共查询到20条相似文献,搜索用时 31 毫秒
1.
Nicolas Crouseilles & Thomas Respaud 《Communications In Computational Physics》2011,10(4):1001-1026
In this report, a charge preserving numerical resolution of the 1D Vlasov-Ampère
equation is achieved, with a forward Semi-Lagrangian method introduced
in [10]. The Vlasov equation belongs to the kinetic way of simulating plasmas evolution, and is coupled with the Poisson's equation, or equivalently under charge conservation, the Ampère's one, which self-consistently rules the electric field evolution. In
order to ensure having proper physical solutions, it is necessary that the scheme preserves charge numerically. B-spline deposition will be used for the interpolation step.
The solving of the characteristics will be made with a Runge-Kutta 2 method and with
a Cauchy-Kovalevsky procedure. 相似文献
2.
In this paper, a new symmetric energy-conserved splitting FDTD scheme
(symmetric EC-S-FDTD) for Maxwell's equations is proposed. The new algorithm inherits
the same properties of our previous EC-S-FDTDI and EC-S-FDTDII algorithms:
energy-conservation, unconditional stability and computational efficiency. It keeps the
same computational complexity as the EC-S-FDTDI scheme and is of second-order accuracy
in both time and space as the EC-S-FDTDII scheme. The convergence and error
estimate of the symmetric EC-S-FDTD scheme are proved rigorously by the energy
method and are confirmed by numerical experiments. 相似文献
3.
New Energy-Conserved Identities and Super-Convergence of the Symmetric EC-S-FDTD Scheme for Maxwell's Equations in 2D
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The symmetric energy-conserved splitting FDTD scheme developed in [1] is
a very new and efficient scheme for computing the Maxwell's equations. It is based on
splitting the whole Maxwell's equations and matching the x-direction and y-direction
electric fields associated to the magnetic field symmetrically. In this paper, we make
further study on the scheme for the 2D Maxwell's equations with the PEC boundary
condition. Two new energy-conserved identities of the symmetric EC-S-FDTD scheme
in the discrete H1-norm are derived. It is then proved that the scheme is unconditionally stable in the discrete H1-norm. By the new energy-conserved identities, the
super-convergence of the symmetric EC-S-FDTD scheme is further proved that it is
of second order convergence in both time and space steps in the discrete H1-norm.
Numerical experiments are carried out and confirm our theoretical results. 相似文献
4.
A fully discrete discontinuous Galerkin method is introduced for solving
time-dependent Maxwell's equations. Distinguished from the Runge-Kutta discontinuous Galerkin method (RKDG) and the finite element time domain method (FETD), in
our scheme, discontinuous Galerkin methods are used to discretize not only the spatial
domain but also the temporal domain. The proposed numerical scheme is proved to be
unconditionally stable, and a convergent rate $\mathcal{O}((∆t)^{r+1}+h^{k+1/2})$ is established under the $L^2$ -norm when polynomials of degree at most $r$ and $k$ are used for temporal and
spatial approximation, respectively. Numerical results in both 2-D and 3-D are provided to validate the theoretical prediction. An ultra-convergence of order $(∆t)^{2r+1}$ in
time step is observed numerically for the numerical fluxes w.r.t. temporal variable at
the grid points. 相似文献
5.
Solving Allen-Cahn and Cahn-Hilliard Equations Using the Adaptive Physics Informed Neural Networks
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Colby L. Wight & Jia Zhao 《Communications In Computational Physics》2021,29(3):930-954
Phase field models, in particular, the Allen-Cahn type and Cahn-Hilliard
type equations, have been widely used to investigate interfacial dynamic problems.
