Numerical Errors Research Articles

A new formulation of the understeer gradient and stability margin to analyse vehicle steady-state cornering behaviour

The understeer gradient is a measure that describes a vehicle's cornering behaviour as either understeer, neutral, or oversteer. Several formulations of this measure have been developed, all of which are reviewed in this paper. It has been found that all of these formulations rely essentially on the lateral acceleration of the vehicle. While the understeer gradient provides a useful connection to vehicle stability, its reliance on lateral acceleration can result in numerical instability and error at high levels of lateral acceleration. To address this issue, a new formulation of the understeer gradient is proposed in this paper. This new formulation is based on vehicle speed and is suitable for all maneuvers. Unlike previous formulations, it does not rely on lateral acceleration or provide a direct connection to vehicle stability. To address the issue of vehicle stability, a new measure called the stability margin is also proposed in this paper. The stability margin provides a more precise measurement of a vehicle's stability than the understeer gradient alone. By combining the stability margin with the new formulation of the understeer gradient, a more comprehensive picture of a vehicle's cornering behaviour and stability can be obtained.

Probabilistic model for the gravitational wave signal from merging black holes

Parametrized models that predict the gravitational-wave (GW) signal from merging black holes are used to extract source properties from GW observations. The majority of research in this area has focused on developing methods capable of producing highly accurate, point estimate, predictions for the GW signal. A key element missing from every model used in the analysis of GW data is an estimate for how confident the model is in its prediction. This omission increases the risk of biased parameter estimation of source properties. Current strategies include running analyses with multiple models to measure systematic bias however, this fails to accurately reflect the true uncertainty in the models. In this work we develop a probabilistic extension to the phenomenological modeling workflow for nonspinning black holes and demonstrate that the model not only produces accurate point estimates for the GW signal but can be used to provide well-calibrated local estimates for its uncertainty. Our analysis highlights that there is a lack of numerical relativity (NR) simulations available at multiple resolutions which can be used to estimate their numerical error and implore the NR community to continue to improve their estimates for the error in NR solutions published. Waveform models that are not only accurate in their point-estimate predictions but also in their error estimates are a potential way to mitigate bias in GW parameter estimation of compact binaries due to unconfident waveform model extrapolations. Published by the American Physical Society 2024

Single-hydrophone-based passive localization for vertically moving targets in shallow water

In this article, we consider underwater passive localization for vertically moving targets in shallow water. A method that utilizes parameters derived from interference structures is introduced, which enables the implementation of both range and depth estimations for targets with a single hydrophone. Underwater moving targets radiate broadband continuous spectrum signals by forming multipath signals through interface reflections, resulting in stable interference structures in the time-frequency domain of the received signals. Ray theory is utilized to derive the trajectory equations of the interference fringes. Considering the periodic characteristics of interference structures, a specialized homomorphic filter is designed to effectively restore the peaks of the fringes to reduce localization errors. Furthermore, with a priori information on the target's initial depth, the Hough Transform is used to extract the target's location information within interference fringes formed by direct signal and surface-reflected signal. Addressing the issue of multi-peaks in parameter domain, parameter information across multiple fringes is fused. The effectiveness and robustness of the proposed method have been demonstrated through numerical simulations, lake trial, and error assessments, even when underwater environments information is not precisely known.

Fast and Reliable Computation of Instantaneous Orbital Collision Probability

Because of the increasing number of objects in low Earth orbit, the fast and reliable estimation of the collision risk is an important challenge for spacecraft owners/operators. Among the available risk indicators, this paper focuses on computing the instantaneous probability of collision, which can be modeled as the integral of a three-dimensional Gaussian probability density function over a Euclidean ball. The authors propose an efficient and accurate method for evaluating this integral. It is based on the combination of two complementary strategies. For the first one, convergent series and numerical error bounds are computed. These bounds provide a tradeoff between the accuracy needed and the number of terms to compute. The second one, using divergent series, approximates the value of the integral with a good accuracy in most cases with only a few terms computed. Based on those two methods, a hybrid algorithm is tested on cases borrowed from the literature and compared against existing methods. Several numerical results and comparisons confirm both the efficiency and robustness of this paper’s approach.

