High-order Method Research Articles

Adaptive MSSRCPF filtering based on constrained optimization and fading memory for SINS/GNSS integrated navigation

To address particle degeneracy and the challenge of selecting importance density functions in traditional particle filters for complex nonlinear systems, we propose an adaptive mixed-order spherical simplex-radial cubature particle filter algorithm based on constrained optimization and fading memory. This algorithm integrates constrained optimization, an adaptive fading memory strategy, adaptive adjustment of particle numbers and weights, and the advantages of mixed-order spherical simplex-radial cubature Kalman filtering. By designing the importance density function using a mixed-order integration method, the algorithm significantly improves filtering accuracy over traditional cubature Kalman filters while maintaining lower computational complexity than high-order cubature Kalman filter methods. The adaptive fading memory strategy dynamically adjusts the covariance matrix, enhancing sensitivity to current measurement information and reducing the influence of historical data. By dynamically adjusting the noise covariance matrix and constraining the ratio between the error covariance and measurement noise covariance, the algorithm improves the convergence speed and accuracy of state estimation. An adaptive particle number and weight adjustment strategy based on effective sample size and entropy regularization dynamically adjusts the number of particles to reduce computational complexity while ensuring filtering accuracy, and employs entropy regularization to suppress weight over-concentration, thereby reducing the impact of outliers on the filtering results. Simulation results demonstrate that in complex SINS/GNSS integrated navigation systems, especially under high-noise and nonlinear conditions, the proposed algorithm significantly improves positioning accuracy compared to the CPF method, with the maximum latitude error reduced by approximately 54.8%, the maximum longitude error reduced by approximately 40.5%, and the maximum altitude error reduced by approximately 68.3%. Compared to the FMCPF method, the proposed algorithm also shows clear advantages in positioning accuracy, convergence speed, and computational efficiency, validating its effectiveness and practicality in complex environments.

The High-Order ADI Difference Method and Extrapolation Method for Solving the Two-Dimensional Nonlinear Parabolic Evolution Equations

In this paper, the numerical solution for two-dimensional nonlinear parabolic equations is studied using an alternating-direction implicit (ADI) Crank–Nicolson (CN) difference scheme. Firstly, we use the CN format in the time direction, and then use the CN format in the space direction before discretizing the second-order center difference quotient. In addition, we strictly prove that the proposed ADI difference scheme has unique solvability and is unconditionally stable and convergent. The extrapolation method is further applied to improve the numerical solution accuracy. Finally, two numerical examples are given to verify our theoretical results.

Discretely Nonlinearly Stable Weight-Adjusted Flux Reconstruction High-Order Method for Compressible Flows on Curvilinear Grids

To achieve genuine predictive capability, an algorithm must consistently deliver accurate results over prolonged temporal integration periods, avoiding the unwarranted growth of aliasing errors that compromise the discrete solution. Provable nonlinear stability bounds the discrete approximation and ensures that the discretization does not diverge. Nonlinear stability is accomplished by satisfying a secondary conservation law, namely for compressible flows; the second law of thermodynamics. For high-order methods, discrete nonlinear stability and entropy stability, have been successfully implemented for discontinuous Galerkin (DG) and residual distribution schemes, where the stability proofs depend on properties of L2-norms. In this paper, nonlinearly stable flux reconstruction (NSFR) schemes are developed for three-dimensional compressible flow in curvilinear coordinates. NSFR is derived by merging the energy stable flux reconstruction (ESFR) framework with entropy stable DG schemes. NSFR is demonstrated to use larger time-steps than DG due to the ESFR correction functions, at the cost of larger error levels at design order convergence for equivalent degrees of freedom, while preserving discrete nonlinear stability. NSFR differs from ESFR schemes in the literature since it incorporates the FR correction functions on the volume terms through the use of a modified mass matrix. We also prove that discrete kinetic energy stability cannot be preserved to machine precision for quadrature rules where the surface quadrature is not a subset of the volume quadrature. This result stems from the inverse mapping from the kinetic energy variables to the conservative variables not existing for the kinetic energy projected variables. This paper also presents the NSFR modified mass matrix in a weight-adjusted form. This form reduces the computational cost in curvilinear coordinates because the dense matrix inversion is approximated by a pre-computed projection operator and the inverse of a diagonal matrix on-the-fly and exploits the tensor product basis functions to utilize sum-factorization. The nonlinear stability properties of the scheme are verified on a nonsymmetric curvilinear grid for the inviscid Taylor-Green vortex problem and the correct orders of convergence were obtained on a curvilinear mesh for a manufactured solution. Lastly, we perform a computational cost comparison between conservative DG, overintegrated DG, and our proposed entropy conserving NSFR scheme, and find that our proposed entropy conserving NSFR scheme is computationally competitive with the conservative DG scheme.

