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1.
《Physics letters. A》1986,118(4):172-176
The Lie-group formalism is applied to deduce the classical symmetries of the nonlinear heat equation, the diffusion-convection equation and the nonlinear wave equations. Some nonclassical symmetries are also presented. 相似文献
2.
Fangfang Fu Linghua Kong & Lan Wang 《advances in applied mathematics and mechanics.》2009,1(5):699-710
In this paper, we establish a family of symplectic integrators for a class
of high order Schrödinger equations with trapped terms. First, we find its symplectic structure and reduce it to a finite dimensional Hamilton system via spatial discretization.
Then we apply the symplectic Euler method to the Hamiltonian system.
It is demonstrated that the scheme not only preserves symplectic geometry structure
of the original system, but also does not require to resolve coupled nonlinear
algebraic equations which is different from the general implicit symplectic schemes.
The linear stability of the symplectic Euler scheme and the errors of the numerical
solutions are investigated. It shows that the semi-explicit scheme is conditionally
stable, first order accurate in time and $2l^{th}$ order accuracy in space. Numerical tests
suggest that the symplectic integrators are more effective than non-symplectic ones,
such as backward Euler integrators. 相似文献
3.
On the Fully Implicit Solution of a Phase-Field Model for Binary Alloy Solidification in Three Dimensions
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Christopher E. Goodyer Peter K. Jimack rew M. Mullis Hongbiao Dong & Yu Xie 《advances in applied mathematics and mechanics.》2012,4(6):665-684
A fully implicit numerical method, based upon a combination of
adaptively refined hierarchical meshes and geometric multigrid, is
presented for the simulation of binary alloy solidification in
three space dimensions. The computational techniques are presented for a
particular mathematical model, based upon the phase-field approach,
however, their applicability is of greater generality than for the
specific phase-field model used here. In particular, an implicit
second order time discretization is combined with the use of second order
spatial differences to yield a large nonlinear system of algebraic
equations as each time step. It is demonstrated that these equations
may be solved reliably and efficiently through the use of a nonlinear
multigrid scheme for locally refined grids. In effect, this paper
presents an extension of earlier research in two space
dimensions (J. Comput. Phys., 225 (2007), pp. 1271-1287) to fully
three-dimensional problems. This extension is validated against earlier
two-dimensional results and against some of the limited results
available in three dimensions, obtained using an explicit scheme.
The efficiency of the implicit approach and the multigrid solver are
then demonstrated and some sample computational results for the
simulation of the growth of dendrite structures are presented. 相似文献
4.
5.
In this Letter, the consensus problem via distributed nonlinear protocols for directed networks is investigated. Its dynamical behaviors are described by ordinary differential equations (ODEs). Based on graph theory, matrix theory and the Lyapunov direct method, some sufficient conditions of nonlinear protocols guaranteeing asymptotical or exponential consensus are presented and rigorously proved. The main contribution of this work is that for nonlinearly coupled networks, we generalize the results for undirected networks to directed networks. Consensus under pinning control technique is also developed here. Simulations are also given to show the validity of the theories. 相似文献
6.
7.
Samet Y. Kadioglu Dana A. Knoll Robert B. Lowrie Rick M. Rauenzahn 《Journal of computational physics》2010,229(22):8313-8332
We present a second order self-consistent implicit/explicit (methods that use the combination of implicit and explicit discretizations are often referred to as IMEX (implicit/explicit) methods , and ) time integration technique for solving radiation hydrodynamics problems. The operators of the radiation hydrodynamics are splitted as such that the hydrodynamics equations are solved explicitly making use of the capability of well-understood explicit schemes. On the other hand, the radiation diffusion part is solved implicitly. The idea of the self-consistent IMEX method is to hybridize the implicit and explicit time discretizations in a nonlinearly consistent way to achieve second order time convergent calculations. In our self-consistent IMEX method, we solve the hydrodynamics equations inside the implicit block as part of the nonlinear function evaluation making use of the Jacobian-free Newton Krylov (JFNK) method , and . This is done to avoid order reductions in time convergence due to the operator splitting. We present results from several test calculations in order to validate the numerical order of our scheme. For each test, we have established second order time convergence. 相似文献
8.