Convergence analysis of a second order numerical scheme for the Flory–Huggins–Cahn–Hilliard–Navier–Stokes system

We present an optimal rate convergence analysis for a second order accurate in time, fully discrete finite difference scheme for the Cahn–Hilliard–Navier–Stokes (CHNS) system, combined with logarithmic Flory–Huggins energy potential. The numerical scheme has been recently proposed, and the positivity-preserving property of the logarithmic arguments, as well as the total energy stability, have been theoretically justified. In this paper, we rigorously prove second order convergence of the proposed numerical scheme, in both time and space. Since the CHNS is a coupled system, the standard ℓ∞(0,T;ℓ2)∩ℓ2(0,T;Hh2) error estimate could not be easily derived, due to the lack of regularity to control the numerical error associated with the coupled terms. Instead, the ℓ∞(0,T;Hh1)∩ℓ2(0,T;Hh3) error analysis for the phase variable and the ℓ∞(0,T;ℓ2) analysis for the velocity vector, which shares the same regularity as the energy estimate, is more suitable to pass through the nonlinear analysis for the error terms associated with the coupled physical process. Furthermore, the highly nonlinear and singular nature of the logarithmic error terms makes the convergence analysis even more challenging, since a uniform distance between the numerical solution and the singular limit values of is needed for the associated error estimate. Many highly non-standard estimates, such as a higher order asymptotic expansion of the numerical solution (up to the third order accuracy in time and fourth order in space), combined with a rough error estimate (to establish the maximum norm bound for the phase variable), as well as a refined error estimate, have to be carried out to conclude the desired convergence result. To our knowledge, it will be the first work to establish an optimal rate convergence estimate for the Cahn–Hilliard–Navier–Stokes system with a singular energy potential.

A numerical B-spline Galerkin method with proper generalized decomposition for reduced order modeling of partial differential equations

A numerical study is presented on reduced order modeling of low and high-dimensional partial differential equations using a new B-spline Galerkin proper generalized decomposition (PGD) method. Galerkin-PGD schemes based on B-splines are numerically investigated for various benchmark problems, including the Poisson equation, linear/nonlinear advection–diffusion equations, and the Navier–Stokes equations. The treatment of inhomogeneous boundary conditions, non-separable terms and nonlinearities is numerically investigated. Numerical errors for high-fidelity B-spline solutions and reduced order solutions are provided to demonstrate the effectiveness and accuracy of the proposed method. Computational costs are reported to illustrate the efficiency of the developed approaches.

Enhancing cubic polynomial solutions: A comprehensive analysis and iterative refinement strategy in response to the versatility of cubic equations of state

The study presents a comprehensive evaluation of ten distinct methods for solving cubic equations, encompassing both analytical, numerical, and linear algebra-based approaches. Among these, an analytical technique involving the transformation of cubic equations into Chebyshev polynomials emerged as the most efficient. To mitigate issues arising from numerical round-off errors, the study proposes an iterative refinement strategy. Furthermore, the research introduces a novel criterion grounded in Vieta's formulas to identify and isolate areas where round-off errors may occur. This criterion aids in the application of the proposed iterative refinement technique. The novel solution method, which involves the conversion of cubic equations into Chebyshev polynomials and subsequent error-checking and iterative refinement, exhibits significant advantages. Firstly, it proves efficient, reducing computational time by 15–20 % in comparison to the well-established Cardano analytical method. It also demonstrates robustness in effortlessly identifying regions of potential error and offers a method for iterative improvement when needed. Additionally, its reliability is underscored by its independence from the specific cubic equation of state under consideration, making it well-suited for the intensive solution of a wide range of cubic equations.

APOST-3D: Chemical concepts from wavefunction analysis.

Open-source APOST-3D software features a large number of wavefunction analysis tools developed over the past 20 years, aiming at connecting classical chemical concepts with the electronic structure of molecules. APOST-3D relies on the identification of the atom in the molecule (AIM), and several analysis tools are implemented in the most general way so that they can be used in combination with any chosen AIM. Several Hilbert-space and real-space (fuzzy atom) AIM definitions are implemented. In general, global quantities are decomposed into one- and two-center terms, which can also be further grouped into fragment contributions. Real-space AIM methods involve numerical integrations, which are particularly costly for energy decomposition schemes. The current version of APOST-3D features several strategies to minimize numerical error and improve task parallelization. In addition to conventional population analysis of the density and other scalar fields, APOST-3D implements different schemes for oxidation state assignment (effective oxidation state and oxidation states localized orbitals), molecular energy decomposition schemes, and local spin analysis. The APOST-3D platform offers a user-friendly interface and a comprehensive suite of state-of-the-art tools to bridge the gap between theory and experiment, representing a valuable resource for both seasoned computational chemists and researchers with a focus on experimental work. We provide an overview of the code structure and its capabilities, together with illustrative examples.

