Accurate Integrator Research Articles

AHKASH: a new Hybrid particle-in-cell code for simulations of astrophysical collisionless plasma.

We introduce Astrophysical Hybrid-Kinetic simulations with the flash code ([Formula: see text]) - a new Hybrid particle-in-cell (PIC) code developed within the framework of the multiphysics code flash. The new code uses a second-order accurate Boris integrator and a predictor-predictor-corrector algorithm for advancing the Hybrid-kinetic equations, using the constraint transport method to ensure that magnetic fields are divergence-free. The code supports various interpolation schemes between the particles and grid cells, with post-interpolation smoothing to reduce finite particle noise. We further implement a [Formula: see text] method to study instabilities in weakly collisional plasmas. The new code is tested on standard physical problems such as the motion of charged particles in uniform and spatially varying magnetic fields, the propagation of Alfvén and whistler waves, and Landau damping of ion acoustic waves. We test different interpolation kernels and demonstrate the necessity of performing post-interpolation smoothing. We couple the turbgen turbulence driving module to the new Hybrid PIC code, allowing us to test the code on the highly complex physical problem of the turbulent dynamo. To investigate steady-state turbulence with a fixed sonic Mach number, it is important to maintain isothermal plasma conditions. Therefore, we introduce a novel cooling method for Hybrid PIC codes and provide tests and calibrations of this method to keep the plasma isothermal. We describe and test the 'hybrid precision' method, which significantly reduces (by a factor [Formula: see text]) the computational cost, without compromising the accuracy of the numerical solutions. Finally, we test the parallel scalability of the new code, showing excellent scaling up to 10,000 cores.

Effective highly accurate time integrators for linear Klein–Gordon equations across the scales

Effective highly accurate time integrators for linear Klein–Gordon equations across the scales

Uniformly accurate nested Picard iterative schemes for nonlinear Schrödinger equation with highly oscillatory potential

The nonlinear Schrödinger equation with a highly oscillatory potential (NLSE-OP) often appears in many multiscale dynamical systems, where the temporal oscillation causes the major numerical difficulties. Recently, the splitting schemes (Su et al., 2020) were analyzed rigorously for solving the NLSE-OP and the error bounds show that they are only uniformly first-order accurate. This obviously can not meet the requirement of high precision in actual calculation. In this paper, in order to obtain a higher uniform convergence order, we study the nested Picard iterative (NPI) schemes for the NLSE-OP with polynomial nonlinearity. Firstly we propose a uniformly accurate first-order exponential wave integrator (EWI) scheme for arbitrary nonlinearity by integrating the potential exactly. Then we construct a uniformly accurate second-order NPI scheme for the NLSE-OP with polynomial nonlinearity. The schemes are fully explicit and very efficient due to the fast Fourier transform (FFT). We give rigorously error analysis and establish error bounds for the numerical solutions without any CFL-type condition constraint. Numerical experiments prove the correctness of our theoretical analysis and the effectiveness of our schemes. Theoretically, our schemes can be extended to uniformly accurate arbitrary high order for the NLSE-OP.

Vlasov–Maxwell equations with spin effects

We present a numerical method to solve the Vlasov–Maxwell equations for spin-1/2 particles, in a semiclassical approximation where the orbital motion is treated classically while the spin variable is fully quantum. Unlike the spinless case, the phase-space distribution function is a $2\times 2$ matrix, which can also be represented, in the Pauli basis, as one scalar function $f_0$ and one three-component vector function $\boldsymbol f$ . The relationship between this ‘vectorial’ representation and the fully scalar representation on an extended phase space first proposed by Brodin et al. (Phys. Rev. Lett., vol. 101, 2008, p. 245002) is analysed in detail. By means of suitable approximations and symmetries, the vectorial spin-Vlasov–Maxwell model can be reduced to two-dimensions in the phase space, which is amenable to numerical solutions using a high-order grid-based Eulerian method. The vectorial model enjoys a Poisson structure that paves the way to accurate Hamiltonian split-time integrators. As an example, we study the stimulated Raman scattering of an electromagnetic wave interacting with an underdense plasma, and compare the results with those obtained earlier with the scalar spin-Vlasov–Maxwell model and a particle-in-cell code.

