High-order Integration Research Articles

Convergence analysis of high-order IMEX-BDF schemes for the incompressible Navier–Stokes equations

In this paper, we consider developing high-order temporal integration schemes for the unsteady incompressible Navier–Stokes equations in bounded two-dimensional domain subjected to the periodic boundary conditions. Utilizing the k-step (k=3,4,5) backward differentiation formula (BDF) coupled with the implicit–explicit (IMEX) treatment of the nonlinear convective term in an anti-symmetry form, a class of IMEX-BDFk schemes up to fifth-order in time are constructed and analyzed. By imposing a zero-mean constrain on the finite-dimensional space for the pressure, the proposed numerical schemes are proven to be uniquely solvable. Based on the recent theoretical framework consisting of a class of discrete orthogonal convolution kernels, rigorous L2 norm error estimates for both the velocity and the pressure are established by using a novel divergence free projection system. The proposed schemes are then implemented in two benchmark experiments, including a Taylor–Green vortex problem and a double shear layer flow at various high Reynolds numbers. Numerical results demonstrate the expected solution accuracy and the computational effectiveness in simulating the realistic flow dynamics.

Exact Inference for Continuous-Time Gaussian Process Dynamics

Many physical systems can be described as a continuous-time dynamical system. In practice, the true system is often unknown and has to be learned from measurement data. Since data is typically collected in discrete time, e.g. by sensors, most methods in Gaussian process (GP) dynamics model learning are trained on one-step ahead predictions. While this scheme is mathematically tempting, it can become problematic in several scenarios, e.g. if measurements are provided at irregularly-sampled time steps or physical system properties have to be conserved. Thus, we aim for a GP model of the true continuous-time dynamics. We tackle this task by leveraging higher-order numerical integrators. These integrators provide the necessary tools to discretize dynamical systems with arbitrary accuracy. However, most higher-order integrators require dynamics evaluations at intermediate time steps, making exact GP inference intractable. In previous work, this problem is often addressed by approximate inference techniques. However, exact GP inference is preferable in many scenarios, e.g. due to its mathematical guarantees. In order to enable direct inference, we propose to leverage multistep and Taylor integrators. We demonstrate how exact inference schemes can be derived for these types of integrators. Further, we derive tailored sampling schemes that allow one to draw consistent dynamics functions from the posterior. The learned model can thus be integrated with arbitrary integrators, just like a standard dynamical system. We show empirically and theoretically that our approach yields an accurate representation of the continuous-time system.

Coupling of the immersed boundary and Fourier pseudo-spectral methods applied to solve fluid–structure interaction problems

The current manuscript addresses fluid–structure interaction (FSI) problems to the analysis of vortex-induced vibration (VIV), employing the methodology designated as IMERSPEC2D. The approach seamlessly integrates the Fourier pseudo-spectral method (FPSM) and the immersed boundary method (IBM) to solve the Navier–Stokes (NS) and continuity equations. Simultaneously, the effects of cylindrical structure, involving 1 and 2 degrees of freedom (d.o.f.) are simulated, by incorporating structural motion equations into NS via the IBM source term and implementing a two-way FSI algorithm. In the paper are presented the comparison of low and high-order temporal integration methods applied to solve the dynamic behavior of the structure and the dimensionless of the cylinder’s structural motion equations, resulting in an increase in computational time step from 1E(-9\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$-9$$\\end{document}) to 1E(-3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$-3$$\\end{document}). Validation results to refined mesh simulations include the accurate representation of the lock-in phenomenon to simulations of 1 d.o.f. obtaining the range between reduced velocity 5.456 and 5.568 and the maximal amplitude is 0.352. Notably, the eight-shape pattern of the 2 d.o.f. cylinder displacement is obtained and the formation of C2S wake patterns is achieved, as presented by other authors. In conclusion, the dimensionless approach reduces computational time, while the high-order both of IMERSPEC methodology and time advancement integration to FSI improve the results of the cylinder’s motion and alters distinct vortex shedding patterns in fluid dynamics.

