Leapfrog Scheme Research Articles

A Leapfrog Scheme for Complying-Divergence Implicit Finite-Difference Time-Domain Method

A leapfrog scheme for the unconditionally stable complying-divergence implicit (CDI) finite-difference time-domain (FDTD) method is presented. The leapfrog CDI-FDTD method is formulated to comprise distinctive implicit and explicit update procedures. The implicit update procedures are in the fundamental form with operator-free right-hand sides (RHS), whereas the explicit ones are compatible and correspond to the familiar explicit FDTD method. Using the Fourier (von Neumann) approach, the stability of the leapfrog CDI-FDTD method is analyzed. The eigenvalues of amplification matrix are obtained to prove the unconditional stability. Furthermore, the leapfrog alternating direction implicit (ADI) FDTD and CDI-FDTD methods are discussed and compared, including the RHS floating-point operations (flops) count, for-loops and memory. The leapfrog CDI-FDTD requires merely half the flops of leapfrog ADI-FDTD at RHS while retaining the same left-hand sides (LHS) of implicit update equations and the same number of for-loops without much extra memory. Discussions and numerical results are presented to demonstrate the advantages of leapfrog CDI-FDTD method including unconditional stability, complying divergence, and efficient leapfrog scheme with reduced RHS flops.

PicFoam: An OpenFOAM based electrostatic Particle-in-Cell solver

picFoam is a fully kinetic electrostatic Particle-in-Cell (PIC) solver, including Monte Carlo Collisions (MCC), for non-equilibrium plasma research in the open-source framework of OpenFOAM. The solver’s modular design, based on the same principles used in OpenFOAM, makes it highly flexible, by allowing the user to choose different methods at run time, and extendable, by building upon templated modular classes. The implementation of the PIC method employing the finite volume method, allows it to simulate on arbitrary geometries in one to three dimensions. OpenFOAM’s barycentric particle tracking is used effectively to perform charge and field weighting from the Lagrangian particle based description to the Eulerian field description and backwards without computational expensive particle searching algorithm. picFoam also includes open and general circuit boundary models for the description of real plasma devices. Program summaryProgram Title: picFoamCPC Library link to program files:http://dx.doi.org/10.17632/bbsm8tjgjy.1Developer’s repository link:https://github.com/TFDzarm/picFoamLicensing provisions: GPLv3Programming language: C++Nature of problem: The description of equilibrium and non-equilibrium plasma with its complex collective behavior between charged species. As well as their interaction with circuits connected to the boundaries of the plasma. The applications are versatile and lie e.g. in the description of plasma phenomena like discharges in arcjet electric propulsion systems, as well as a general study of plasma phenomena.Solution method: Implementation of the Particle-in-Cell method with Monte Carlo Collisions [1] in the open-source numerical toolbox OpenFOAM [2]. By this the newly implemented solver gains access to OpenFOAM’s powerful tool sets and an easy to set up simulation case structure. Fields are solved by employing OpenFOAM’s cell-centered finite volume solvers, this allows for the simulation on arbitrary structured meshes. The leapfrog scheme [1] is employed for the integration of the particle’s equation of motion combined with relativistic and non-relativistic integration schemes for Newton’s second law of motion [3]. Monte Carlo Collision methods include models for the collision of electrons and neutral species, considered collision events are elastic collision events and inelastic excitation and ionization events. Additional models include Coulomb collisions and isotropic scattering of neutral species [4]. The solver is able to bind circuit models, include open circuit, ideal voltage and current sources, as well as general purpose circuits with an ideal voltage source and a resistance, an impedance and a capacity in series, to the plasma domain [5].Additional comments including restrictions and unusual features: The finite volume method is a relatively rarely used method for implementations of the PIC method, here by employing this method we gain the ability to use arbitrary structured meshes in one to three dimensions. OpenFOAM’s barycentric particle tracking is a novel approach for moving the particles through the mesh. With this method we are able to interpolate (weight) the charges efficiently to the mesh without using computational expensive algorithms to locate the particle’s positions beforehand.

