High-order Accuracy Research Articles

Ensemble machine learning using hydrometeorological information to improve modeling of quality parameter of raw water supplying treatment plants

Source and raw water quality may deteriorate due to rainfall and river flow events that occur in watersheds. The effects on raw water quality are normally detected in drinking water treatment plants (DWTPs) with a time-lag after these events in the watersheds. Early warning systems (EWSs) in DWTPs require models with high accuracy in order to anticipate changes in raw water quality parameters. Ensemble machine learning (EML) techniques have recently been used for water quality modeling to improve accuracy and decrease variance in the outcomes. We used three decision-tree-based EML models (random forest [RF], gradient boosting [GB], and eXtreme Gradient Boosting [XGB]) to predict two critical parameters for DWTPs, raw water Turbidity and UV absorbance (UV254), using rainfall and river flow time series as predictors. When modeling raw water turbidity, the three EML models (rRF−Tu2=0.87, rGB−Tu2=0.80 and rXGB−Tu2=0.81) showed very good performance metrics. For raw water UV254, the three models (rRF−UV2=0.89, rGB−UV2=0.85 and rXGB−UV2=0.88) again showed very good performance metrics. Results from this study suggest that EML approaches could be used in EWSs to anticipate changes in the quality parameters of raw water and enhance decision-making in DWTPs.

Finite Elements with Switch Detection for direct optimal control of nonsmooth systems

This paper introduces Finite Elements with Switch Detection (FESD), a numerical discretization method for nonsmooth differential equations. We consider the Filippov convexification of these systems and a transformation into dynamic complementarity systems introduced by Stewart (Numer Math 58(1):299–328, 1990). FESD is based on solving nonlinear complementarity problems and can automatically detect nonsmooth events in time. If standard time-stepping Runge–Kutta (RK) methods are naively applied to a nonsmooth ODE, the accuracy is at best of order one. In FESD, we let the integrator step size be a degree of freedom. Additional complementarity conditions, which we call cross complementarities, enable exact switch detection, hence FESD can recover the high order accuracy that the RK methods enjoy for smooth ODE. Additional conditions called step equilibration allow the step size to change only when switches occur and thus avoid spurious degrees of freedom. Convergence results for the FESD method are derived, local uniqueness of the solution and convergence of numerical sensitivities are proven. The efficacy of FESD is demonstrated in several simulation and optimal control examples. In an optimal control problem benchmark with FESD, we achieve up to five orders of magnitude more accurate solutions than a standard time-stepping approach for the same computational time.

Approximation of piecewise smooth functions by nonlinear bivariate C2 quartic spline quasi-interpolants on criss-cross triangulations

In this paper we focus on the space of C2 quartic splines on uniform criss-cross triangulations and we propose a method based on weighted essentially non-oscillatory techniques and obtained by modifying classical spline quasi-interpolants in order to approximate piecewise smooth functions avoiding Gibbs phenomenon near discontinuities and, at the same time, maintaining the high-order accuracy in smooth regions. We analyse the convergence properties of the proposed quasi-interpolants and we provide some numerical and graphical tests confirming the theoretical results.

Finite Difference Method for Solving Second-Order Boundary Value Problems with High-Order Accuracy

When researching and solving practical problems in continuous environments, through modeling methods, the vast majority of problems lead to models described by equations containing differential operators. In case the problem model is not complicated, we often obtain simple partial differential equations, then the solution of the problem can be obtained directly through analytical methods. Most complex problems, through the method of approximating differential operators by difference operators, from which differential problems are approximated by corresponding difference schemes and approximate solutions will be obtained. is achieved through solving systems of difference equations based on the tools of electronic computers. Then, building difference schemes to approximate the differential problem with high-order accuracy will play an important role in the accuracy of the obtained approximate solution. In this paper, we propose two difference schemes with high-order accuracy to solve second-order differential problems with Dirichlet and mixed boundary conditions. Theoretical results and experimental calculations have confirmed the accuracy of the proposed schemes.