On the Origin of Görtler Vortices in Flow over a Multi-Element Airfoil

The flow characteristics of a 30P30N three-element high-lift airfoil at low Reynolds numbers O104 are examined through three-dimensional simulations using a high-order spectral element method. This study primarily investigates the flow structures of the slat cove and Görtler vortices formed on the upper surface of the main airfoil. Spanwise instability grows exponentially in the slat cove with a constant wavelength, corresponding to that of the subsequently formed Görtler vortices. Görtler number calculations show that curvature-induced centrifugal instability at the slat cusp leads to the subsequent formation of Görtler vortices. Proper orthogonal decomposition (POD) is used to analyze the development of flow structures in the slat cove in different time ranges. At early time, the flow in the slat cove is dominated by shear layers that evolve into spanwise perturbations. These perturbations further evolve into distinct bell-shaped structures close to the slat cusp and are advected to the upper surface of the main airfoil, leading to the formation of Görtler vortices.

Numerical research of functionally graded magneto-electro-elastic structures under thermal environment via the thermo-mechanical-electro-magnetic enriched finite element method

Abstract In this work, the functionally graded magneto-electro-elastic (FG-MEE) structures were used to study mechanical performance under a thermal environment using the thermo-mechanical-electro-magnetic enriched finite element method (TMEM-EFEM). Through enhancing the traditional solution space of the finite element method (FEM), additional degrees of freedom are introduced without modifying the mesh structure. Therefore, as a class of higher-order finite element methods, field variables with high gradients can be captured without a relatively finer mesh. The accuracy of TMEM-EFEM is improved by using interpolation covers. The feasibility of TMEM-EFEM is proven by numerical examples. The proposed TMEM-EFEM is proven to be accurate under the relatively coarse mesh. Through numerical examples, the high accuracy of TMEM-EFEM is demonstrated. Therefore, TMEM-EFEM can be a good alternative for the statics study of FG-MEE structures.

Physics-informed neural networks and higher-order high-resolution methods for resolving discontinuities and shocks: A comprehensive study

Addressing discontinuities in fluid flow problems is inherently difficult, especially when shocks arise due to the nonlinear nature of the flow. While handling discontinuities is a well-established practice in computational fluid dynamics (CFD), it remains a major challenge when applying physics-informed neural networks (PINNs). In this study, we compare the shock-resolving capabilities of traditional CFD methods with those of PINNs, highlighting the advantages of the latter. Our findings show that PINNs exhibit less dissipative behavior compared to conventional techniques. We evaluated the performance of both PINNs and traditional methods on linear and nonlinear test cases, demonstrating that PINNs offer superior shock-resolving properties. Notably, PINNs can accurately resolve inviscid shocks with just three grid points, whereas traditional methods require at least seven points. This suggests that PINNs are more effective at resolving shocks and discontinuities when using the same grid for both PINN and CFD simulations. However, it’s important to note that PINNs, in this context, are computationally more expensive than traditional methods on a given grid.

High-Fidelity OC-Seislet Stacking Method for Low-SNR Seismic Data

Seismic stacking is a core technique in seismic data processing, aimed at enhancing the signal-to-noise ratio (SNR) of data by utilizing seismic data acquisition with multifold geometry. Traditional stacking methods always have certain limitations, such as the reliance on the accuracy of velocity analysis for dip moveout (DMO) in common midpoint (CMP) stacking. The seislet transform, a compression technique tailored to nonstationary seismic data, can compress and stack along the prediction direction of seismic data, which provides a new technical idea for high-fidelity seismic imaging based on the effectiveness of the compression. This paper introduces a high-order OC-seislet stacking method for low-SNR seismic data, capable of achieving the high-fidelity stacking of reflection and diffraction waves simultaneously. With the multi-scale analysis advantages of the seislet transform, this method addresses the dependency of DMO stacking on velocity analysis accuracy. In the frequency–wavenumber–scale domain, the correction compensation of the high-order CDF 9/7 basis function is used to obtain the compression coefficients of the high-order OC-seislet transform. This approach simultaneously stacks frequency–wavenumber information of reflection and diffraction waves with high fidelity while implementing DMO processing. After normalizing the weighting coefficients and applying soft thresholding for denoising, the final result is transformed back to the original time–space domain, yielding high-fidelity stacking sections. The results of applying this method to both synthetic and field data show that, compared with conventional DMO stacking methods, the high-order OC-seislet stacking technique reasonably represents dipping layers and fault amplitudes, and it can achieve a balance of a high SNR and high fidelity in stacked profiles.