Designing accurate, efficient, and stable numerical algorithms for solving the phase
field models has been an active field for decades. In this paper, we focus on using
the deep neural network to design an automatic numerical solver for the Allen-Cahn
and Cahn-Hilliard equations by proposing an improved physics informed neural network (PINN). Though the PINN has been embraced to investigate many differential
equation problems, we find a direct application of the PINN in solving phase-field
equations won't provide accurate solutions in many cases. Thus, we propose various
techniques that add to the approximation power of the PINN. As a major contribution of this paper, we propose to embrace the adaptive idea in both space and time
and introduce various sampling strategies, such that we are able to improve the efficiency and accuracy of the PINN on solving phase field equations. In addition, the
improved PINN has no restriction on the explicit form of the PDEs, making it applicable to a wider class of PDE problems, and shedding light on numerical approximations
of other PDEs in general. 相似文献
6.
Nathan L. Gibson 《Communications In Computational Physics》2015,18(5):1234-1263
Electromagnetic wave propagation in complex dispersive media is governed
by the time dependent Maxwell's equations coupled to equations that describe the
evolution of the induced macroscopic polarization. We consider "polydispersive" materials
represented by distributions of dielectric parameters in a polarization model.
The work focuses on a novel computational framework for such problems involving
Polynomial Chaos Expansions as a method to improve the modeling accuracy of the
Debye model and allow for easy simulation using the Finite Difference Time Domain
(FDTD) method. Stability and dispersion analyses are performed for the approach
utilizing the second order Yee scheme in two spatial dimensions. 相似文献
7.
Xue Jiang Linbo Zhang & Weiying Zheng 《Communications In Computational Physics》2013,13(2):559-582
In this paper, hp-adaptive finite element methods are studied for time-harmonic Maxwell's equations. We propose the parallel hp-adaptive algorithms on conforming unstructured tetrahedral meshes based on residual-based a posteriori error estimates. Extensive numerical experiments are reported to investigate the efficiency of the hp-adaptive methods for point singularities, edge singularities, and an engineering benchmark problem of Maxwell's equations. The hp-adaptive methods show much better performance than the h-adaptive method. 相似文献
8.
Na Liu Luis Tobó n Yifa Tang & Qing Huo Liu 《Communications In Computational Physics》2015,17(2):458-486
It is well known that conventional edge elements in solving vector Maxwell's
eigenvalue equations by the finite element method will lead to the presence of spurious
zero eigenvalues. This problem has been addressed for the first order edge element
by Kikuchi by the mixed element method. Inspired by this approach, this paper
describes a higher order mixed spectral element method (mixed SEM) for the computation
of two-dimensional vector eigenvalue problem of Maxwell's equations. It
utilizes Gauss-Lobatto-Legendre (GLL) polynomials as the basis functions in the finite-element
framework with a weak divergence condition. It is shown that this method
can suppress all spurious zero and nonzero modes and has spectral accuracy. A rigorous
analysis of the convergence of the mixed SEM is presented, based on the higher
order edge element interpolation error estimates, which fully confirms the robustness
of our method. Numerical results are given for homogeneous, inhomogeneous, L-shape,
coaxial and dual-inner-conductor cavities to verify the merits of the proposed
method. 相似文献
9.
A Numerical Method and Its Error Estimates for the Decoupled Forward-Backward Stochastic Differential Equations
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In this paper, a new numerical method for solving the decoupled forward-backward stochastic differential equations (FBSDEs) is proposed based on some specially derived reference equations. We rigorously analyze errors of the proposed method
under general situations. Then we present error estimates for each of the specific cases
when some classical numerical schemes for solving the forward SDE are taken in the
method; in particular, we prove that the proposed method is second-order accurate
if used together with the order-2.0 weak Taylor scheme for the SDE. Some examples
are also given to numerically demonstrate the accuracy of the proposed method and
verify the theoretical results. 相似文献
10.
Construction,Analysis and Application of Coupled Compact Difference Scheme in Computational Acoustics and Fluid Flow Problems
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Jitenjaya Pradhan Amit Bikash Mahato Satish D. Dhandole & Yogesh G. Bhumkar 《Communications In Computational Physics》2015,18(4):957-984
In the present work, a new type of coupled compact difference scheme has
been proposed for the solution of computational acoustics and flow problems. The
proposed scheme evaluates the first, the second and the fourth derivative terms simultaneously.