Fast, high-order numerical evaluation of volume potentials via polynomial density interpolation

This article presents a high-order accurate numerical method for the evaluation of singular volume integral operators, with attention focused on operators associated with the Poisson and Helmholtz equations in two dimensions. Following the ideas of the density interpolation method for boundary integral operators, the proposed methodology leverages Green's third identity and a local polynomial interpolant of the density function to recast the volume potential as a sum of single- and double-layer potentials and a volume integral with a regularized (bounded or smoother) integrand. The layer potentials can be accurately and efficiently evaluated everywhere in the plane by means of existing methods (e.g. the density interpolation method), while the regularized volume integral can be accurately evaluated by applying elementary quadrature rules. Compared to straightforwardly computing corrections for every singular and nearly-singular volume target, the method significantly reduces the amount of required specialized quadrature by pushing all singular and near-singular corrections to near-singular layer-potential evaluations at target points in a small neighborhood of the domain boundary. Error estimates for the regularization and quadrature approximations are provided. The method is compatible with well-established fast algorithms, being both efficient not only in the online phase but also to set-up. Numerical examples demonstrate the high-order accuracy and efficiency of the proposed methodology; applications to inhomogeneous scattering are presented.

A high order cut-cell method for solving the shallow-shelf equations

In this paper we present a novel method for solving the shallow-shelf equations in the presence of grounding lines. The shallow-self equations are a two-dimensional system of nonlinear elliptic PDEs with variable coefficients that are discontinuous across the grounding line, which we treat as a sharp interface between grounded and floating ice. The grounding line is “reconstructed” from ice thickness and basal topography data to provide necessary geometric information for our cut-cell, finite volume discretization. Our discretization enforces jump conditions across the grounding line and achieves high-order accuracy using stencils constructed with a weighted least-squares method. We demonstrate second and fourth order convergence of the velocity field, driving stress, and reconstructed geometric information.

A Long-Term Video Tracking Method for Group-Housed Pigs

Pig tracking provides strong support for refined management in pig farms. However, long and continuous multi-pig tracking is still extremely challenging due to occlusion, distortion, and motion blurring in real farming scenarios. This study proposes a long-term video tracking method for group-housed pigs based on improved StrongSORT, which can significantly improve the performance of pig tracking in production scenarios. In addition, this research constructs a 24 h pig tracking video dataset, providing a basis for exploring the effectiveness of long-term tracking algorithms. For object detection, a lightweight pig detection network, YOLO v7-tiny_Pig, improved based on YOLO v7-tiny, is proposed to reduce model parameters and improve detection speed. To address the target association problem, the trajectory management method of StrongSORT is optimized according to the characteristics of the pig tracking task to reduce the tracking identity (ID) switching and improve the stability of the algorithm. The experimental results show that YOLO v7-tiny_Pig ensures detection applicability while reducing parameters by 36.7% compared to YOLO v7-tiny and achieving an average video detection speed of 435 frames per second. In terms of pig tracking, Higher-Order Tracking Accuracy (HOTA), Multi-Object Tracking Accuracy (MOTP), and Identification F1 (IDF1) scores reach 83.16%, 97.6%, and 91.42%, respectively. Compared with the original StrongSORT algorithm, HOTA and IDF1 are improved by 6.19% and 10.89%, respectively, and Identity Switch (IDSW) is reduced by 69%. Our algorithm can achieve the continuous tracking of pigs in real scenarios for up to 24 h. This method provides technical support for non-contact pig automatic monitoring.

Error analysis for local discontinuous Galerkin semidiscretization of Richards’ equation

Abstract This paper concerns an error analysis of the space semidiscrete scheme for the Richards’ equation modeling flows in variably saturated porous media. This nonlinear parabolic partial differential equation can degenerate; namely, we consider the case where the time derivative term can vanish, i.e., the fast-diffusion type of degeneracy. We discretize the Richards’ equation by the local discontinuous Galerkin method, which provides high order accuracy and preserves stability. Due to the nonlinearity of the problem, special techniques for numerical analysis of the scheme are required. In particular, we combine two partial error bounds using continuous mathematical induction and derive a priori error estimates with respect to the spatial discretization parameter and the Hölder coefficient of the nonlinear temporal derivative. Finally, the theoretical results are supported by numerical experiments, including cases beyond the assumptions of the theoretical results.

Direct/split invariant-preserving Fourier pseudo-spectral methods for the rotation-two-component Camassa–Holm system with peakon solitons

The Fourier pseudo-spectral method is well suited to solve PDEs under the periodic boundary condition due to its high-order accuracy and easy-to-implement feature. In this paper, we explore as well as comparatively study four classes of Fourier pseudo-spectral schemes for solving the rotation-two-component Camassa–Holm system which possibly owns peakon solitons. Via exploiting inherent structural properties of the system, we reformulate it into two kinds of different equivalent forms and then apply the Fourier pseudo-spectral method to derive two spatial semi-discrete systems, both of which are proved to preserve the corresponding invariants including mass, momentum and energy. Subsequently, we construct two linearly implicit schemes based on Strang splitting technique and two nonlinear schemes, respectively, for both semi-discrete systems. Owing to the different equivalent forms in the structure, one of the nonlinear schemes preserves discrete mass and momentum, while the other one is shown to preserve all three invariants. Numerical results under the situation of smooth/nonsmooth initial values are provided for distinct types of solutions to test the accuracy in long time simulation and to verify the capacity of predicting water wave propagation, as well as advantages in preserving these invariants. For instance, the present schemes are shown to be at least 14 significant digits, improving upon 10 from ones in previous references.