On the accurate computation of the Newton form of the Lagrange interpolant

AbstractIn recent years many efforts have been devoted to finding bidiagonal factorizations of nonsingular totally positive matrices, since their accurate computation allows to numerically solve several important algebraic problems with great precision, even for large ill-conditioned matrices. In this framework, the present work provides the factorization of the collocation matrices of Newton bases—of relevance when considering the Lagrange interpolation problem—together with an algorithm that allows to numerically compute it to high relative accuracy. This further allows to determine the coefficients of the interpolating polynomial and to compute the singular values and the inverse of the collocation matrix. Conditions that guarantee high relative accuracy for these methods and, in the former case, for the classical recursion formula of divided differences, are determined. Numerical errors due to imprecise computer arithmetic or perturbed input data in the computation of the factorization are analyzed. Finally, numerical experiments illustrate the accuracy and effectiveness of the proposed methods with several algebraic problems, in stark contrast with traditional approaches.

Emergence of Microphysical Bulk Viscosity in Binary Neutron Star Postmerger Dynamics

In nuclear matter in isolated neutron stars, the flavor content (e.g., proton fraction) is subject to weak interactions, establishing flavor (β-)equilibrium. However, there can be deviations from this equilibrium during the merger of two neutron stars. We study the resulting out-of-equilibrium dynamics during the collision by incorporating direct and modified Urca processes (in the neutrino-transparent regime) into general-relativistic hydrodynamics simulations with a simplified neutrino transport scheme. We demonstrate how weak-interaction-driven bulk viscosity in postmerger simulations can emerge and assess the bulk viscous dynamics of the resulting flow. We further place limits on the impact of the postmerger gravitational-wave strain. Our results show that weak-interaction-driven bulk viscosity can potentially lead to a phase shift of the postmerger gravitational-wave spectrum, although the effect is currently on the same level as the numerical errors of our simulation.

DeepOTF: Learning Equations-constrained Prediction for Electromagnetic Behavior

High-quality passive devices are becoming increasingly important for the development of mobile devices and telecommunications, but obtaining such devices through simulation and analysis of electromagnetic (EM) behavior is time-consuming. To address this challenge, artificial neural network (ANN) models have emerged as an effective tool for modeling EM behavior, with NeuroTF being a representative example. However, these models are limited by the specific form of the transfer function, leading to discontinuity issues and high sensitivities. Moreover, previous methods have overlooked the physical relationship between distributed parameters, resulting in unacceptable numeric errors in the conversion results. To overcome these limitations, we propose two different neural network architectures: DeepOTF and ComplexTF. DeepOTF is a data-driven deep operator network for automatically learning feasible transfer functions for different geometric parameters. ComplexTF utilizes complex-valued neural networks to fit feasible transfer functions for different geometric parameters in the complex domain while maintaining causality and passivity. Our approach also employs an Equations-constraint Learning scheme to ensure the strict consistency of predictions and a dynamic weighting strategy to balance optimization objectives. The experimental results demonstrate that our framework shows superior performance than baseline methods, achieving up to 1700 × higher accuracy.

An improved two‐dimensional (2,4) finite‐difference time‐domain method for Lorentz dispersive media

AbstractThe credible solution of discretized Maxwell's equations in spaces occupied by Lorentz dispersive media is the main subject of this work. Specifically, we introduce a finite‐difference time‐domain (FDTD) algorithm with a typical (2,4) structure that features dispersion‐relation‐preserving characteristics and produces reduced numerical errors in two‐dimensional electromagnetic simulations, compared to the standard approach with similar computational requirements. We consider the case of dispersive media with non‐vanishing absorption coefficients and investigate different options for the suitable modification of the spatial approximations, so that the accomplished accuracy is optimized for a given computational overhead. The properties of the proposed FDTD technique are thoroughly examined, both theoretically and in numerical tests, and the performance upgrade compared with the conventional solution is assessed.

DUGKS-GPU: An efficient parallel GPU code for 3D turbulent flow simulations using Discrete Unified Gas Kinetic Scheme

