multilinear PageRank vector using tensor splitting and vector extrapolation methods.

High-Order Well-Balanced Methods for the Euler Equations of Gas Dynamics with Gravitational Forces and the Ripa Model

Different well-balanced high-order finite-volume numerical methods for the one-dimensional compressible Euler equations of gas dynamics with gravitational force and for the Ripa model have been proposed in the literature. Most of them preserve either a given family of hydrostatic stationary solutions exactly or all of them approximately. The goal of this paper is to design a general methodology to obtain high-order finite-volume numerical methods for a class of one-dimensional hyperbolic systems of balance laws that preserve approximately all the hydrostatic equilibria and exactly a given family of them. Many fluid models for which the velocity is an eigenvalue of the system belong to this class, the Euler equations and the Ripa model among them. The methods proposed here are based on the design of well-balanced reconstruction operators that require the exact or the approximate computation of local hydrostatic equilibria. To check the efficiency and the well-balancedness of the methods, a number of numerical tests have been performed: the numerical results confirm the theoretical ones.

Non-Invasive and Accurate Blood Glucose Detection Based on an Equivalent Bioimpedance Spectrum

Accurate blood glucose monitoring is a key issue for the diagnosis and treatment of diabetes. For this purpose, a non-invasive blood glucose detection method is proposed, which makes use of the equivalent bioelectric impedance spectrum. An impedance detection platform is designed using an automatic balance bridge technique, which can acquire an impedance spectrum within the range of dispersion. Then, the K-nearest neighbor algorithm is used to extract the characteristics of the impedance spectrum. Furthermore, higher-order multiple regression methods are used to establish a blood glucose–electrical impedance spectrum model. Experimental results show that the proposed blood glucose–electrical impedance spectrum model can estimate the change in blood glucose and reliably identify the high blood glucose samples. The correlation between the proposed method and the biochemical blood glucose values can reach 0.89 and 0.87 in personal and multi-person blood glucose tests, respectively. Thus, the proposed method provides a feasible solution for non-invasive blood glucose detection and can help us identify diabetes mellitus.

Accelerating composite sandwich plate analysis: A hybrid higher-order XFEM and ANN approach for natural frequency prediction

Free vibration analysis of laminated composites plays a critical role in design optimization, material selection, structural integrity assessment, vibration suppression, and fatigue life prediction. A novel coupled higher-order extended finite element method (HO-XFEM) and artificial neural network (ANN) approach is implemented for predicting the natural frequency of sandwich plates with a composite layout of 0°/θ°/0°/core/0°/θ°/0°, where θ varies from 15 to 90 degrees. A total of 557 data points are generated through HO-XFEM, which is then utilized to construct an optimized ANN model for predicting nondimensionalized natural frequency. The ANN model, designed specifically to predict the first mode of natural frequency, is optimized with a structure of 4-90-90-1 and softmax transfer functions in intermediate layers. It predicts the nondimensional natural frequency with the average, minimum, and maximum mean-squared error values of 0.0984 %, 0.00339 %, and 0.49 %, respectively. The optimized ANN model computes the natural frequency for different sandwich plate configurations a few million times faster compared to HO-XFEM and thus establishes a transformative framework for real-time structural integrity assessment.

Polynomial quasi-Trefftz DG for PDEs with smooth coefficients: elliptic problems

Abstract Trefftz schemes are high-order Galerkin methods whose discrete spaces are made of elementwise exact solutions of the underlying partial differential equation (PDE). Trefftz basis functions can be easily computed for many PDEs that are linear, homogeneous and have piecewise-constant coefficients. However, if the equation has variable coefficients, exact solutions are generally unavailable. Quasi-Trefftz methods overcome this limitation relying on elementwise ‘approximate solutions’ of the PDE, in the sense of Taylor polynomials. We define polynomial quasi-Trefftz spaces for general linear PDEs with smooth coefficients and source term, describe their approximation properties and, under a nondegeneracy condition, provide a simple algorithm to compute a basis. We then focus on a quasi-Trefftz DG method for variable-coefficient elliptic diffusion–advection–reaction problems, showing stability and high-order convergence of the scheme. The main advantage over standard DG schemes is the higher accuracy for comparable numbers of degrees of freedom. For nonhomogeneous problems with piecewise-smooth source term we propose to construct a local quasi-Trefftz particular solution and then solve for the difference. Numerical experiments in two and three space dimensions show the excellent properties of the method both in diffusion-dominated and advection-dominated problems.

Shannon Entropy Computations in Navier-Stokes Flow Problems Using the Stochastic Finite Volume Method.

The main aim of this study is to achieve the numerical solution for the Navier-Stokes equations for incompressible, non-turbulent, and subsonic fluid flows with some Gaussian physical uncertainties. The higher-order stochastic finite volume method (SFVM), implemented according to the iterative generalized stochastic perturbation technique and the Monte Carlo scheme, are engaged for this purpose. It is implemented with the aid of the polynomial bases for the pressure-velocity-temperature (PVT) solutions, for which the weighted least squares method (WLSM) algorithm is applicable. The deterministic problem is solved using the freeware OpenFVM, the computer algebra software MAPLE 2019 is employed for the LSM local fittings, and the resulting probabilistic quantities are computed. The first two probabilistic moments, as well as the Shannon entropy spatial distributions, are determined with this apparatus and visualized in the FEPlot software. This approach is validated using the 2D heat conduction benchmark test and then applied for the probabilistic version of the 3D coupled lid-driven cavity flow analysis. Such an implementation of the SFVM is applied to model the 2D lid-driven cavity flow problem for statistically homogeneous fluid with limited uncertainty in its viscosity and heat conductivity. Further numerical extension of this technique is seen in an application of the artificial neural networks, where polynomial approximation may be replaced automatically by some optimal, and not necessarily polynomial, bases.