A class of arbitrarily high-order energy-preserving method for nonlinear Klein–Gordon–Schrödinger equations

In this paper, we develop a class of arbitrarily high-order energy-preserving time integrators for the nonlinear Klein–Gordon–Schrödinger equations. We employ Fourier pseudo-spectral method for spatial discretization, resulting in a semi-discrete system. Subsequently, we employ the Petrov-Galerkin method in time to obtain a fully-discrete system. We rigorously demonstrate that the proposed scheme preserves the original energy of the target system. Furthermore, we have proved that the mass of the system is also approximately preserved. To assess the accuracy of our method, we provide a simple estimate of the local error, revealing that the proposed approach achieves a temporal order of 2s. Additionally, we extend the proposed methods to the damped Klein–Gordon–Schrödinger equations, and the corresponding fully-discrete scheme preserves the original energy dissipation law of the system. We present numerical examples to verify the accuracy and robustness of the proposed scheme.

A gradient reproducing kernel based stabilized collocation method for the 5th order Korteweg–de Vries equations

This paper presents a gradient reproducing kernel based stabilized collocation method (GRK-SCM) to solve the generalized nonlinear 5th order Korteweg–de Vries (KdV) equations. By introducing gradient reproducing kernel (GRK) approximations, one can circumvent the intricacy of high-order derivatives in RK approximations, while still satisfying high-order consistency requirements. This leads to the high accuracy and efficiency. GRKs are very good approximation candidates for addressing high order partial differential equations that require higher order derivatives formulated as strong form. The stabilized collocation method (SCM) effectively meets high-order integration constraints which can achieve accurate integration in the subdomains. This ensures both high precision and optimal convergence. Von Neumann analysis is utilized to establish the stability criteria for GRK-SCM when combined with forward difference temporal discretization. Numerical solutions for Sawada–Kotera (SK) equation and Kaup-Kupershmidt (KK) equation are studied where the solitary wave migration and collision, and periodic waves are represented. A fifth order forced KdV equation is also considered. The presented method's high accuracy and convergence are demonstrated through comparative studies with analytical solutions, and investigations of invariants and corresponding errors determine the good conservation properties of this algorithm.

High-order geometric integrators for the local cubic variational Gaussian wavepacket dynamics.

Gaussian wavepacket dynamics has proven to be a useful semiclassical approximation for quantum simulations of high-dimensional systems with low anharmonicity. Compared to Heller's original local harmonic method, the variational Gaussian wavepacket dynamics is more accurate, but much more difficult to apply in practice because it requires evaluating the expectation values of the potential energy, gradient, and Hessian. If the variational approach is applied to the local cubic approximation of the potential, these expectation values can be evaluated analytically, but they still require the costly third derivative of the potential. To reduce the cost of the resulting local cubic variational Gaussian wavepacket dynamics, we describe efficient high-order geometric integrators, which are symplectic, time-reversible, and norm-conserving. For small time steps, they also conserve the effective energy. We demonstrate the efficiency and geometric properties of these integrators numerically on a multidimensional, nonseparable coupled Morse potential.

Exploring Higher-Order Thinking Skills Integration in Mathematics Curriculum: A Study of Six Lebanese Schools

This article investigates the promotion of higher-order thinking skills in the written curriculum of six diverse schools in Lebanon, focusing on Grades 1, 2, and 3 mathematics outcomes. The educational programs encompass the International Baccalaureate’s Primary Years Programme, the Lebanese National Curriculum, and the French National Curriculum. Employing Bloom’s Taxonomy (1956) as the theoretical framework, the study meticulously categorizes learning objectives into six cognitive domains, shedding light on the emphasis placed on lower and higher-order thinking skills. Results reveal nuanced patterns across schools and grade levels. In Grade 1, School 4 demonstrates a notable focus on “analyze,” while School 2 excels in “create.” Grade 2 highlights the International Baccalaureate’s Primary Years Programme, particularly School 2, for emphasizing the three higher domains of Bloom’s Taxonomy (1956). The Lebanese National Curriculum exhibits variations, with School 4 emphasizing “analyze.” This research holds significance amidst Lebanon’s educational challenges, marked by outdated curricula, declining academic performance, and recent disruptive events. The study advocates for a paradigm shift, emphasizing the need for comprehensive educational philosophies, curriculum revisions, and student-centered teaching approaches. The findings provide valuable insights for policymakers and educators to navigate the complexities of Lebanon’s educational landscape, fostering a transformative approach for resilient and innovative students amid challenging circumstances.