Derived compact difference scheme has a significant spectral resolution and
a physical dispersion relation preserving (DRP) ability over a considerable wavenumber
range when a fourth order four stage Runge-Kutta scheme is used for the time
integration. Central stencil has been used for the present numerical scheme to evaluate
spatial derivative terms. Derived scheme has the capability of adding numerical
diffusion adaptively to attenuate spurious high wavenumber oscillations responsible
for numerical instabilities. The DRP nature of the proposed scheme across a wider
wavenumber range provides accurate results for the model wave equations as well
as computational acoustic problems. In addition to the attractive feature of adaptive
diffusion, present scheme also helps to control spurious reflections from the domain
boundaries and is projected as an alternative to the perfectly matched layer (PML)
technique. 相似文献
11.
A Weighted Runge-Kutta Discontinuous Galerkin Method for 3D Acoustic and Elastic Wave-Field Modeling
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Numerically solving 3D seismic wave equations is a key requirement for
forward modeling and inversion. Here, we propose a weighted Runge-Kutta discontinuous Galerkin (WRKDG) method for 3D acoustic and elastic wave-field modeling. For this method, the second-order seismic wave equations in 3D heterogeneous anisotropic media are transformed into a first-order hyperbolic system, and
then we use a discontinuous Galerkin (DG) solver based on numerical-flux formulations for spatial discretization. The time discretization is based on an implicit diagonal Runge-Kutta (RK) method and an explicit iterative technique, which avoids
solving a large-scale system of linear equations. In the iterative process, we introduce
a weighting factor. We investigate the numerical stability criteria of the 3D method in
detail for linear and quadratic spatial basis functions. We also present a 3D analysis of
numerical dispersion for the full discrete approximation of acoustic equation, which
demonstrates that the WRKDG method can efficiently suppress numerical dispersion
on coarse grids. Numerical results for several different 3D models including homogeneous and heterogeneous media with isotropic and anisotropic cases show that the 3D
WRKDG method can effectively suppress numerical dispersion and provide accurate
wave-field information on coarse mesh. 相似文献
12.
Huangxin Chen Jingzhi Li Weifeng Qiu & Chao Wang 《Communications In Computational Physics》2021,29(4):1125-1151
The quad-curl problem arises in the resistive magnetohydrodynamics
(MHD) and the electromagnetic interior transmission problem. In this paper we study
a new mixed finite element scheme using Nédélec's edge elements to approximate
both the solution and its curl for quad-curl problem on Lipschitz polyhedral domains.
We impose element-wise stabilization instead of stabilization along mesh interfaces.
Thus our scheme can be implemented as easy as standardNédélec's methods for
Maxwell's equations. Via a discrete energy norm stability due to element-wise stabilization, we prove optimal convergence under a low regularity condition. We also
extend the mixed finite element scheme to the quad-curl eigenvalue problem and provide corresponding convergence analysis based on that of source problem. Numerical
examples are provided to show the viability and accuracy of the proposed method for
quad-curl source problem. 相似文献
13.
Analysis and Application of Single Level,Multi-Level Monte Carlo and Quasi-Monte Carlo Finite Element Methods for Time-Dependent Maxwell's Equations with Random Inputs
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This article is devoted to three quadrature methods for the rapid solution
of stochastic time-dependent Maxwell's equations with uncertain permittivity, permeability and initial conditions. We develop the mathematical analysis of the error estimate for single level Monte Carlo method, multi-level Monte Carlo method, and the
quasi-Monte Carlo method. The theoretical results are supplemented by numerical
experiments. 相似文献
14.
A Nominally Second-Order Cell-Centered Finite Volume Scheme for Simulating Three-Dimensional Anisotropic Diffusion Equations on Unstructured Grids
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Pascal Jacq Pierre-Henri Maire & R& eacute mi Abgrall 《Communications In Computational Physics》2014,16(4):841-891
We present a finite volume based cell-centered method for solving diffusion
equations on three-dimensional unstructured grids with general tensor conduction.
