Standard Leapfrog Scheme Research Articles

Development and Research of a Modified Upwind Leapfrog Scheme for Solving Transport Problems

Modeling complex hydrodynamic processes in coastal systems is an important problem of mathematical modeling that cannot be solved analytically. The approximation of convective terms is difficult from the point of view of error reduction. This paper proposes a difference scheme based on a linear combination of the Upwind Leapfrog scheme with 2/3 weight coefficient, and the Standard Leapfrog scheme with 1/3 weight coefficient. The weight coefficients are obtained as a result of solving the problem of minimizing the approximation error. Numerical experiments show the advantage of the developed scheme in comparison with other modifications of the Upwind Leapfrog scheme in the case when the convective transport prevails over the diffusion one. The proposed difference scheme solves transport problems more effectively than classical difference schemes in the case when the Péclet number falls in the range from 2 to 20. It follows that the considered difference scheme allows hydrodynamic problems to be solved in regions of complex shape effectively.

Explicit Time Stepping for the Wave Equation using CutFEM with Discrete Extension

In this note we develop a fully explicit cut finite element method for the wave equation. The method is based on using a standard leap frog scheme combined with an extension operator that defines the nodal values outside of the domain in terms of the nodal values inside the domain. We show that the mass matrix associated with the extended finite element space can be lumped leading to a fully explicit scheme. We derive stability estimates for the method and provide optimal order a priori error estimates. Finally, we present some illustrating numerical examples.

Accelerating NBODY6 with a graphics processing unit-enabled Particle-Particle Particle-Tree scheme

ABSTRACTWe describe a modified version of the nbody6 code for simulating star clusters which greatly improves computational efficiency while sacrificing little in the way of accuracy. The distant force calculator is replaced by a graphics processing unit-enabled Barnes–Hut code, and integration is done with a standard leap frog scheme. Short-range forces continue to use the CPU-based fourth-order Hermite predictor-corrector scheme of nbody6. Our code outperforms nbody6 for systems with more than 3 × 105 particles and runs more than a factor 2 faster for systems of 106 particles with similar energy conservation. Our code should be useful for simulating realistic dense stellar clusters, such as globular clusters or galactic nuclei.

О разностных схемах кабаре и крест

Для решения задачи переноса в статье предложено использовать схему, построенную на основе линейной комбинации разностной схемы кабаре (англ. Upwind Leapfrog) и крест (англ. Standard Leapfrog) с весовыми коэффициентами, полученными в результате минимизации погрешности аппроксимации. Проведено сравнение расчетов для задачи переноса на основе предложенной схемы с результатами, полученными с использованием схемы, построенной на основе линейной комбинации схемы с центральными разностями и схемы кабаре, и двухпараметрической разностной схемы третьего порядка точности. In order to solve the transfer problem, it is proposed to use the scheme based on a linear combination of the upwind and standard leapfrog difference schemes with weighting coefficients obtained by minimizing the approximation error. The estimate of the approximation error of the proposed difference scheme shows that, for small Courant numbers, this scheme whose approximation error is $O(ch^2)$, where the constant $c$ is significantly less than unity, is preferable to use than the original upwind and standard leapfrog schemes whose approximation errors are $O(h^2)$. The numerical results for the transfer problem based on the proposed scheme are compared with the results obtained using the following schemes: (i) the scheme based on a linear combination of the standard leapfrog scheme and the upwind leapfrog sscheme and (ii) the two-parameter difference scheme of the third order of accuracy.

Implicit scheme for Maxwell equations solution in case of flat 3D domains

We present a new finite-difference scheme for Maxwell's equations solution for three-dimensional domains with different scales in different directions. The stability condition of the standard leap-frog scheme requires decreasing of the time-step with decreasing of the minimal spatial step, which depends on the minimal domain size. We overcome the conditional stability by modifying the standard scheme adding implicitness in the direction of the smallest size. The new scheme satisfies the Gauss law for the electric and magnetic fields in the final- differences. The approximation order, the maintenance of the wave amplitude and propagation speed, the invariance of the wave propagation on angle with the coordinate axes are analyzed.

Explicit local time-stepping methods for Maxwell’s equations

Explicit local time-stepping methods are derived for time dependent Maxwell equations in conducting and non-conducting media. By using smaller time steps precisely where smaller elements in the mesh are located, these methods overcome the bottleneck caused by local mesh refinement in explicit time integrators. When combined with a finite element discretisation in space with an essentially diagonal mass matrix, the resulting discrete time-marching schemes are fully explicit and thus inherently parallel. In a non-conducting source-free medium they also conserve a discrete energy, which provides a rigorous criterion for stability. Starting from the standard leap-frog scheme, local time-stepping methods of arbitrarily high accuracy are derived for non-conducting media. Numerical experiments with a discontinuous Galerkin discretisation in space validate the theory and illustrate the usefulness of the proposed time integration schemes.

Numerical solution of nonlinear wave equations in stratified dispersive media

Nonlinear wave motion in dispersive media is solved numerically. The model applies to laser propagation in a relativistic plasma. The latter causes, besides dispersion, nonlinear effects due to relativistic mass variation in the presence of strong laser pulses. A new variant of the Gautschi-type integrator for reducing the number of time steps is proposed as a fast solver for such nonlinear wave-equations. In order to reduce the number of spatial grid points, a physically motivated quasi-envelope approach (QEA) is introduced. The new method turns out to reduce the computational time significantly compared to the standard leap-frog scheme.