Strength Analysis and Optimization Recommendations for the Cargo Hold Sections of Ultra-Large Container Ships Based on an Improved High-Order Finite Element Method

Strength Analysis and Optimization Recommendations for the Cargo Hold Sections of Ultra-Large Container Ships Based on an Improved High-Order Finite Element Method

Numerical Advancements: A Duel between Euler-Maclaurin and Runge-Kutta for Initial Value Problem

This work is dedicated to advancing the approximation of initial value problems through the introduction of an innovative and superior method inspired by the Euler-Maclaurin formula. This results in a higher-order implicit corrected method that outperforms the Runge-Kutta method in terms of accuracy. We derive an error bound for the Euler-Maclaurin higher-order method, showcasing its stability, convergence, and greater efficiency compared to the conventional Runge-Kutta method. To substantiate our claims, numerical experiments are provided, highlighting the exceptional efficiency of our proposed method over the traditional well-known methods. In conclusion, the proposed method consistently outperforms the Runge-Kutta method experimentally in all practical problems.

High-order numerical method and error analysis based on a mixed scheme for fourth-order problem in a ball

High-order numerical method and error analysis based on a mixed scheme for fourth-order problem in a ball

High-order numerical method for the fractional Korteweg-de Vries equation using the discontinuous Galerkin method

High-order numerical method for the fractional Korteweg-de Vries equation using the discontinuous Galerkin method

A High-Order Isogeometric Collocation Method with Adaptive Refinement and Multi-patch Geometric Modeling

A High-Order Isogeometric Collocation Method with Adaptive Refinement and Multi-patch Geometric Modeling

Numerical Evaluation of Strongly Near-Singular Integrals in High-Order Method of Moments

Numerical Evaluation of Strongly Near-Singular Integrals in High-Order Method of Moments

On novel Hexa-compound combination synchronization over nineteen n-dimensional category-B chaotic systems and electronic circuit schematic

Abstract Synchronization of chaotic models involving multiple drives and responses has numerous practical applications in cryptography and information processing. Existing research on synchronizing multiple chaotic systems is currently limited to twelve components. This study introduces a novel higher-order synchronization method, hexa compound combination, that synchronizes an assembly of nineteen n-dimensional chaotic models. Well-known synchronization methods, such as double compound, triple compound, and quad compound, serve as particular instances of this new strategy. Thus, our research significantly advances the understanding of multi-leveled chaos synchronization. In addition, we also present a non-uniformly conservative system classified into a rare category B, analyze its dynamic properties, and utilize it for achieving the proposed synchronization. Numerical results are provided through graphical representations to illustrate the efficacy of the new synchronization approach by comparing it with other techniques. Furthermore, we emulate the corresponding virtual schematic circuit of the newly designed system to evaluate its real-world applicability and utility.

PENALTY FUNCTION METHOD FOR MODELING OF CYLINDER FLOW WITH SUBSONIC COMPRESSIBLE FLOW

Numerical modelling of compressible flows around moving solids is important for engineering applications such as aerodynamic flutter, rocket engines and landing gear. The penalty function method is particularly effective when using orthogonal structural meshes within a finite difference scheme and is widely used to solve both laminar and turbulent flow problems. The method is based on the direct application of the Navier-Stokes equations with added sources, which allows the boundary conditions to be set indirectly. This method facilitates the imposition of Dirichlet boundary conditions but complicates the application of Neumann conditions. Nevertheless, the method works well with both types of boundary conditions, making it suitable for thermal and compressible flows where Neumann conditions are often used. Despite its flexibility, the method requires a high degree of data management and additional coding. This paper presents results of a recently developed higher-order method for compressible subsonic flows, demonstrating accurate modeling of moving objects without numerical noise. The method has been tested on stationary and moving objects over a wide range of Reynolds and Mach numbers.

Compact scheme with two-sided estimates for nonlinear convection-diffusion-reaction equations

It is desirable that a numerical method is high-order accurate, compact, efficient and able to simulate purely convection problems. Most existing high-order methods for convection-diffusion-reaction equations (CDREs) either cannot simulate purely-convection problems, or reduce to first order when they can, or require larger stencil to maintain high-order accuracy. This is challenging, especially due to the convection term which can easily lead to oscillatory solutions if naively approximated. This paper proposes a spatially second-order scheme which is able to simulate purely-convection problems with high-order accuracy on minimal stencil. Our idea is based on non-standard central discretization of the convection term. We first discretize the diffusion term in both space and time but discretize the convection term in space only. Next, the semi-discrete convection term is split into positive and negative parts. Transport coefficients are evaluated explicitly in time while different spatial operators are discretized either implicitly or explicitly such that positivity of canonical form is guaranteed. This led to a second-order scheme with all the above desirable properties. Under smoothness assumptions consistency is proved, discrete maximum principle is used to derived two-sided bounds on the numerical solution, and convergence is proved in maximum norm. Several numerical examples are provided to verify the second-order spatial accuracy, convergence and the ability to simulate purely convection problems.

Preface to Focused Section on Efficient High-Order Time Discretization Methods for Partial Differential Equations

Preface to Focused Section on Efficient High-Order Time Discretization Methods for Partial Differential Equations