Sparse-grid discontinuous Galerkin methods for the Vlasov–Poisson–Lenard–Bernstein model

Sparse-grid methods have recently gained interest in reducing the computational cost of solving high-dimensional kinetic equations. In this paper, we construct adaptive and hybrid sparse-grid methods for the Vlasov–Poisson–Lenard–Bernstein (VPLB) model. This model has applications to plasma physics and is simulated in two reduced geometries: a 0x3v space homogeneous geometry and a 1x3v slab geometry. We use the discontinuous Galerkin (DG) method as a base discretization due to its high-order accuracy and ability to preserve important structural properties of partial differential equations. We utilize a multiwavelet basis expansion to determine the sparse-grid basis and the adaptive mesh criteria. We analyze the proposed sparse-grid methods on a suite of three test problems by computing the savings afforded by sparse-grids in comparison to standard solutions of the DG method. The results are obtained using the adaptive sparse-grid discretization library ASGarD.

High-order accurate transient and free-vibration analysis of plates and shells

The limited availability of analytical solutions and the high cost associated with experimental testing motivate the use of computational tools to assess the dynamic behavior of load-bearing components, especially when a wide design space must be explored, as is often the case with composite structures. In this context, a novel high-order accurate discontinuous Galerkin formulation for transient and free-vibration analysis of multilayered plates and shells is presented and numerical validated. The starting point of the formulation is a generalized structural theory for multilayered shells with arbitrary curvature based on the expansion of the displacement covariant components throughout the shell thickness. The variational statement of three-dimensional elastodynamics allows deriving the strong form of the governing differential equations, which form the basis to obtain the corresponding discontinuous Galerkin weak statements. As the order of the through-the-thickness expansion and the order of the discontinuous Galerkin basis functions are free parameters, the proposed approach allows tuning the order of accuracy of the computed solution throughout both the shell thickness and the shell modeling domain. Numerical results are reported and discussed for several validation test cases in terms of h- and p-convergence analyses, demonstrating the high-order accuracy, robustness, and computational savings of the formulation.

A bound- and positivity-preserving discontinuous Galerkin method for solving the γ-based model

In this work, a bound- and positivity-preserving quasi-conservative discontinuous Galerkin (DG) method is proposed for the γ-based model of compressible two-medium flows. The contribution of this paper mainly includes three parts. On one hand, the DG method with the extended Harten-Lax-van Leer contact flux is proposed to solve the γ-based model, and satisfies the equilibrium-preserving property which preserves uniform velocity and pressure fields at an isolated material interface. On the other hand, an affine-invariant weighted essentially non-oscillatory (Ai-WENO) limiter is adopted to suppress oscillations near the discontinuities. The limiter with the Ai-WENO reconstruction method to the conservative variables not only is able to maintain the equilibrium property, but also generates sharper results around the locations of shock waves in contrast to that applying to the primitive variables. Last but not least, a flux-based bound- and positivity-preserving limiting strategy is introduced and analyzed, which preserves the physical bounds for auxiliary variables in the non-conservative governing equations, and the positivity for density and internal energy. Extensive numerical experiments in both one and two space dimensions show that the proposed method performs well in simulating compressible two-medium flows with high-order accuracy, equilibrium-preserving and bound-preserving properties.

A multi-order candidates weighting framework for discontinuity capturing of hyperbolic conservation laws

Weighted compact nonlinear schemes (WCNSs) represent a popular family of high-order methods used for simulating compressible flows. Within these methods, the nonlinear interpolation process is crucial for the scheme's ability to capture shocks. In this study, we introduce a generalized weighting framework that utilizes candidates of varying orders, named multi-order candidates (MOC) weighting, to design nonlinear schemes capable of capturing discontinuities. Analyzing within the MOC framework reveals that the Z nonlinear weights are equivalent to the JS nonlinear weights, supplemented by an additional fifth-order candidate. The convergence accuracy of the MOC weighting is examined, indicating that higher-order candidates in the MOC weighting require smoothness indicators with higher-order leading terms for better accuracy. Based on this convergence analysis, consistently high-order smoothness indicators (CHOSI) are proposed for the MOC weighting, which are essential for achieving the designed high-order accuracy and high resolution. It is found that many existing nonlinear weighting methods can be integrated into the MOC framework. Three of these methods are analyzed and further improved within the MOC framework. Finally, the newly proposed schemes are tested across a range of benchmark problems to showcase their enhanced performance, also illustrating the MOC framework's capability in developing innovative schemes.