Our main motivation concerns the numerical simulation of the coupling between fluid
flows and heat transfers. The corresponding numerical scheme is characterized by
cell-centered unknowns and a local stencil. Namely, the scheme results in a global
sparse diffusion matrix, which couples only the cell-centered unknowns. The space
discretization relies on the partition of polyhedral cells into sub-cells and on the partition of cell faces into sub-faces. It is characterized by the introduction of sub-face
normal fluxes and sub-face temperatures, which are auxiliary unknowns. A sub-cell-based variational formulation of the constitutive Fourier law allows to construct an
explicit approximation of the sub-face normal heat fluxes in terms of the cell-centered
temperature and the adjacent sub-face temperatures. The elimination of the sub-face
temperatures with respect to the cell-centered temperatures is achieved locally at each
node by solving a small and sparse linear system. This system is obtained by enforcing
the continuity condition of the normal heat flux across each sub-cell interface impinging at the node under consideration. The parallel implementation of the numerical
algorithm and its efficiency are described and analyzed. The accuracy and the robustness of the proposed finite volume method are assessed by means of various numerical
test cases. 相似文献
15.
A Simple Solver for the Two-Fluid Plasma Model Based on PseudoSpectral Time-Domain Algorithm
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Benoit Morel Remo Giust Kazem Ardaneh & Francois Courvoisier 《Communications In Computational Physics》2021,29(3):955-978
We present a solver of 3D two-fluid plasma model for the simulation of
short-pulse laser interactions with plasma. This solver resolves the equations of the
two-fluid plasma model with ideal gas closure. We also include the Bhatnagar-Gross-Krook collision model. Our solver is based on PseudoSpectral Time-Domain (PSTD)
method to solve Maxwell's curl equations. We use a Strang splitting to integrate Euler
equations with source term: while Euler equations are solved with a composite scheme
mixing Lax-Friedrichs and Lax-Wendroff schemes, the source term is integrated with
a fourth-order Runge-Kutta scheme. This two-fluid plasma model solver is simple to
implement because it only relies on finite difference schemes and Fast Fourier Transforms. It does not require spatially staggered grids. The solver was tested against
several well-known problems of plasma physics. Numerical simulations gave results
in excellent agreement with analytical solutions or with previous results from the literature. 相似文献
16.
Avijit Chatterjee 《Communications In Computational Physics》2015,17(3):703-720
An algebraic multilevel method is proposed for efficiently simulating linear
wave propagation using higher-order numerical schemes. This method is used in
conjunction with the Finite Volume Time Domain (FVTD) technique for the numerical
solution of the time-domain Maxwell's equations in electromagnetic scattering
problems. In the multilevel method the solution is cycled through spatial operators
of varying orders of accuracy, while maintaining highest-order accuracy at coarser approximation
levels through the use of the relative truncation error as a forcing function.
Higher-order spatial accuracy can be enforced using the multilevel method at a
fraction of the computational cost incurred in a conventional higher-order implementation.
The multilevel method is targeted towards electromagnetic scattering problems
at large electrical sizes which usually require long simulation times due to the use of
very fine meshes dictated by point-per-wavelength requirements to accurately model
wave propagation over long distances. 相似文献
17.
A Second-Order Finite Difference Method for Two-Dimensional Fractional Percolation Equations
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A finite difference method which is second-order accurate in time and in
space is proposed for two-dimensional fractional percolation equations. Using the
Fourier transform, a general approximation for the mixed fractional derivatives is analyzed.
An approach based on the classical Crank-Nicolson scheme combined with
the Richardson extrapolation is used to obtain temporally and spatially second-order
accurate numerical estimates. Consistency, stability and convergence of the method
are established. Numerical experiments illustrating the effectiveness of the theoretical
analysis are provided. 相似文献
18.
Omar al-Khayat & Hans Petter Langtangen 《Communications In Computational Physics》2012,12(4):1257-1274
First introduced in [2], the lumped particle framework is a flexible and numerically efficient framework for the modelling of particle transport in fluid flow.