A nonsmooth modified symplectic integration scheme for frictional contact dynamics of rigid–flexible multibody systems

Constructing an efficient high-order nonsmooth time-stepping integration scheme for rigid–flexible multibody systems with frictional contacts and impacts is a challenging task. This paper presents a nonsmooth modified symplectic integration scheme that can automatically achieve second-order convergence if possible. This scheme is obtained in three main steps. Firstly, the position coordinates are used to interpolate midpoint approximated integrals, which include velocity jumps implicitly; Secondly, a two-timestep central difference scheme is formulated to approximate the measure differential inclusions; Thirdly, composing two half-condensed two-timestep schemes leads to a one-timestep scheme, which retains the structure of a second-order scheme. Furthermore, while bilateral constraints are satisfied at the position level, unilateral constraints at the velocity level are subject to constraint drifts. To alleviate this issue, a classified cone complementarity problem is formulated to restrain penetration from deteriorating and sticking contacts from fictitious slipping. After time discretization, a set of nonlinear equations with cone complementarity conditions at each timestep is solved by a two-layer iterative computational strategy, where the outer layer is Newton iteration and the inner layer is a primal–dual interior point method. Several numerical examples demonstrate the high accuracy attained by the proposed method. Nonsmooth dynamics of a rigid–flexible robot involving high-speed and high-friction impacts can be simulated faster than real-time.

A High Order Integration Enhanced Zeroing Neural Network for Time-Varying Convex Quadratic Program Against Nonlinear Noise

A High Order Integration Enhanced Zeroing Neural Network for Time-Varying Convex Quadratic Program Against Nonlinear Noise

Predefined-Time Consensus of High-Order Integrator Multi-Agent Systems With Prescribed Output-Norm Constraints

Predefined-Time Consensus of High-Order Integrator Multi-Agent Systems With Prescribed Output-Norm Constraints

Performance of affine-splitting pseudo-spectral methods for fractional complex Ginzburg-Landau equations

We evaluate the performance of novel numerical methods for solving one-dimensional nonlinear fractional dispersive and dissipative evolution equations. The methods are based on affine combinations of time-splitting integrators and pseudo-spectral discretizations using Hermite and Fourier expansions. We show the effectiveness of the proposed methods by numerically computing the dynamics of soliton solutions of the standard and fractional variants of the nonlinear Schrödinger equation (NLSE) and the complex Ginzburg-Landau equation (CGLE), and by comparing the results with those obtained by standard splitting integrators. An exhaustive numerical investigation shows that the new technique is competitive when compared to traditional composition-splitting schemes for the case of Hamiltonian problems both in terms accuracy and computational cost. Moreover, it is applicable straightforwardly to irreversible models, outperforming high-order symplectic integrators which could become unstable due to their need of negative time steps. Finally, we discuss potential improvements of the numerical methods aimed to increase their efficiency, and possible applications to the investigation of dissipative solitons that arise in nonlinear optical systems of contemporary interest. Overall, the method offers a promising alternative for solving a wide range of evolutionary partial differential equations.

Stabilization With Prescribed Instant for High-Order Integrator Systems.

This article develops a new controller design approach to stabilize system states onto the equilibrium at an arbitrarily selected time instant irrespective of the initial system states and parameters. By the stabilization approach, the actual convergence time (not the bound of actual convergence time) is independent of the initial value of system states. This feature differentiates our proposed prescribed-instant stability from conventional fixed, predefined, and prescribed time stability. In this work, we propose the controller design method for the prescribed-instant stability of n-order integrator systems. The proposed control is bounded and can gradually go to zero at an arbitrarily selected time instant, at which the system states reach zero simultaneously. This special stability of the controlled system is analyzed by reduction to absurdity. In simulations, an example of comparison with frequently used prescribed-time control is presented to show the difference. Moreover, the proposed stabilization method is validated by a magnetic suspension system with matched disturbances.

Uniform L∞-bounds for energy-conserving higher-order time integrators for the Gross–Pitaevskii equation with rotation

Abstract In this paper, we consider an energy-conserving continuous Galerkin discretization of the Gross–Pitaevskii equation with a magnetic trapping potential and a stirring potential for angular momentum rotation. The discretization is based on finite elements in space and time and allows for arbitrary polynomial orders. It was first analyzed by O. Karakashian and C. Makridakis (SIAM J. Numer. Anal., 36(6),1779–1807, 1999) in the absence of potential terms and corresponding a priori error estimates were derived in $2D$. In this work we revisit the approach in the generalized setting of the Gross–Pitaevskii equation with rotation and we prove uniform $L^{\infty }$-bounds for the corresponding numerical approximations in $2D$ and $3D$ without coupling conditions between the spatial mesh size and the time step size. With this result at hand, we are particularly able to extend the previous error estimates to the $3D$ setting while avoiding artificial CFL conditions.