Positivity-preserving discontinuous Galerkin scheme for linear hyperbolic equations with characteristics-informed augmentation

This paper presents a positivity-preserving discontinuous Galerkin (DG) scheme for the linear hyperbolic problem with variable coefficients on structured Cartesian domains. The standard DG spaces are augmented with either polynomial or non-polynomial basis functions. The primary purpose of these augmented basis functions is to ensure that the cell average from the unmodulated DG scheme remains positive. We explicitly obtain suitable basis functions by inspecting the method of characteristics on an auxiliary problem. A key result is proved which demonstrates that the unmodulated augmented DG scheme will retain a positive cell average, provided that the inflow, source term, and variable coefficients are positive. A simple scaling limiter can then be leveraged to produce a high-order conservative positivity-preserving DG scheme. Numerical experiments demonstrate the scheme is able to retain high-order accuracy as well as robustness for variable coefficients. To improve efficiency, an inexact augmented basis function can be obtained rather than a analytic non-polynomial solution to the auxiliary problem from the method of characteristics.

A Time-Domain Wavenumber Integration Model for Underwater Acoustics Based on the High-Order Finite Difference Method

Simulating the acoustic field excited by pulse sound sources holds significant practical value in computational ocean acoustics. Two primary methods exist for modeling underwater acoustic propagation in the time domain: the Fourier synthesis technique based on frequency decomposition and the time-domain underwater acoustic propagation model (TD-UAPM). TD-UAPMs solve the wave equation in the time domain without requiring frequency decomposition, providing a more intuitive explanation of the physical process of sound energy propagation over time. However, time-stepping numerical methods can accumulate numerical errors, making it crucial to improve the algorithm’s accuracy for TD-UAPMs. Herein, the time-domain wavenumber integration model SPARC was improved by replacing the second-order finite element method (FEM) with the high-order accuracy finite difference method (FDM). Furthermore, the matched interface and boundary (MIB) method was used to process the seabed more accurately. The improved model was validated using three classic underwater acoustic benchmarks. By comparing the acoustic solutions obtained using the FDM and the FEM, it is evident that the improved model requires fewer grid points while maintaining the same level of accuracy, leading to lower computational costs and faster processing compared to the original model.

Application of an Extended Cubic B-Spline to Find the Numerical Solution of the Generalized Nonlinear Time-Fractional Klein–Gordon Equation in Mathematical Physics

A B-spline function is a series of flexible elements that are managed by a set of control points to produce smooth curves. By using a variety of points, these functions make it possible to build and maintain complicated shapes. Any spline function of a certain degree can be expressed as a linear combination of the B-spline basis of that degree. The flexibility, symmetry and high-order accuracy of the B-spline functions make it possible to tackle the best solutions. In this study, extended cubic B-spline (ECBS) functions are utilized for the numerical solutions of the generalized nonlinear time-fractional Klein–Gordon Equation (TFKGE). Initially, the Caputo time-fractional derivative (CTFD) is approximated using standard finite difference techniques, and the space derivatives are discretized by utilizing ECBS functions. The stability and convergence analysis are discussed for the given numerical scheme. The presented technique is tested on a variety of problems, and the approximate results are compared with the existing computational schemes.

A new type of modified MR-WENO schemes with new troubled cell indicators for solving hyperbolic conservation laws in multi-dimensions