Research on Nanometer Precision Measurement Method of High Order Even Aspheres

Optical aspheres are demanded with extremely high precision to meet functional requirements in space telescopes, extreme ultraviolet lithography, and other modern large optical systems. The nano-precision fabrication of optical aspheres requires high-precision measurements to guide deterministic optical processing. Null test is the preferred method for high-precision measurements. Null optics are required to compensate for the incident wavefront in the null test of optical aspheres. However, wavefront aberrations caused by the transmission flat or transmission sphere of interferometer and null optics can limit measurement accuracy and need to be separated. A nano-precision measurement method is proposed for the even optical aspheres of high order in this paper. A computer-generated hologram is used as a null optic to realize a null test on optical aspheres. Mapping distortion correction is performed on the measurement results to ensure that the transverse coordinates of the measurement results correspond correctly to those of the test surface. Absolute testing is applied to separate the wavefront aberrations caused by the computer-generated hologram and interferometer optics. Finally, the results obtained by this method were used to guide deterministic optical processing, enabling the nano-precision fabrication of optical aspheres.

Mathematical Challenges for the Theory of Hyperbolic Balance Laws in Fluid Mechanics: Complexity, Scales, Randomness

AbstractUnderstanding the dynamics of hyperbolic balance laws is of paramount interest in the realm of fluid mechanics. Nevertheless, fundamental questions on the analysis and the numerics for distinctive hyperbolic features related to turbulent flow motion remain vastly open. Recent progress on the mathematical side reveals novel routes to face these concerns. This includes findings about the failure of the entropy principle to ensure uniqueness, the use of structure-preserving concepts in high-order numerical methods, and the advent of tailored probabilistic approaches. Whereas each of these three directions on hyperbolic modelling are of completely different origin they are all linked to small- or subscale features in the solutions which are either enhanced or depleted by the hyperbolic nonlinearity. Thus, any progress in the field might contribute to a deeper understanding of turbulent flow motion on the basis of the continuum-scale mathematical models. We present an overview on the mathematical state-of-the-art in the field and relate it to the scientific work in the DFG Priority Research Programme 2410. As such, the survey is not necessarily targeting at readers with comprehensive knowledge on hyperbolic balance laws but instead aims at a general audience of reseachers which are interested to gain an overview on the field and associated challenges in fluid mechanics.

A high-order isogeometric vibrational method for adaptive analysis of athlete diving equipment using advanced kinematic and geometric modeling

Isogeometric high-order analysis of a sandwich beam is presented in this article. The kinematic and geometric modeling is applied using the higher order modeling based on the stretchable models. The suggested model in this article can be applied for the sport equipment specially for analysis of diving beams boards and beams. The Variational-based approach is extended to derive governing equations. The results are analytically obtained in order to seek impact of significant parameters of the problem on the responses. The results are confirmed through comparison with available ones in the literature.

High-order finite-difference ghost-point methods for elliptic problems in domains with curved boundaries

Abstract In this article, a fourth-order finite-difference ghost-point method for the Poisson equation on regular Cartesian mesh is presented. The method can be considered the high-order extension of the second-order ghost method introduced earlier by the authors. Three different discretizations are considered, which differ in the stencil that discretizes the Laplacian and the source term. It is shown that only two of them provide a stable method. The accuracy of such stable methods is numerically verified on several test problems.

An energy stable high-order cut cell discontinuous Galerkin method with state redistribution for wave propagation

Cut meshes are a type of mesh that is formed by allowing embedded boundaries to “cut” a simple underlying mesh resulting in a hybrid mesh of cut and standard elements. While cut meshes can allow complex boundaries to be represented well regardless of the mesh resolution, their arbitrarily shaped and sized cut elements can present issues such as the small cell problem, where small cut elements can result in a severely restricted CFL condition. State redistribution, a technique developed by Berger and Giuliani in [1], can be used to address the small cell problem. In this work, we pair state redistribution with a high-order discontinuous Galerkin scheme that is L2 energy stable under arbitrary quadrature. We prove that state redistribution can be added to a provably L2 energy stable discontinuous Galerkin method on a cut mesh without damaging the scheme's L2 stability. We numerically verify the high order accuracy and stability of our scheme on two-dimensional wave propagation problems.