A fourth-order energy-preserving and symmetric average vector field integrator with low regularity assumption

It is known that the average vector field (AVF) integrator is energy-preserving but only with second-order accuracy. Motivated by the construction of the classical AVF integrator, we devoted to constructing and analysing of high-order energy-preserving integrators for the second-order Hamiltonian systems in this work. Firstly, a novel improved average vector field (IAVF) integrator is derived by modifying the average vector field of the AVF integrator. Then the energy preservation and symmetry of the derived IAVF integrator are rigorously studied. Another significant investigation of the paper is that the IAVF integrator is shown to converge with order four under low regularity assumption q(t)∈C3([t0,T]). Numerical experiments are conducted to demonstrate the remarkable superiority in accuracy and efficiency over the traditional AVF integrator. Numerical results confirm the conclusions of our theoretical analysis.

High-order composite implicit time integration schemes based on rational approximations for elastodynamics

A new approach for developing implicit composite time integration schemes is established starting with rational approximations of the matrix exponential in the solution of the equations of motion. The rational approximations are designed to have the same effective stiffness matrix in all sub-steps. An efficient algorithm is devised so that the implicit equation for a sub-step is in the same form as that of the trapezoidal method. The proposed M-schemes are Mth order accurate using M sub-steps. The amount of numerical dissipation is controlled by a user-specified parameter ρ∞. The (M+1)-schemes that are (M+1)th order accurate using M sub-steps can also be constructed, but the amount of numerical dissipation is built in the schemes and not adjustable. The order of accuracy of all the schemes is not affected by external forces and physical damping. Numerical examples demonstrate that the proposed high-order composite schemes are effective for introducing numerical dissipation and allow the use of large time step sizes in wave propagation problems. The source code written in MATLAB is available for download at: https://github.com/ChongminSong/CompsiteTimeIntegration.

Propagators for molecular dynamics in a magnetic field

ABSTRACT Ab initio molecular dynamics in a magnetic field requires solving equations of motion with velocity-dependent forces – namely, the Lorentz force arising from the nuclear charges moving in a magnetic field and the Berry force arising from the shielding of these charges from the magnetic field by the surrounding electrons. In this work, we revisit two existing propagators for these equations of motion, the auxiliary-coordinates-and-momenta (ACM) propagator and the Tajima propagator (TAJ), and compare them with a new exponential (EXP) propagator based on the Magnus expansion. Additionally, we explore limits (for example, the zero-shielding limit), the implementation of higher-order integration schemes, and series truncation to reduce computational cost by carrying out simulations of a HeH + model system for a wide range of field strengths. While being as efficient as the TAJ propagator, the EXP propagator is the only propagator that converges to both the schemes of Spreiter and Walter (derived for systems without shielding of the Lorentz force) and to the exact cyclotronic motion of a charged particle. Since it also performs best in our model simulations, we conclude that the EXP propagator is the recommended propagator for molecules in magnetic fields.

LibFastMesh: An optimized finite-volume framework for computational aeroacoustics

This paper presents an optimized framework for simulating aeroacoustic flow on unstructured meshes. The libFastMesh (LFM) framework is based on a low-dissipation finite–volume discretization, together with high-order explicit time integration, for direct computation of aeroacoustic fields. These methods were previously implemented in caafoam, an OpenFOAM-based solver developed by D'Alessandro et al. [1] to resolve aeroacoustic flows. The new framework is specifically developed from scratch to alleviate two of the main bottlenecks currently limiting the performance of OpenFOAM-based solvers at large scale: memory bandwidth and inter-node communication. Compact data structures and compute kernel fusion are employed to enhance cache utilization and re-use, respectively. Separate treatment of inter-process boundary cells, together with non-blocking MPI communication enable effective overlap between communication and flux computations. The accuracy and robustness of libFastMesh are established via simulations of well-established aeroacoustic flow benchmarks. Comparison to previous simulation results obtained with caafoam, and to DNS data serve to validate the code. Finally, the effectiveness of the aforementioned optimizations is demonstrated through rigorous analysis of the proposed solver performance in single-node and multi-node operation modes. The obtained results show that libFastMesh offers a speed-up of up to 20x with respect to the previously developed OpenFOAM-based solver, caafoam, in large-scale computations. Moreover, LFM is shown to scale extremely for cell-per-core ratios of less than 500. These results are particularly appealing given the highly resource-demanding nature of direct computational aeroacoustics (CAA). Consequently, LFM could be an extremely attractive tool for scientists who wish to conduct large-scale CAA simulations on modern, exascale architectures. Program summaryProgram Title: libFastMeshCPC Library link to program files:https://doi.org/10.17632/7w9fy2xtcf.1Developer's repository link:https://github.com/TRC-HPC/LFM_PublicLicensing provisions: GPLv3Programming language: C++Nature of problem: This software solves compressible Navier–Stokes equations. The adopted numerics and flow models allow to capture with high accuracy aerodinamically generated sound. The solver is also designed to take advantage of the massively parallel computing systems which are strictly required for facing real–life aeroacoustic problems.Solution method: The libFastMesh (LFM) framework is based on a low-dissipation finite–volume discretization, together with high-order explicit time integration, for direct computation of aeroacoustic fields on unstructured meshes. Convective terms can be approximated using either the Kurganov–Noelle–Petrova (KNP) scheme or Pirozzoli's energy conserving approach. Time integration is fully–explicit and relies on a low-storage, 4th order Runge–Kutta scheme. The solver is specifically developed to run efficiently on modern many-core CPUs, and at large-scale where it exhibits excellent strong scaling features. This is made possible thanks to several optimizations that alleviate memory bandwidth and inter-node communication bottlenecks.Additional comments: OpenFOAM-v2112+ is an installation prerequisite. Standard OpenFOAM pre-processing and post-processing tools, as well as mesh decomposition and reordering methods may be used with the solver.