Resource-Aware Discretization of Accelerated Optimization Flows: The Heavy-Ball Dynamics Case

This article proposes a methodology for discretizing accelerated optimization flows while retaining their convergence properties. Inspired by the success of resource-aware control in developing efficient closed-loop feedback implementations on digital systems, we view the last sampled state of the system as the resource to be aware of. We illustrate our design methodology for discretization on a newly introduced continuous-time dynamics, the heavy-ball dynamics with displaced gradient. Our algorithm design employs techniques from resource-aware control that, in this context, have interesting parallelisms with the discrete-time implementation of optimization algorithms. These include derivative- and performance-based triggers to monitor the evolution of the Lyapunov function as a way of determining the stepsize, exploiting sampled information to enhance performance, and employing high-order holds using more accurate integrators of the original dynamics. Our approach gives rise to variable-stepsize discrete-time algorithms that retain by design the monotonically decreasing properties of the Lyapunov certificate of the continuous-time heavy-ball dynamics with displaced gradient.

A Consistent Time Level Implementation Preserving Second-Order Time Accuracy via a Framework of Unified Time Integrators in the Discrete Element Approach

In this work, a consistent and physically accurate implementation of the general framework of unified second-order time accurate integrators via the well-known GSSSS framework in the Discrete Element Method is presented. The improved tangential displacement evaluation in the present implementation of the discrete element method has been derived and implemented to preserve the consistency of the correct time level evaluation during the time integration process in calculating the algorithmic tangential displacement. Several numerical examples have been used to validate the proposed tangential displacement evaluation; this is in contrast to past practices which only seem to attain the first-order time accuracy due to inconsistent time level implementation with different algorithms for normal and tangential directions. The comparisons with the existing implementation and the superiority of the proposed implementation are given in terms of the convergence rate with improved numerical accuracy in time. Moreover, several schemes via the unified second-order time integrators within the framework of the GSSSS family have been carried out based on the proposed correct implementation. All the numerical results demonstrate that using the existing state-of-the-art implementation reduces the time accuracy to be first-order accurate in time, while the proposed implementation preserves the correct time accuracy to yield second-order.

A Regularized Real-Time Integrator for Data-Driven Control of Heating Channels

In many contexts of scientific computing and engineering science, phenomena are monitored over time and data are collected as time-series. Plenty of algorithms have been proposed in the field of time-series data mining, many of them based on deep learning techniques. High-fidelity simulations of complex scenarios are truly computationally expensive and a real-time monitoring and control could be efficiently achieved by the use of artificial intelligence. In this work we build accurate data-driven models of a two-phase transient flow in a heated channel, as usually encountered in heat exchangers. The proposed methods combine several artificial neural networks architectures, involving standard and transposed deep convolutions. In particular, a very accurate real-time integrator of the system has been developed.

Uniformly accurate integrators for Klein–Gordon–Schrödinger systems from the classical to non-relativistic limit regime

In this paper we present a novel class of asymptotic consistent exponential-type integrators for Klein–Gordon–Schrödinger systems that capture all regimes from the slowly varying classical regime up to the highly oscillatory non-relativistic limit regime. We achieve convergence of order one and two that is uniform in c without any time step size restrictions. In particular, we establish an explicit relation between gain in negative powers of the potentially large parameter c in the error constant and loss in derivative.

A uniformly accurate exponential wave integrator Fourier pseudo-spectral method with structure-preservation for long-time dynamics of the Dirac equation with small potentials

A uniformly accurate exponential wave integrator Fourier pseudo-spectral method with structure-preservation for long-time dynamics of the Dirac equation with small potentials

Uniformly Accurate Low Regularity Integrators for the Klein--Gordon Equation from the Classical to NonRelativistic Limit Regime

We propose a novel class of uniformly accurate integrators for the Klein--Gordon equation which capture classical $c=1$ as well as highly-oscillatory non-relativistic regimes $c\gg1$ and, at the same time, allow for low regularity approximations. In particular, the schemes converge with order $\tau$ and $\tau^2$, respectively, under lower regularity assumptions than classical schemes, such as splitting or exponential integrator methods, require. The new schemes in addition preserve the nonlinear Schrodinger (NLS) limit on the discrete level. More precisely, we will design our schemes in such a way that in the limit $c\to \infty$ they converge to a recently introduced class of low regularity integrators for NLS.