In this paper, the framework is expanded to simulate multicomponent particle-laden
fluid flow. This is accomplished by introducing simulation protocols to model particles
over a wide range of length and time scales. Consequently, we present a time ordering
scheme and an approximate approach for accelerating the computation of evolution of
different particle constituents with large differences in physical scales. We apply the
extended framework on the temporal evolution of three particle constituents in sand-laden flow, and horizontal release of spherical particles. Furthermore, we evaluate the
numerical error of the lumped particle model. In this context, we discuss the Velocity-Verlet numerical scheme, and show how to apply this to solving Newton's equations
within the framework. We show that the increased accuracy of the Velocity-Verlet
scheme is not lost when applied to the lumped particle framework. 相似文献
19.
A New Family of High Order Unstructured MOOD and ADER Finite Volume Schemes for Multidimensional Systems of Hyperbolic Conservation Laws
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Rapha& euml l Loub& egrave re Michael Dumbser & Steven Diot 《Communications In Computational Physics》2014,16(3):718-763
In this paper, we investigate the coupling of the Multi-dimensional Optimal
Order Detection (MOOD) method and the Arbitrary high order DERivatives (ADER)
approach in order to design a new high order accurate, robust and computationally
efficient Finite Volume (FV) scheme dedicated to solving nonlinear systems of hyperbolic conservation laws on unstructured triangular and tetrahedral meshes in two and
three space dimensions, respectively. The Multi-dimensional Optimal Order Detection
(MOOD) method for 2D and 3D geometries has been introduced in a recent series of
papers for mixed unstructured meshes. It is an arbitrary high-order accurate Finite
Volume scheme in space, using polynomial reconstructions with a posteriori detection
and polynomial degree decrementing processes to deal with shock waves and other
discontinuities. In the following work, the time discretization is performed with an
elegant and efficient one-step ADER procedure. Doing so, we retain the good properties of the MOOD scheme, that is to say, the optimal high-order of accuracy is reached
on smooth solutions, while spurious oscillations near singularities are prevented. The
ADER technique not only reduces the cost of the overall scheme as shown
on a set of numerical tests in 2D and 3D, but also increases the stability of the overall scheme. A systematic comparison between classical unstructured ADER-WENO
schemes and the new ADER-MOOD approach has been carried out for high-order
schemes in space and time in terms of cost, robustness, accuracy and efficiency. The
main finding of this paper is that the combination of ADER with MOOD generally
outperforms the one of ADER and WENO either because at given accuracy MOOD isless expensive (memory and/or CPU time), or because it is more accurate for a given
grid resolution. A large suite of classical numerical test problems has been solved
on unstructured meshes for three challenging multi-dimensional systems of conservation laws: the Euler equations of compressible gas dynamics, the classical equations
of ideal magneto-Hydrodynamics (MHD) and finally the relativistic MHD equations
(RMHD), which constitutes a particularly challenging nonlinear system of hyperbolic
partial differential equation. All tests are run on genuinely unstructured grids composed of simplex elements. 相似文献
20.
Bao Zhu Jiefu Chen Wanxie Zhong & Qing Huo Liu 《Communications In Computational Physics》2011,9(3):828-842
A quasi non-overlapping hybrid scheme that combines the finite-difference
time-domain (FDTD) method and the finite-element time-domain (FETD) method with
nonconforming meshes is developed for time-domain solutions of Maxwell's equations.
The FETD method uses mixed-order basis functions for electric and magnetic
fields, while the FDTD method uses the traditional Yee's grid; the two methods are
joined by a buffer zone with the FETD method and the discontinuous Galerkin method
is used for the domain decomposition in the FETD subdomains. The main features of
this technique is that it allows non-conforming meshes and an arbitrary numbers of
FETD and FDTD subdomains. The hybrid method is completely stable for the time
steps up to the stability limit for the FDTD method and FETD method. Numerical
results demonstrate the validity of this technique. 相似文献