We present a fully second order implicit/explicit time integration technique for solving hydrodynamics coupled with nonlinear heat conduction problems. The idea is to hybridize an implicit and an explicit discretization in such a way to achieve second order time convergent calculations. In this scope, the hydrodynamics equations are discretized explicitly making use of the capability of well-understood explicit schemes. On the other hand, the nonlinear heat conduction is solved implicitly. Such methods are often referred to as IMEX methods [2], [1], [3]. The Jacobian-Free Newton Krylov (JFNK) method (e.g. [10], [9]) is applied to the problem in such a way as to render a nonlinearly iterated IMEX method. We solve three test problems in order to validate the numerical order of the scheme. For each test, we established second order time convergence. We support these numerical results with a modified equation analysis (MEA) [21], [20]. The set of equations studied here constitute a base model for radiation hydrodynamics. 相似文献
9.
Efficiency enhancement of a two-beam free-electron laser using a nonlinearly tapered wiggler
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A nonlinear and non-averaged model of a two-beam free-electron laser (FEL) wiggler that is tapered nonlinearly in the absence of slippage is presented. The two beams are assumed to have different energies, and the fundamental resonance of the higher energy beam is at the third harmonic of the lower energy beam. By using Maxwell's equations and the full Lorentz force equation of motion for the electron beams, coupled differential equations are derived and solved numerically by the fourth-order Runge-Kutta method. The amplitude of the wiggler field is assumed to decrease nonlinearly when the saturation of the third harmonic occurs. By simulation, the optimum starting point of the tapering and the slopes for reducing the wiggler amplitude are found. This technique can be applied to substantially improve the efficiency of the two-beam FEL in the XUV and X-ray regions. The effect of tapering on the dynamical stability of the fast electron beam is also studied. 相似文献
10.
Efficiency enhancement of a two-beam free-electron laser using a nonlinearly tapered wiggler
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A nonlinear and non-averaged model of a two-beam free-electron laser(FEL) wiggler that is tapered nonlinearly in the absence of slippage is presented.The two beams are assumed to have different energies,and the fundamental resonance of the higher energy beam is at the third harmonic of the lower energy beam.By using Maxwell’s equations and the full Lorentz force equation of motion for the electron beams,coupled differential equations are derived and solved numerically by the fourth-order Runge-Kutta method.The amplitude of the wiggler field is assumed to decrease nonlinearly when the saturation of the third harmonic occurs.By simulation,the optimum starting point of the tapering and the slopes for reducing the wiggler amplitude are found.This technique can be applied to substantially improve the efficiency of the two-beam FEL in the XUV and X-ray regions.The effect of tapering on the dynamical stability of the fast electron beam is also studied. 相似文献
11.
This paper discusses a novel fully implicit formulation for a one-dimensional electrostatic particle-in-cell (PIC) plasma simulation approach. Unlike earlier implicit electrostatic PIC approaches (which are based on a linearized Vlasov–Poisson formulation), ours is based on a nonlinearly converged Vlasov–Ampére (VA) model. By iterating particles and fields to a tight nonlinear convergence tolerance, the approach features superior stability and accuracy properties, avoiding most of the accuracy pitfalls in earlier implicit PIC implementations. In particular, the formulation is stable against temporal (Courant–Friedrichs–Lewy) and spatial (aliasing) instabilities. It is charge- and energy-conserving to numerical round-off for arbitrary implicit time steps (unlike the earlier “energy-conserving” explicit PIC formulation, which only conserves energy in the limit of arbitrarily small time steps). While momentum is not exactly conserved, errors are kept small by an adaptive particle sub-stepping orbit integrator, which is instrumental to prevent particle tunneling (a deleterious effect for long-term accuracy). The VA model is orbit-averaged along particle orbits to enforce an energy conservation theorem with particle sub-stepping. As a result, very large time steps, constrained only by the dynamical time scale of interest, are possible without accuracy loss. Algorithmically, the approach features a Jacobian-free Newton–Krylov solver. A main development in this study is the nonlinear elimination of the new-time particle variables (positions and velocities). Such nonlinear elimination, which we term particle enslavement, results in a nonlinear formulation with memory requirements comparable to those of a fluid computation, and affords us substantial freedom in regards to the particle orbit integrator. Numerical examples are presented that demonstrate the advertised properties of the scheme. In particular, long-time ion acoustic wave simulations show that numerical accuracy does not degrade even with very large implicit time steps, and that significant CPU gains are possible. 相似文献
12.
The nonlinear evolution
of the kink instability of a plasma
with an elliptic magnetic stagnation line is studied by means
of an amplitude expansion of the ideal magnetohydrodynamic equations.