This paper presents a parallel implementation of the Discrete Unified Gas Kinetic Scheme (DUGKS) on the GPU system using the CUDA Fortran and CUDA C++ programming languages. Firstly, we conducted an extensive revision of our original CPU-based code, resulting in a threefold decrease in memory usage. This new implementation is also paired with a novel approach to compute cell face flux using trilinear interpolation. It is shown analytically that the interpolation-based approach to flux calculation is more accurate compared to the one used in the original DUGKS. The initial simulation results using this new approach suggest that trilinear interpolation can reduce numerical errors on a coarse mesh. For example, in the case of the decaying Taylor-Green vortex flow at a 1283 mesh resolution, the relative numerical error in the energy dissipation rate at t⁎=2, using the spectral simulation result as the benchmark, is approximately 30% lower than that of the original implementation. The improved GPU DUGKS method is applied to laminar and turbulent flows in periodic and wall-bounded boundary configurations. A performance comparison of the GPU implementation is also presented and compared to the previous CPU implementation. A maximum speedup of 7.64x was achieved on a desktop-level GPU compared to a 32-core CPU. The strong scaling test, conducted on an eight-GPU node, demonstrated the efficient utilization of available multiple GPU resources by the code. Program summaryProgram Title: DUGKS-GPUCPC Library link to program files:https://doi.org/10.17632/yykv5s9g2n.1Developer's repository link:https://github.com/kairzhan/DUGKS-GPUCode Ocean capsule:https://codeocean.com/capsule/3b1e4f74-cdd3-4781-8923-8514d5923dfb/Licensing provisions: GNU General Public License 3Programming language: CUDA C++, CUDA FortranSupplementary material:Nature of problem: DUGKS-GPU is an advanced numerical code designed to accelerate the simulation of turbulent fluid flows through the application of the kinetic method known as the Discrete Unified Gas Kinetic Scheme (DUGKS). This computational tool comprises a CUDA Fortran version, tailored for a single GPU, as well as a CUDA C++ version developed for multi-GPU platforms.Solution method: The code employs a second-order finite-volume discretization of the discrete velocity Boltzmann equation with the BGK collision model. It can be used to simulate small to medium-scale problems on modern multi-GPU platforms with CUDA architecture.

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.

A gas-mechanical-damage coupling model based on the TLF-SPH method and its application to gas seepage in fractured coal

The coupled calculation of gas seepage in fractured coal is a challenge in computational mechanics. In this paper, a gas seepage model of fractured coal is developed based on smoothed particle hydrodynamics (SPH). Then, the accuracy of the seepage model based on the partial differential equation (PDE) with a constant coefficient is verified by a reference example. Moreover, the seepage characteristics of gas in homogeneous coal based on a PDE with constant and variable coefficients are analyzed, and their numerical errors are compared. Then, based on the gas seepage model of fractured coal, the effects of heterogeneity, defect opening and inclination on gas seepage in coal are studied. In addition, considering the influence of coal deformation and preexisting defects on gas seepage, a gas-mechanical-damage coupling model for gas seepage simulation is proposed by using the SPH method with the total Lagrangian formula (TLF-SPH). Subsequently, the permeability evolution and gas seepage characteristics of borehole surrounding rock under different conditions (such as overburden load, extraction pressure, borehole radius and initial gas pressure) are investigated, revealing the seepage-mechanical coupling mechanism of fractured coal during gas extraction. This work provides important theoretical support for mine gas extraction and gas outburst disaster prevention technology.

Simulations and analysis of underwater acoustic wave propagation model based on Helmholtz equation

In this study, we explore the effectiveness of elliptic partial differential equations (PDEs) in two and three dimensional space based on Helmholtz equation for simulations of acoustic sound in a very complex environment of propagation. For this purpose, we use an advance and robust numerical technique by utilizing the properties of shifted Chebyshev spectral collocation method. This technique is an extension of the traditional Chebyshev polynomials, incorporating a shift in their argument to enhance flexibility across a wider domain, while retaining an extraordinary numerical characteristic such as orthogonality and spectral convergence making them exceptionally effective in finding the approximate solutions. The exponential order of convergence of the proposed approach is shown both through theoretical and numerical approaches. We provide a number of numerical experiments to verify the theoretical results. The spectral convergence has been substantially enhanced by these numerical examples. The exponential order is further validated by numerical error behaviour in both L 2 and L ∞ norms.

Mitigation of numerical issues appearing in transient analyses when applying fractional derivative approximations

Due to the potential decrease of the computation time for problems with fractional order derivatives, and due to the extension of the range of applicable solvers for a given problem, the approximation of fractional derivatives (e.g., using Oustaloup’s method) is an important and frequently discussed topic. A significant problem that can occur (although one that is not discussed very often) when approximations are applied concerns the potential numerical issues that can arise when applying these approximations in a transient analysis, e.g., as shown in the paper, in a time-stepping solver. This paper examines five different methods for approximating the fractional derivative (Oustaloup’s method, a refined Oustaloup method, a modified continued fraction expansion method, Matsuda’s method and Charef’s method). An important preliminary step in the analysis was the establishment of the general form of FDAE (fractional differential-algebraic equations), the general form of the applied approximations and the general form of the DAE (differential-algebraic equations) resulting from the approximation. This allowed for a precise description of the equations according to which the transformation from the FDAE to the approximating DAE takes place. Three problems derived from studies in electrical engineering have been introduced into the analysis (a linear circuit problem, a circuit problem with nonlinear fractional elements, and a problem featuring nonlinearities and pregenerated noise). For these problems, the conditions in which numerical issues appear have been studied in detail. They are the main motivation of this study as these issues are so significant that they often result in zero-solutions or solutions tending to infinity (giving an impression as if the system is unstable). Modifications and alternatives of the transformation into the DAE, that aim at the mitigation of these numerical errors, are mentioned later. The final result is very satisfactory, where the algorithm for the transformation of the FDAE to the approximating DAE practically eliminates the most important barriers (which was the impossibility of using approximations above a certain order due to the mentioned numerical issues). The study presented in this paper is motivated by problems in electrical engineering but due to its generality it is also applicable in other fields where fractional calculus is used.