A uniformly accurate exponential wave integrator Fourier pseudo-spectral method with energy-preservation for long-time dynamics of the nonlinear Klein-Gordon equation

For the nonlinear Klein-Gordon (NKG) equation with weak nonlinearity characterized by a small parameter ε∈(0,1], the numerical methods for long-time dynamics have received more and more attention. For the NKG equation up to the time at O(1/ε2), the error bounds of classical numerical methods depend significantly on the small parameters ε which creates serious numerical burdens as ε→0+. Recently, an exponential wave integrator Fourier pseudo-spectral (EWIFP) method for the NKG equation has been proposed (Feng et al., 2021) which is uniformly accurate about ε and of course performs well over the classical methods. However, the EWIFP method can not preserve the discrete energy, which is an important structural feature of the true solution in the long-time dynamics from the perspective of geometric numerical integration. In addition, the EWIFP method only is conditionally stable under specific stability condition and the authors did not give a convergence analysis for the method without any CFL condition restrictions on the grid ratio. In this work, we propose an energy-preserving EWIFP (EPEWIFP) method. The proposed method is proved to be time symmetric, unconditionally stable and preserves the discrete energy. We carry out a rigorously error analysis and give uniform error bounds of the method at O(hm0+ε2−βτ2) up to the time at O(1/εβ) with β∈[0,2], mesh size h, time step τ and an integer m0 determined by the regularity conditions. In general, the NKG equation with weak nonlinearity can be converted to an oscillatory NKG equation with wavelength at O(ε2) in time. It is easy to extend the error bounds and energy-preservation properties to the oscillatory NKG equation. Numerical results confirm our error bounds and energy-preservation properties.

A mixed finite element formulation for energy‐momentum time integrations of composites with fiber bending stiffness

AbstractWe aim at dynamic simulations of 3D‐fiber‐reinforced materials in light‐weight structures, which are reinforced by fiber rovings comprised of thousands of filaments. Each roving possesses a separate bending stiffness, which has a large influence on the material behaviour on the macro scale. For the description of this material behavior, we extend a hyperelastic anisotropic Cauchy continuum formulation, which models only the macro‐scopic behaviour of a reinforcement by filaments, by a higher‐order gradient of the deformation mapping as an independent field. The new mixed finite element formulation is based on well‐known mixed finite elements for anisotropic polyconvex material formulations, but include new additional mixed fields concernd with the fiber bending stiffness. As we aim at dynamic long‐term simulations, we apply higher‐order accurate time integrators. Here, we include the new mixed finite element formulation in a variational‐based time finite element method. As numerical examples serve a simple cantilever beam. Here, we first analyze the influence of the fiber bending stiffness.

Merlin++, a flexible and feature-rich accelerator physics and particle tracking library

Merlin++ is a C++ charged-particle tracking library developed for the simulation and analysis of complex beam dynamics within high energy particle accelerators. Accurate simulation and analysis of particle dynamics is an essential part of the design of new particle accelerators, and for the optimization of existing ones. Merlin++ is a feature-full library with focus on long-term tracking studies. A user may simulate distributions of protons or electrons in either single particle or sliced macro-particle bunches. The tracking code includes both straight and curvilinear coordinate systems allowing for the simulation of either linear or circular accelerator lattice designs, and uses a fast and accurate explicit symplectic integrator. Physics processes for common design studies have been implemented, including RF cavity acceleration, synchrotron radiation damping, on-line physical aperture checks and collimation, proton scattering, wakefield simulation, and spin-tracking. Merlin++ was written using C++ object orientated design practices and has been optimized for speed using multicore processors. This article presents an account of the program, including its functionality and guidance for use. Program summaryProgram Title: Merlin++CPC Library link to program files:https://doi.org/10.17632/4x4nsbhz37.1Developer's repository link: 10.5281/zenodo.3700155Licensing provisions: GPLv2+Programming language: C++Nature of problem: Complexity of particle accelerators beam dynamics over extensive tracking distances.Solution method: Long-term particle accelerator and tracking simulations utilizing explicit symplectic integrators.Additional comments including restrictions and unusual features: For further information see github.com/Merlin-Collaboration

Forced Variational Integrators for the Formation Control of Multiagent Systems

Formation control of autonomous agents can be seen as a physical system of individuals interacting with local potentials, and whose evolution can be described by a Lagrangian function. In this article, we construct and implement forced variational integrators for the formation control of autonomous agents modeled by double integrators. In particular, we provide an accurate numerical integrator with a lower computational cost than traditional solutions. We find error estimations for the rate of the energy dissipated along with the agents' motion to achieve desired formations. Consequently, this permits to providing sufficient conditions on the time step for the convergence of discrete formation control systems such as the consensus problem in discrete systems. We present practical applications such as the rapid estimation of regions of attraction to desired shapes in distance-based formation control.