Wahlberg et al. [12] have shown that, near marginal stability,
the nonlinear evolution of the stability can be described in terms
of a two-dimensional potential U(X,Y), where X and Y represent
the amplitudes of the perturbations with positive and negative
helical polarization. The potential U(X,Y) is found to be nonlinearly
stabilizing for all values of the polarization. In our paper a Lagrangian
and an invariant variational principle for two coupled nonlinear ordinal
differential equations describing the nonlinear evolution of the stagnation
line instability with arbitrary polarization are given. Using a trial function
in a rectangular box we find the functional integral. The general case for
the two box potential can be obtained on the basis of a different ansatz
where we approximate the Jost function by polynomials of order n instead
of a piecewise linear function. An example for the second order is
given to illustrate the general case. Some considerations concerning
solar filaments and filament bands (circular or straight) are
indicated as possible applications besides laboratory experiments
with cusp geometry corresponding to quadripolar cusp geometries for
some clouds and thunderstorms. 相似文献
13.
There are many interesting methods can be utilized to construct special solutions of nonlinear differential equations with constant coefficients. However, most of these methods are not applicable to nonlinear differential equations with variable coefficients. A new method is presented in this Letter, which can be used to find special solutions of nonlinear differential equations with variable coefficients. This method is based on seeking appropriate Bernoulli equation corresponding to the equation studied. Many well-known equations are chosen to illustrate the application of this method. 相似文献
14.
15.
M. Tokman 《Journal of computational physics》2011,230(24):8762-8778
We propose a new class of the exponential propagation iterative methods of Runge–Kutta-type (EPIRK). The EPIRK schemes are exponential integrators that can be competitive with explicit and implicit methods for integration of large stiff systems of ODEs. Introducing the new, more general than previously proposed, ansatz for EPIRK schemes allows for more flexibility in deriving computationally efficient high-order integrators. Recent extension of the theory of B-series to exponential integrators [1] is used to derive classical order conditions for schemes up to order five. An algorithm to systematically solve these conditions is presented and several new fifth order schemes are constructed. Several numerical examples are used to verify the order of the methods and to illustrate the performance of the new schemes. 相似文献
16.
Rafael Cortell 《Physics letters. A》2008,372(5):631-636
This Letter endeavours to complete an earlier numerical analysis for flow and heat transfer in a viscous fluid over a sheet nonlinearly stretched by extending the investigation in two directions. On one side, the effects of thermal radiation are included in the energy equation, and, on the other hand, the prescribed wall heat flux case (PHF case) is also analyzed. The governing partial differential equations are converted into nonlinear ordinary differential equations by a similarity transformation. The variations of dimensionless surface temperature as well as flow and heat-transfer characteristics with the governing dimensionless parameters of the problem, which include a nonlinearly stretching sheet, thermal radiation, viscous dissipation and power-law index of the wall temperature parameters, are graphed and tabulated. 相似文献
17.
Hybrid functions for nonlinear initial-value problems with applications to Lane-Emden type equations
A numerical method for solving nonlinear initial-value problems is proposed. The Lane-Emden type equations which have many applications in mathematical physics are then considered. The method is based upon hybrid function approximations. The properties of hybrid of block-pulse functions and Lagrange interpolating polynomials are presented and are utilized to reduce the computation of nonlinear initial-value problems to a system of non-algebraic equations. The method is easy to implement and yields very accurate results. 相似文献
18.
This paper obtains solitons as well as other solutions to a few nonlinear evolution equations that appear in various areas of mathematical physics. The two analytical integrators that are applied to extract solutions are tan–cot method and functional variable approaches. The soliton solutions can be used in the further study of shallow water waves in (1+1) as well as (2+1) dimensions. 相似文献
19.
This paper describes two new techniques which give improved exponential finite difference solutions of Burgers’ equation. These techniques are called implicit exponential finite difference method and fully implicit exponential finite difference method for solving Burgers’ equation. As the Burgers’ equation is nonlinear, the scheme leads to a system of nonlinear equations. At each time-step, Newton’s method is used to solve this nonlinear system. The results are compared with exact values and it is clearly shown that results obtained using both the methods are precise and reliable. 相似文献
20.
It is shown that expansion waves for the compressible Navier-Stokes equations are nonlinearly stable. The expansion waves are constructed for the compressible Euler equations based on the inviscid Burgers equation. Our result shows that Navier-Stokes equations and Euler equations are time-asymptotically equivalent on the level of expansion waves. The result is proved using the energy method, making essential use of the expansion of the underlining nonlinear waves and the specific form of the constitutive eqution for a polytropic gas.Supported in part by NSF Grant DMS-87-03971 and Army Grant DAAL03-87-K-0063Supported in part by Army Grant DAAL03-87-K-0063 相似文献