Automatic design of interpretable control laws through parametrized Genetic Programming with adjoint state method gradient evaluation

This work investigates the application of a Local Search (LS) enhanced Genetic Programming (GP) algorithm to the control scheme’s design task. The combination of LS and GP aims to produce an interpretable control law as similar as possible to the optimal control scheme reference. Inclusive Genetic Programming (IGP), a GP heuristic capable of promoting and maintaining the population diversity, is chosen as the GP algorithm since it proved successful on the considered task. IGP is enhanced with the Operators Gradient Descent (OPGD) approach, which consists of embedding learnable parameters into the GP individuals. These parameters are optimized during and after the evolutionary process. Moreover, the OPGD approach is combined with the adjoint state method to evaluate the gradient of the objective function. The original OPGD was formulated by relying on the backpropagation technique for the gradient’s evaluation, which is impractical in an optimization problem involving a dynamical system because of scalability and numerical errors. On the other hand, the adjoint method allows for overcoming this issue. Two experiments are formulated to test the proposed approach, named Operator Gradient Descent - Inclusive Genetic Programming (OPGD-IGP): the design of a Proportional–Derivative (PD) control law for a harmonic oscillator and the design of a Linear Quadratic Regulator (LQR) control law for an inverted pendulum on a cart. OPGD-IGP proved successful in both experiments, being capable of autonomously designing an interpretable control law similar to the optimal ones, both in terms of shape and control gains.

Effects of plasma resistivity in FELTOR simulations of three-dimensional full-F gyro-fluid turbulence

A full-F, isothermal, electromagnetic, gyro-fluid model is used to simulate plasma turbulence in a COMPASS-sized, diverted tokamak. A parameter scan covering three orders of magnitude of plasma resistivity and two values for the ion to electron temperature ratio with otherwise fixed parameters is setup and analysed. Two transport regimes for high and low plasma resistivities are revealed. Beyond a critical resistivity the mass and energy confinement reduces with increasing resistivity. Further, for high plasma resistivity the direction of parallel acceleration is swapped compared to low resistivity.Three-dimensional visualisations using ray tracing techniques are displayed and discussed. The field-alignment of turbulent fluctuations in density and parallel current becomes evident. Relative density fluctuation amplitudes increase from below 1% in the core to 15% in the edge and up to 40% in the scrape-off layer.Finally, the integration of exact conservation laws over the closed field line region allows for an identification of numerical errors within the simulations. The electron force balance and energy conservation show relative errors on the order of 10−3 while the particle conservation and ion momentum balance show errors on the order of 10−2.All simulations are performed with a new version of the FELTOR code, which is fully parallelized on GPUs. Each simulation covers a couple of milliseconds of turbulence.

Transient analyses of underwater acoustic propagation with the modified meshfree radial point interpolation method and newmark time integration techniques

Underwater acoustic propagation plays a very important role in underwater communication and ocean engineering field. The radial point interpolation method (RPIM) is an effective numerical approach to solve underwater acoustic propagations though considerable numerical errors can always be caused due to the discontinuously differentiable nodal interpolation function in one integration cell. The origin of this numerical error primarily comes from the mismatch between the background integration cell and the nodal interpolation function support. To address this issue, a modified meshfree RPIM is developed in this work for transient analyses of underwater acoustic propagations. In the proposed modified RPIM, the construction of the interpolation function supports are dependent on the integration cells. In consequence, in one background cell all the quadrature points share one mutual support domain and the continuously differentiable interpolation functions can be obtained. Therefore, the numerical integration errors can be evidently decreased and the related numerical dispersion effects in wave analysis can be largely suppressed. Owing to the adequately small spatial dispersion errors from the proposed modified RPIM, it is observed that the modified RPIM with the Newmark time domain discretization scheme can monotonically improve the solution precision by directly reducing the temporal discretization interval. This property makes the modified RPIM clearly outperform the original RPIM and is especially suitable for handling very complex underwater acoustic propagations.