Parallel 3-D hybrid particle-in-cell model for electromagnetic plasma

We have presented here the parallelization of a three-dimensional electromagnetic hybrid plasma particle-in-cell model and its performance on a eight-node symmetric multiprocessor machine. One-dimensional domain decomposition has been used to parallelize the hybrid particle-in-cell model where the electromagnetic fields are updated by solving the coupled set of Maxwell's equations using the finite-difference-time-domain technique and the particles are advanced spatially using the standard leap-frog scheme. Inter-processor communication between the adjacent processor domain has been achieved by using several common message-passing-interface routines. The performance results presented here are based on the problem of solar wind interaction with the earth's magnetic field.

An Alternative Leapfrog Scheme for Surface Gravity Wave Equations

An alternative leapfrog scheme using a staggered time grid system is proposed to solve surface gravity wave equations. In addition to the nondissipative second-order accuracy scheme that is inherent in the standard leapfrog scheme, the alternative leapfrog scheme has the advantages that 1) no separation of the solution occurs at the even and odd time levels, 2) it is twice as efficient as the standard leapfrog scheme, and 3) it requires only the one time level storage. Numerical experiments show its numerical properties and computational efficiency. The wave amplitude is not damped and the total energy is conserved. The alternative leapfrog scheme is one of the most efficient schemes. It is applied to the surface gravity wave terms in ocean circulation models, and its usefulness is shown.

PARALLELIZATION AND PERFORMANCE OF A 3-D ELECTROMAGNETIC PLASMA PIC CODE

International Journal of Computational Engineering ScienceVol. 03, No. 02, pp. 117-127 (2002) No AccessPARALLELIZATION AND PERFORMANCE OF A 3-D ELECTROMAGNETIC PLASMA PIC CODESAIKAT SAHASAIKAT SAHADepartment of Electrical and Computer Engineering, The University of Alabama in Huntsville, Huntsville, AL 35899, USA Search for more papers by this author https://doi.org/10.1142/S1465876302000587Cited by:0 PreviousNext AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail AbstractA fully 3 dimensional parallel explicit electromagnetic plasma Particle-in-Cell code is developed and is tested on 64 processors of SGI-Origin 2000 supercomputer at JPL / Caltech. The SGI-Origin 2000 allows one to run simulations as large as 256 × 128 × 32, i.e. approximately 106 grid cells and 18 × 106 particles of each species (electrons and ions) with 18 particles per cell. The particles in this code are pushed using the standard leapfrog scheme and the self-consistent electromagnetic fields are updated by solving the time dependent Maxwell's equations using the Finite-Difference-Time-Domain technique. We have adopted the one dimensional domain decomposition technique to parallelize the sequential electromagnetic particle-in-cell code. Testing and performance analyses of this code were done by generating a narrow current sheet in the system as seen in magnetic reconnection phenomenon.Keywords:Parallel3-DElectromagneticPlasmaPIC FiguresReferencesRelatedDetails Recommended Vol. 03, No. 02 Metrics History KeywordsParallel3-DElectromagneticPlasmaPICPDF download

A stable interpolation technique for FDTD on non-orthogonal grids

The application of the FDTD algorithm on generalized non-orthogonal meshes, following the basic ideas of Holland (1983), has been investigated by many authors for several years now, and detailed dispersion analysis as well as convergence studies have been published. Already in 1992 also a general stability criterion was given for the time integration using the standard leap-frog scheme (Lee et al.). Many authors, however, still propose some damped time stepping algorithms to work around unexpected instabilities in the discretization method. In this paper the origin of this type of instability is revealed, and a technique to obtain a stable discretization of Maxwell's equations on non-orthogonal grids is proposed. To obtain more insight into the stability properties of the method, it is reformulated according to the matrix–vector notation of the Finite Integration Technique. © 1998 John Wiley & Sons, Ltd.

Explicit upwind schemes for lossy MTLs with linear terminations

The time domain multiconductor transmission line (MTL) equations are written as a general first order system of partial differential equations and a characteristic decomposition is used to obtain first order and second order accurate upwind differencing schemes. Linear boundary conditions in the form of generalized Thevenin equivalent sources are incorporated into the scheme. These schemes are compared with the standard time-space centered second order accurate leapfrog scheme where the current and voltage variables are interlaced in space and time. For any general explicit numerical scheme, for a given MTL, only the fastest propagating TEM mode can be solved for at the Courant limit of the scheme. This causes the other slower modes to disperse. The results of our comparisons, show that at the Courant number both upwind schemes produce less numerical dispersion for the slower propagating modes than the standard leapfrog scheme under the same conditions. In addition, the Courant number of the second order upwind scheme is twice that of the leapfrog scheme. These advantages make the upwind schemes better tools to model inhomogeneous MTLs with linear terminations.

Accuracy of a Lax–Wendroff scheme for the wave equation

In this paper, an investigation of the behavior of a finite difference scheme for solving initial value problems for the wave equation is reported. The scheme is a Lax–Wendroff method, which is second-order accurate. The objective of this work is to assess its accuracy in comparison to the well-known leapfrog method. To this end, spectral analysis is performed, which has a natural interpretation in terms of numerical dispersion relation. It is found that the scheme is highly dispersive, dissipative, and anisotropic. Numerical experiments were carried out to compare the performance of this scheme with the leapfrog scheme. It is apparent from our numerical tests and analysis that the standard leapfrog scheme outperforms a Lax–Wendroff scheme in solving linear wave equations.