A unified approach to the construction of higher-order derivative-free iterative methods for solving systems of nonlinear equations

In this article, we introduce a unified approach to constructing a higher-order derivative-free scheme based on the approximations of F'(zk)-1. A family of order p=6,7 derivative-free method is proposed and compared to some well-known methods. The necessary and sufficient condition for p-th order of convergence are given in terms of parameter matrices τ(k) and α(k) . Some good choices of and are offered. Numerical experiments were carried out to confirm the theoretical results.

A Nonlinear Approach in the Quantification of Numerical Uncertainty by High-Order Methods for Compressible Turbulence with Shocks

This is a comprehensive overview on our research work to link interdisciplinary modeling and simulation techniques to improve the predictability and reliability simulations (PARs) of compressible turbulence with shock waves for general audiences who are not familiar with our nonlinear approach. This focused nonlinear approach is to integrate our “nonlinear dynamical approach” with our “newly developed high order entropy-conserving, momentum-conserving and kinetic energy-preserving methods” in the quantification of numerical uncertainty in highly nonlinear flow simulations. The central issue is that the solution space of discrete genuinely nonlinear systems is much larger than that of the corresponding genuinely nonlinear continuous systems, thus obtaining numerical solutions that might not be solutions of the continuous systems. Traditional uncertainty quantification (UQ) approaches in numerical simulations commonly employ linearized analysis that might not provide the true behavior of genuinely nonlinear physical fluid flows. Due to the rapid development of high-performance computing, the last two decades have been an era when computation is ahead of analysis and when very large-scale practical computations are increasingly used in poorly understood multiscale data-limited complex nonlinear physical problems and non-traditional fields. This is compounded by the fact that the numerical schemes used in production computational fluid dynamics (CFD) computer codes often do not take into consideration the genuinely nonlinear behavior of numerical methods for more realistic modeling and simulations. Often, the numerical methods used might have been developed for weakly nonlinear flow or different flow types other than the flow being investigated. In addition, some of these methods are not discretely physics-preserving (structure-preserving); this includes but is not limited to entropy-conserving, momentum-conserving and kinetic energy-preserving methods. Employing theories of nonlinear dynamics to guide the construction of more appropriate, stable and accurate numerical methods could help, e.g., (a) delineate solutions of the discretized counterparts but not solutions of the governing equations; (b) prevent numerical chaos or numerical “turbulence” leading to FALSE predication of transition to turbulence; (c) provide more reliable numerical simulations of nonlinear fluid dynamical systems, especially by direct numerical simulations (DNS), large eddy simulations (LES) and implicit large eddy simulations (ILES) simulations; and (d) prevent incorrect computed shock speeds for problems containing stiff nonlinear source terms, if present. For computation intensive turbulent flows, the desirable methods should also be efficient and exhibit scalable parallelism for current high-performance computing. Selected numerical examples to illustrate the genuinely nonlinear behavior of numerical methods and our integrated approach to improve PARs are included.

Novel conformal structure-preserving schemes for the linearly damped nonlinear Schrödinger equation

In this work, by utilizing the Strang splitting technique, some advanced conformal structure-preserving schemes are developed for solving the damped nonlinear Schrödinger equation (DNLSE). The proposed innovative numerical approaches, namely the high-order compact conformal multi-symplectic method and the conformal momentum-preserving method, have two key computational advantages. Firstly, these two approaches excel in conserving local structures, and more importantly, they are capable of maintaining exact energy dissipation rates in any time-space region, particularly under periodic boundary conditions. To validate the theoretical properties and demonstrate the efficacy and stability of these approaches in long-time integrations, we conduct through extensive numerical simulations involving both bright and dark solitons.

A fault location method for DC transmission line based on the distributed parameter model considering line parameter uncertainty

Addressing the impact of whether line parameters can be accurately obtained and the errors due to environmental influences on the accuracy of fault location for DC transmission line, this paper proposes a fault location method based on the distributed parameter model that accounts for uncertainty in line parameter. To address the challenge of accurately obtaining DC transmission line parameters, a high-order derivative-based parameter estimation method is introduced to determine unknown equivalent parameters of the DC transmission line. Additionally, an optimization algorithm is incorporated to dynamically correct parameter errors, thereby enhancing parameter stability and constructing an accurate line model. Based on this, theoretical derivations confirm the applicability of the Tellegen's quasi-power theorem in the normal operation and internal fault equivalent models of DC transmission line. A fault location function is established based on the relationship between the line's Tellegen quasi-power characteristic and fault distance. The proposed method's effectiveness and accuracy are validated using a DC system model built on the PSCAD/EMTDC simulation platform. Simulation results indicate that the proposed method can locate faults under different fault conditions.