Higher order Hamiltonian Monte Carlo sampling for cosmological large-scale structure analysis

ABSTRACT We investigate higher order symplectic integration strategies within Bayesian cosmic density field reconstruction methods. In particular, we study the fourth-order discretization of Hamiltonian equations of motion (EoM). This is achieved by recursively applying the basic second-order leap-frog scheme (considering the single evaluation of the EoM) in a combination of even numbers of forward time integration steps with a single intermediate backward step. This largely reduces the number of evaluations and random gradient computations, as required in the usual second-order case for high-dimensional cases. We restrict this study to the lognormal-Poisson model, applied to a full volume halo catalogue in real space on a cubical mesh of 1250 h−1 Mpc side and 2563 cells. Hence, we neglect selection effects, redshift space distortions, and displacements. We note that those observational and cosmic evolution effects can be accounted for in subsequent Gibbs-sampling steps within the COSMIC BIRTH algorithm. We find that going from the usual second to fourth order in the leap-frog scheme shortens the burn-in phase by a factor of at least ∼30. This implies that 75–90 independent samples are obtained while the fastest second-order method converges. After convergence, the correlation lengths indicate an improvement factor of about 3.0 fewer gradient computations for meshes of 2563 cells. In the considered cosmological scenario, the traditional leap-frog scheme turns out to outperform higher order integration schemes only when considering lower dimensional problems, e.g. meshes with 643 cells. This gain in computational efficiency can help to go towards a full Bayesian analysis of the cosmological large-scale structure for upcoming galaxy surveys.

Shadow Lagrangian dynamics for superfluidity

Motivated by a similar approach for Born-Oppenheimer molecular dynamics, this paper proposes an extended shadow Lagrangian density for quantum states of superfluids. The extended Lagrangian contains an additional field variable that is forced to follow the wave function of the quantum state through a rapidly oscillating extended harmonic oscillator. By considering the adiabatic limit for large frequencies of the harmonic oscillator, we can derive the two equations of motions, a Schrodinger-type equation for the quantum state and a wave equation for the extended field variable. The equations are coupled in a nonlinear way, but each equation individually is linear with respect to the variable that it defines. The computational advantage of this new system is that it can be easily discretized using linear time stepping methods, where we propose to use a Crank-Nicolson-type approach for the Schrodinger equation and an extended leapfrog scheme for the wave equation. Furthermore, the difference between the quantum state and the extended field variable defines a consistency error that should go to zero if the frequency tends to infinity. By coupling the time-step size in our discretization to the frequency of the harmonic oscillator we can extract an easily computable consistency error indicator that can be used to estimate the numerical error without additional costs. The findings are illustrated in numerical experiments.

Discrete transparent boundary conditions for the two-dimensional leap-frog scheme: approximation and fast implementation

We develop a general strategy in order to implement approximate discrete transparent boundary conditions for finite difference approximations of the two-dimensional transport equation. The computational domain is a rectangle equipped with a Cartesian grid. For the two-dimensional leap-frog scheme, we explain why our strategy provides with explicit numerical boundary conditions on the four sides of the rectangle and why it does not require prescribing any condition at the four corners of the computational domain. The stability of the numerical boundary condition on each side of the rectangle is analyzed by means of the so-called normal mode analysis. Numerical investigations for the full problem on the rectangle show that strong instabilities may occur when coupling stable strategies on each side of the rectangle. Other coupling strategies yield promising results.

ON A STUDY OF STAGGERED LEAPFROG SCHEME FOR LINEAR SHALLOW WATER EQUATIONS

In this paper, we present an analytical and numerical study of staggered leapfrog scheme for linear shallow water equation. It is shown that the scheme is stable when Courant number < 1, has second order accurate in both time and space, and there is no damping error in this scheme. We implement the scheme to simulate standing wave in a closed basin to show that the surface motions stay zero in a node and have constant amplitude at the antinode. For an external force given into the basin, it will induce a resonance, which cause the wave amplitude is getting bigger at the position of antinode. Moreover, we simulate a wave in a tidal basin, and show that the model has infinite spin up time. For a linear shallow water equation with linear friction, it is shown that the model has finite spin up time.