In this paper, a new type of increasingly high-order modified multi-resolution weighted essentially non-oscillatory (MMR-WENO) schemes with new troubled cell indicators is designed in the finite difference framework for solving hyperbolic conservation laws in one, two, and three dimensions. It is a first time to design new troubled cell indicators that based on two high-degree reconstruction polynomials, which are defined on the three-point, five-point, seven-point, and nine-point spatial stencils, respectively. Such new troubled cell indicators can automatically identify the discontinuous solutions without manually adjusting the parameters related to the problems. Subsequently, a series of MMR-WENO schemes are designed by using these new troubled cell indicators, which use the simple linear upwind schemes in smooth areas and the sophisticated MR-WENO schemes in discontinuous areas, thus achieving the goal of inheriting the excellent characteristics of original MR-WENO schemes while reducing computational costs. The new modified methodology is divided into two parts: if all extreme points of two reconstruction polynomials defined on the big spatial stencil are outward the smallest interval [xi−1/2,xi+1/2], the numerical flux is straightforwardly approximated by a high-degree reconstruction polynomial and the approximation has a high-order accuracy. Otherwise, the high-order MR-WENO spatial reconstruction procedures are adopted. The main benefits of new MMR-WENO schemes are their robustness and efficiency in comparison to original MR-WENO schemes, since the MMR-WENO schemes could save at most 63%-79% CPU time than the same-order MR-WENO schemes do for some numerical examples.

Nektar++: Development of the compressible flow solver for jet aeroacoustics

A recently developed computational framework for jet noise predictions is presented. The framework consists of two main components, focusing on source prediction and noise propagation. To compute the noise sources, the turbulent jet is simulated using the compressible flow solver implemented in the open-source spectral/hp element framework Nektar++, which solves the unfiltered Navier-Stokes equations on unstructured grids using the high-order discontinuous Galerkin method. This allows high-order accuracy to be achieved on unstructured grids, which in turn is important in order to accurately simulate industrially relevant geometries. For noise propagation, the Ffowcs Williams - Hawkings method is used to propagate the noise between the jet and the far-field. The paper provides a detailed description of the computational framework, including how the different components fit together and how to use them. To demonstrate the framework, two configurations of a single stream subsonic jet are considered. In the first configuration, the jet is treated in isolation, whereas in the second configuration, it is installed under a wing. The aerodynamic results for these two jets show strong agreement with experimental data, while some discrepancies are observed in the acoustic results, which are discussed. In addition to this, we demonstrate close to linear scaling beyond 100,000 processors on the ARCHER2 supercomputer.

Developing Higher-Order Unconditionally Positive Finite Difference Methods for the Advection Diffusion Reaction Equations

This study introduces the higher-order unconditionally positive finite difference (HUPFD) methods to solve the linear, nonlinear, and system of advection–diffusion–reaction (ADR) equations. The stability and consistency of the developed methods are analyzed, which are necessary and sufficient for the numerical approach to converge to the exact solution. The problem under consideration is of the Cauchy type, and hence, Von Neumann stability analysis is used to analyze the stability of the proposed schemes. The HUPFD’s efficacy and efficiency are investigated by calculating the error, convergence rate, and computing time. For validation purposes, the higher-order unconditionally positive finite difference solutions are compared to analytical calculations. The numerical results demonstrate that the proposed methods produce accurate solutions to solve the advection diffusion reaction equations. The results also show that increasing the order of the unconditionally positive finite difference leads an implicit scheme that is conditionally stable and has a higher order of accuracy with respect to time and space.

Inverse Differential Quadrature Based Model for Static Behaviour of Variable Stiffness Curved Composite Beams

Recent advancements in fibre placement technologies have expanded the potential applications of variable stiffness curved composite beams in industries such as aerospace, automotive, and naval engineering. Accurate solution techniques for examining these beams, especially those composed of advanced composite materials, are indispensable. In view of this demand, this study proposes a new high-order computational tool that combines the higher-order accuracy of the emerging inverse differential quadrature method (iDQM) and the simple kinematics of the Timoshenko beam theory for efficient and accurate prediction of the static behaviour of both constant stiffness and variable stiffness curved beam structures. This novel application of iDQM to curved beam analysis is leveraged upon its excellent potential to mitigate differentiation-induced errors by using the so-called indirect approximation strategy. Simple procedures for implementing different orders of iDQM models are presented to analyse curved beam problems, and are independently benchmarked against closed-form Navier's solutions, as well as numerical solutions obtained through the differential quadrature method (DQM) and finite element method (FEM), demonstrating excellent spectral accuracy. Furthermore, the iDQM scheme offers outstanding potential in recovering transverse shear stress, achieving superior accuracy over lower-order FEM in approximating higher-order derivatives. Remarkably, iDQM predictions for variable stiffness curved beams exhibit satisfactory agreement with the Strong Unified Formulation, achieving over 98% computational efficiency. Finally, convergence analysis of iDQM solutions reveal up to three orders of improved accuracy and faster convergence rates compared to the DQM, constituting a new benchmark for curved beam analysis.