Uniformly accurate nested Picard integrators for a system of oscillatory ordinary differential equations

A uniformly second order accurate nested Picard iterative integrator (NPI) is proposed to solve a system of oscillatory ordinary equations involving a parameter $$0<\varepsilon \le 1$$ . The solution of this system propagates wave with $$O(\varepsilon ^2)$$ wavelength and $$O(\varepsilon ^4)/O(\varepsilon ^2)$$ amplitude for well-prepared/ill-prepared initial data. Based on the NPI and the exponential integrator, a uniformly accurate (w.r.t $$\varepsilon $$ ) second order numerical scheme with $$O(\tau ^2)$$ error has been developed. The method can be generalized to higher order. Numerical tests are presented to confirm our error estimates.

An accurate and time-parallel rational exponential integrator for hyperbolic and oscillatory PDEs

Rational exponential integrators (REXI) are a class of numerical methods that are well suited for the time integration of linear partial differential equations with imaginary eigenvalues. Since these methods can be parallelized in time (in addition to the spatial parallelization that is commonly performed) they are well suited to exploit modern high performance computing systems. In this paper, we propose a novel REXI scheme that drastically improves accuracy and efficiency. The chosen approach will also allow us to easily determine how many terms are required in the approximation in order to obtain accurate results. We provide comparative numerical simulations for a shallow water equation that highlight the efficiency of our approach and demonstrate that REXI schemes can be efficiently implemented on graphic processing units.

Noether Symmetries and Decay Laws in Formation Control of Multi-agent Systems

Abstract Distance-based formation control of agents with double integrator dynamics can be seen as a stabilization system whose evolution can be described by a time-dependent Lagrangian function. In this paper, a Noether theorem for this class of systems is obtained, giving rise to an intrinsic geometric understanding of the exponential decay for the constants of the motion of the agents, in particular, linear and angular momentum. An interesting family of geometric integrators, called variational integrators, is defined by using discretizations of the Hamilton’s principle of critical action. The variational integrators preserve some geometric features such as the momentum map, and the decay of the system’s energy presents a good behavior. We derive variational integrators for time-dependent Lagrangian systems that can be employed in the context of distance-based formation control algorithms. In particular, we provide an accurate numerical integrator preserving the exponential decay of the constants of motion.

A class of structure-preserving discontinuous Galerkin variational time integrators for Birkhoffian systems

Accurate time integrators that preserving Birkhoffian structure are of great practical use for Birkhoffian systems. In this paper, a class of structure-preserving discontinuous Galerkin variational integrators (DGVIs) is presented. Start from the Pfaff action functional, the technique of variational integrators combined with discontinuous Galerkin time discretization is used to derive numerical schemes for Birkhoffian systems. For the derived DGVIs, symplecticity is proved rigorously through the preserving of particular 2-forms induced by these integrators. Linear stability and order of accuracy of the DGVIs are illustrated considering the example of linear damped oscillators. The order of accuracy and the property of preserving conserved quantities of the developed DGVIs are also confirmed by numerical examples. Comparisons are made with several numerical schemes such as backward/forward Euler, Runge–Kutta and RBF methods to show the advantages of DGVIs in preserving the Birkhoffians.

Forced variational integrator for distance-based shape control with flocking behavior of multi-agent systems

A multi-agent system designed to achieve distance-based shape control with flocking behavior can be seen as a mechanical system described by a Lagrangian function and subject to additional external forces. Forced variational integrators are given by the discretization of Lagrange-d’Alembert principle for systems subject to external forces, and have proved useful for numerical simulation studies of complex dynamical systems. We derive forced variational integrators that can be employed in the context of control algorithms for distance-based shape with velocity consensus. In particular, we provide an accurate numerical integrator with a lower computational cost than traditional solutions, while preserving the configuration space and symmetries. We also provide an explicit expression for the integration scheme in the case of an arbitrary number of agents with double integrator dynamics. For a numerical comparison of the performances, we use a planar formation consisting of three autonomous agents.

Splitting Methods for Rotations: Application to Vlasov Equations

In this work, a splitting strategy is introduced to approximate two-dimensional rotation motions. Unlike standard approaches based on directional splitting which usually lead to a wrong angular velocity and then to large error, the splitting studied here turns out to be exact in time. Combined with spectral methods, the so-obtained numerical method is able to capture the solution to the associated partial differential equation with a very high accuracy. A complete numerical analysis of this method is given in this work. Then, the method is used to design highly accurate time integrators for Vlasov type equations: the Vlasov-Maxwell system and the Vlasov-HMF model. Finally , several numerical illustrations and comparisons with methods from the literature are discussed.