High-order geometric integrators for the variational Gaussian approximation.

Among the single-trajectory Gaussian-based methods for solving the time-dependent Schrödinger equation, the variational Gaussian approximation is the most accurate one. In contrast to Heller's original thawed Gaussian approximation, it is symplectic, conserves energy exactly, and may partially account for tunneling. However, the variational method is also much more expensive. To improve its efficiency, we symmetrically compose the second-order symplectic integrator of Faou and Lubich and obtain geometric integrators that can achieve an arbitrary even order of convergence in the time step. We demonstrate that the high-order integrators can speed up convergence drastically compared to the second-order algorithm and, in contrast to the popular fourth-order Runge-Kutta method, are time-reversible and conserve the normand the symplectic structure exactly, regardless of the time step. To show that the method is not restricted to low-dimensional systems, we perform most of the analysis on a non-separable twenty-dimensional model of coupled Morse oscillators. We also show that the variational method may capture tunneling and, in general, improves accuracy over the non-variational thawed Gaussian approximation.

Application-Oriented Homogeneous Control Protocol Design for Multi-Agent Systems Under Input Constraints

An application-oriented homogeneous control protocol is proposed to solve the consensus problem of multi-agent system (MAS) modeled by higher-order integrator. This nonlinear control protocol is able to homogenize the linear system with a special degree called homogeneity degree. This homogeneous control protocol ensures asymptotically/finite-time stable multi-agent systems (MASs) (or fixed-time attractive to compact sets containing the origin) by selecting different homogeneity degrees. If the linear control protocol for each agent is provided, the proposed nonlinear homogeneous control protocol in this paper can be easily implemented without requiring any tuning of the control parameters. A bounded homogeneous control protocol, which is a special form of the controller proposed, is also introduced to address the same problem with input constraints. Finally, numerical simulations are conducted to demonstrate the effectiveness of the proposed approach.

Neural coding of numerousness

Perception of numerousness, i.e. number of items in a set, is an important cognitive ability, which is present in several animal taxa. In spite of obvious differences in neuroanatomy, insects, fishes, reptiles, birds, and mammals all possess a “number sense”. Furthermore, information regarding numbers can belong to different sensory modalities: animals can estimate a number of visual items, a number of tones, or a number of their own movements. Given both the heterogeneity of stimuli and of the brains processing these stimuli, it is hard to imagine that number cognition can be traced back to the same evolutionary conserved neural pathway. However, neurons that selectively respond to the number of stimuli have been described in higher-order integration brain centres both in primates and in birds, two evolutionary distant groups. Although most probably not of the same evolutionary origin, these number neurons share remarkable similarities in their response properties. Instead of homology, this similarity might result from computational advantages of the underlying coding mechanism. This means that one might expect numerousness information to undergo similar steps of neural processing even in evolutionary distant neural pathways. Following this logic, in this review we summarize our current knowledge of how numerousness is processed in the brain from sensory input to coding of abstract information in the higher-order integration centres. We also propose a list of key open questions that might promote future research on number cognition.