Higher-order stochastic averaging for investigating a vehicle suspension system with nonlinear damping and stiffness

The paper deals with a quarter-car model with nonlinear damping and stiffness under white noise base excitation using the higher-order stochastic averaging method for analyzing approximate responses. Recently, a novel higher-order averaging procedure has been developed to find analytically the first-, second-, and third-order stationary joint probability density functions (PDF) of amplitude and full phase by solving the corresponding Fokker-Planck-Kolmogorov (FPK) equation, and it will be extended to the nonlinear quarter-car model. Accordingly, the mean-square responses such as the displacement, and velocity of the sprung mass can be obtained analytically. The influences of excitation intensity on the dynamic responses, as well as the variations of linear and nonlinear damping, are analyzed. A very satisfactory agreement is found between the accuracy of the solutions corresponding to higher-order stochastic averaging and that of Monte Carlo simulation.

A Sub-1 ppm/°C Reference Voltage Source with a Wide Input Range

With the continuous advancement of electronic technology, the application of high-voltage integrated circuits is becoming increasingly prevalent in fields such as power systems, medical devices, and industrial automation. The reference circuit within high-voltage integrated circuits must not only exhibit insensitivity to temperature variations but also maintain stability across a broad voltage supply. This paper presents a bandgap reference (BGR) source capable of operating over a wide input range. This BGR employs a high-order curvature compensation method to eliminate nonlinear voltage terms, resulting in minimal temperature drift. The circuit achieves an impressive temperature coefficient (TC) of 0.88 ppm/°C over a temperature range from −40 °C to 130 °C. To ensure stable operation within a 4–40 V range, the design incorporates a pre-regulation circuit that stabilizes the supply voltage of the BGR core at a fixed value, thereby enhancing the ability to withstand variations in power supply voltage.

Physics-informed neural networks for efficient aeroacoustic computational based on two time sequence prediction models

Physics-informed neural networks (PINNs) have been proven powerful for solving partial differential equations (PDEs). However, conventional PINNs fail handling time/space variables in the training data distinctly and overlook the temporal relationships among data points, which is critical for accurately modeling dynamic systems. To address this issue, this work proposes two novel PINN models based on the transformer and the convolutional long short-term memory (ConvLSTM) architectures, respectively. An encoder-decoder neural network structure is adopted to enhance handling of time-space variables and a high-order finite-difference method is used to compute partial derivatives. These computations are implemented using the convolution within the PyTorch framework that incorporates both physical and mathematical information into the neural networks. The efficacy of the proposed methods has been evaluated for the wave equation and the Reynolds-Averaged Navier-Stokes equations. Results suggest that the transformer-based PINN outperforms both the conventional and ConvLSTM-based PINN in terms of accuracy, efficiency and generalizability.

Flux reconstruction approach for long-time calculations of the linearized Euler equation

The Linearized Euler Equations (LEE) are one of the fundamental governing equations in acoustics research. Due to the relatively small magnitude of acoustic disturbances compared to the physical quantities of the background flow, ensuring highly precise solutions without dissipation and dispersion over time is essential. The Flux Reconstruction (FR) method, recognized as a high-order numerical method with flexible construction capabilities, has been extensively utilized in recent years for solving partial differential equations numerically. This paper introduces an offsetting numerical flux for one-dimensional LEE. Subsequently, it develops a flux reconstruction scheme with energy-conserving characteristics, referred to as the ECFR scheme, aimed at achieving high-precision numerical solutions for prolonged computations. The energy-conserving feature of the scheme is substantiated through theoretical derivation and further corroborated by numerical validation. Moreover, the impact of Mach numbers and the parameter of the offsetting flux on the scheme's performance is discussed. The stability of the scheme is also analyzed, with the limit of the CFL number being provided. The novel ECFR scheme facilitates obtaining high-precision acoustic numerical solutions, thereby providing strong support for the study of complex engineering problems such as aircraft, submarine, and automobile noise.