Two-grid scheme of expanded mixed finite element method for semilinear parabolic integro-differential equations

ABSTRACT In this paper, we present a two-grid scheme of expanded mixed finite element method combined with two second-order time discretization schemes for semilinear parabolic integro-differential equations and give a detailed convergence analysis. On the coarse grid space, we first use the Crank–Nicolson scheme to solve the original nonlinear problem at the first time step, then we utilize the Leap-Frog scheme at the rest time levels. Next, we make use of the known coarse mesh solution and Taylor expansion to infer the solution on the fine mesh space. Thus, we only need to solve the nonlinear problem once on the coarse grid space of the two-grid scheme. Finally, a numerical example is presented to verify the effectiveness of the proposed two-grid scheme.

EM modelling of arbitrary shaped dispersive chiral dielectric objects using a 3D leapfrog scheme on unstructured meshes

The standard Yee FDTD algorithm is widely used in computational electromagnetics because of its simplicity and divergence free nature. A generalization of this classical scheme to 3D unstructured co-volume meshes is adopted, based on the use of a Delaunay primal mesh and its high quality Voronoi dual. This circumvents the problem of accuracy losses, which are normally associated with the use of a staircased representation of curved material interfaces in the standard Yee scheme. The procedure has been successfully employed for modelling problems involving both isotropic and anisotropic lossy materials. Here, we consider the novel extension of this approach to allow for the challenging modelling of chiral materials, where the material parameters are frequency dependent. To adequately model the dispersive behaviour, the Z-transform is employed, using second order Padé approximations to maintain the accuracy of the basic scheme. To validate the implementation, the numerical results produced are compared with available analytical solutions. The stability of the chiral algorithm is also studied.

Numerical analysis of CNLF modular Grad–Div stabilization method for time-dependent Navier–Stokes equations

This paper presents a modular Grad–Div(GD) algorithm based on Crank–Nicolson LeapFrog (CNLF) scheme for the non-stationary Navier–Stokes model, which is second-order accurate. The CNLF modular Grad–Div(CNLF-MGD) method has the obvious advantage that the computational efficiency will not crash or slow down with the large parameters. Stability and error estimate of the numerical method are given. Some numerical tests are proposed to justify the theoretical analysis.

Two-dimensional leapfrog scheme for trajectories of relativistic charged particles in static axisymmetric electric and magnetic field

A method for the calculation of two-dimensional particle trajectories is proposed in this work. It makes use of the cylindrical symmetry and the simplification of the static electric field, so that there should be no systematic error for the centered large-orbit rotations nor for the acceleration or deceleration in a uniform electric field. The method also shows a lower error level than the standard Boris method in many cases. Typical applications of this method are for example, electron microscopes, electron guns and collectors of gyro-devices as well as of other vacuum tubes, which can be described in axisymmetric cylindrical coordinates. Besides, the proposed method enforces the conservation of canonical angular momentum by construction, which is expected to show its advantages in the simulation of cusp electron guns and other components relying on non-adiabatic transitions in the externally applied static magnetic field.

How fast do young star clusters expel their natal gas? Estimating the upper limit of the gas expulsion time-scale

ABSTRACT Formation of massive stars within embedded star clusters starts a complex interplay between their feedback, inflowing gas, and stellar dynamics, which often includes close stellar encounters. Hydrodynamical simulations usually resort to substantial simplifications to model embedded clusters. Here, we address the simplification which approximates the whole star cluster by a single sink particle, which completely neglects the internal stellar dynamics. In order to model the internal stellar dynamics, we implement a Hermite predictor–corrector integration scheme to the hydrodynamic code flash. As we illustrate by a suite of tests, this integrator significantly outperforms the current leap-frog scheme, and it is able to follow the dynamics of small compact stellar systems without the necessity to soften the gravitational potential. We find that resolving individual massive stars instead of representing the whole cluster by a single energetic source has a profound influence on the gas component: for clusters of mass less than $\approx3 \times 10^3 \, \mathrm{M}_{\odot }$ , it slows gas expulsion by a factor of ≈5 to $\approx 1 \, \mathrm{Myr}$, and it results in substantially more complex gas structures. With increasing cluster mass (up to $\approx 3\times 10^3 \, \mathrm{M}_{\odot }$), the gas expulsion time-scale slightly decreases. However, more massive clusters ($\gtrsim 5\times 10^3 \, \mathrm{M}_{\odot }$) are unable to clear their natal gas with photoionizing radiation and stellar winds only if they form with a star formation efficiency (SFE) of 1/3. This implies that the more massive clusters are either cleared with another feedback mechanism or they form with an SFE higher than 1/3.

Unconditionally Stable Implicit Schemes for Transient Analysis of Lossy Multiconductor Lines

This article shows the advantages provided by implicit time stepping schemes for the study of multiconductor transmission lines. Implicit schemes allow for choosing time and space steps independently by attending only to resolution criteria, without the limitation imposed by the Courant-Friedrich-Levy stability condition. Hence, they behave favorably whenever a large time step would be preferable, for instance when a massive number of simulations is required for sensitivity analyses. We focus our attention on two common schemes in electromagnetics, Crank-Nicolson and Newmark-β methods, that are shown to have the same accuracy and to lead to the same solving-equation structure. We illustrate how to include frequency-dependent losses in the algorithms, and we show the influence of dispersion on fast transients. Results show that implicit formulations can be favorably compared with the classical explicit leap-frog scheme.

A leap-frog meshless method with radial basis functions for simulating electromagnetic wave splitter and rotator

In this paper, we develop a meshless method by using radial basis functions (RBFs) to solve time domain Maxwell’s equations resulting from simulating wave propagation in electromagnetic wave splitter and rotator devices. To simulate wave propagation in these devices, we introduce a perfectly matched layer (PML) to reduce the unbounded physical domain problem to a bounded domain problem. Using PML leads to a multi-physics problem with different governing equations in different subdomains, which makes the simulation quite challenging. By following the idea of leap-frog FDTD scheme, we develop the leap-frog RBF meshless method to solve these complicated coupled modeling equations. Extensive numerical results by using multiquadric RBF and Gaussian RBF are presented to demonstrate the effectiveness of our meshless method.

A Continuous-Stage Modified Leap-Frog Scheme for High-Dimensional Semi-Linear Hamiltonian Wave Equations

A Continuous-Stage Modified Leap-Frog Scheme for High-Dimensional Semi-Linear Hamiltonian Wave Equations

A second order, linear, unconditionally stable, Crank–Nicolson–Leapfrog scheme for phase field models of two-phase incompressible flows

In this article we propose a second order, linear, unconditionally stable, implicit–explicit scheme based on the Crank–Nicolson–Leapfrog discretization and the artificial compression method for solving phase field models of two-phase incompressible flows. We show that the scheme is unconditionally long-time stable. Numerical examples are provided to demonstrate the accuracy and long-time stability.

Analysis of Metal Oxide Varistor Arresters for Protection of Multiconductor Transmission Lines Using Unconditionally-Stable Crank–Nicolson FDTD

Surge arresters may represent an efficient choice for limiting lightning surge effects, significantly reducing the outage rate of power lines. The present work firstly presents an efficient numerical approach suitable for insulation coordination studies based on an implicit Crank–Nicolson finite difference time domain method; then, the IEEE recommended surge arrester model is reviewed and implemented by means of a local implicit scheme, based on a set of non-linear equations, that are recast in a suitable form for efficient solution. The model is proven to ensure robustness and second-order accuracy. The implementation of the arrester model in the implicit Crank–Nicolson scheme represents the added value brought by the present study. Indeed, its preserved stability for larger time steps allows reducing running time by more than 60 % compared to the well-known finite difference time domain method based on the explicit leap-frog scheme. The reduced computation time allows faster repeated solutions, which need to be looked for on assessing the lightning performance (randomly changing, parameters such as peak current, rise time, tail time, location of the vertical leader channel, phase conductor voltages, footing resistance, insulator strength, etc. would need to be changed thousands of times).

Construction and analysis of fourth order, energy consistent, family of explicit time discretizations for dissipative linear wave equations

This paper deals with the construction of a family of fourth order, energy consistent, explicit time discretizations for dissipative linear wave equations. The schemes are obtained by replacing the inversion of a matrix, that comes naturally after using the technique of the Modified Equation on the second order Leap Frog scheme applied to dissipative linear wave equations, by explicit approximations of its inverse. The stability of the schemes are studied using an energy analysis and a convergence analysis is carried out. Numerical results in 1D illustrate the space/time convergence properties of the schemes and their efficiency is compared to more classical time discretizations.

3D particle-in-cell simulations of electron magnetic confinement in electron cyclotron resonance ion sources.

The electron confinement in an electron cyclotron resonance ion source is considered a critical factor that has a great influence on the working efficiency. In this paper, a three-dimensional electromagnetic particle-in-cell code is presented for simulations of the electron confinement in minimum-B structures. Electron dynamics under the effect of electromagnetic fields is modeled through a leapfrog scheme by solving Maxwell's equations and the particle equations of motion iteratively. Some characteristics of electron magnetic confinement are obtained through a benchmarking example. Preliminary results of the simulation show good agreement with the well-known effects.

Superclose analysis of a two-grid finite element scheme for semilinear parabolic integro-differential equations

<p style='text-indent:20px;'>In this paper, a two-grid finite element scheme for semilinear parabolic integro-differential equations is proposed. In the two-grid scheme, continuous linear element is used for spatial discretization, while Crank-Nicolson scheme and Leap-Frog scheme are ultilized for temporal discretization. Based on the combination of the interpolation and Ritz projection technique, some superclose estimates between the interpolation and the numerical solution in the <inline-formula><tex-math id="M1">$ H^1 $</tex-math></inline-formula>-norm are derived. Notice that we only need to solve nonlinear problem once in the two-grid scheme, namely, the first time step on the coarse-grid space. A numerical example is presented to verify the effectiveness of the proposed two-grid scheme.</p>

Study of the accuracy and applicability of the difference scheme for solving the diffusion-convection problem at large grid Péclet numbers

The work is devoted to the study of a difference scheme for solving the diffusion-convection problem at large grid Péclet numbers. The suspension transport problem numerical solving is carried out using the improved Upwind Leapfrog difference scheme. Its difference operator is a linear combination of the operators of Upwind and Standard Leapfrog difference schemes, while the modified scheme is obtained from schemes with optimal weighting coefficients. At certain values of the weighting coefficients, this combination leads to mutual compensation of approximation errors, and the resulting scheme gets better properties than the original schemes. In addition, it includes a cell filling function that allows simulating problems in areas with complex geometry. Computational experiments were carried out to solve the suspension transport problem, which arises, for example, during the propagation of suspended matter plumes in an aquatic environment and changes in the bottom topography due to the deposition of suspended soil particles into the sediment during soil unloading into a reservoir (dumping). The results of modeling the suspension transport problem at various values of the grid Péclet number are presented. The algorithm implementation was carried out using the software and hardware architecture of parallel computing: on a central processing unit (Central Processing Unit - CPU) and on a graphics accelerator (Graphics Processing Unit - GPU). The solution to the applied problem has shown its efficiency on the CPU with small computational grids and, if it is necessary to decrease the space steps, then the GPU solution is preferable. It was found that, when using the modified Upwind Leapfrog scheme, an increase in the speed of the water flow does not lead to a loss of solution accuracy due to dissipative sources and is accompanied by an insignificant increase in computational labor costs.