28,167 research outputs found
A linearly implicit conservative difference scheme for the generalized Rosenau-Kawahara-RLW equation
This paper concerns the numerical study for the generalized
Rosenau-Kawahara-RLW equation obtained by coupling the generalized Rosenau-RLW
equation and the generalized Rosenau-Kawahara equation. We first derive the
energy conservation law of the equation, and then develop a three-level
linearly implicit difference scheme for solving the equation. We prove that the
proposed scheme is energy-conserved, unconditionally stable and second-order
accurate both in time and space variables. Finally, numerical experiments are
carried out to confirm the energy conservation, the convergence rates of the
scheme and effectiveness for long-time simulation.Comment: accepted in Applied Mathmatics and Computation
An energy preserving finite difference scheme for the Poisson-Nernst-Planck system
In this paper, we construct a semi-implicit finite difference method for the
time dependent Poisson-Nernst-Planck system. Although the Poisson-Nernst-Planck
system is a nonlinear system, the numerical method presented in this paper only
needs to solve a linear system at each time step, which can be done very
efficiently. The rigorous proof for the mass conservation and electric
potential energy decay are shown. Moreover, mesh refinement analysis shows that
the method is second order convergent in space and first order convergent in
time. Finally we point out that our method can be easily extended to the case
of multi-ions.Comment: 13 pages, 7 Postscript figures, uses elsart1p.cl
A three-level linearized difference scheme for the coupled nonlinear fractional Ginzburg-Landau equation
In this paper, the coupled fractional Ginzburg-Landau equations are first
time investigated numerically. A linearized implicit finite difference scheme
is proposed. The scheme involves three time levels, is unconditionally stable
and second-order accurate in both time and space variables. The unique
solvability, the unconditional stability and optimal pointwise error estimates
are obtained by using the energy method and mathematical induction. Moreover,
the proposed second-order method can be easily extended into the fourth-order
method by using an average finite difference operator for spatial fractional
derivatives and Richardson extrapolation for time variable. Finally, numerical
results are presented to confirm the theoretical results.Comment: 17 pages, 2 figure
Some symmetric -congruences
We prove some symmetric -congruences.Comment: 11 pages. This is a very very preliminary manuscript. And some
results will be added in the future verision
Divisibility of some binomial sums
With help of -congruence, we prove the divisibility of some binomial sums.
For example, for any integers , Comment: This is a very preliminary, which maybe contains some minor mistake
A new extrapolation cascadic multigrid method for 3D elliptic boundary value problems on rectangular domains
In this paper, we develop a new extrapolation cascadic multigrid
(ECMG) method, which makes it possible to solve 3D elliptic boundary
value problems on rectangular domains of over 100 million unknowns on a desktop
computer in minutes. First, by combining Richardson extrapolation and
tri-quadratic Serendipity interpolation techniques, we introduce a new
extrapolation formula to provide a good initial guess for the iterative
solution on the next finer grid, which is a third order approximation to the
finite element (FE) solution. And the resulting large sparse linear system from
the FE discretization is then solved by the Jacobi-preconditioned Conjugate
Gradient (JCG) method. Additionally, instead of performing a fixed number of
iterations as cascadic multigrid (CMG) methods, a relative residual stopping
criterion is used in iterative solvers, which enables us to obtain conveniently
the numerical solution with the desired accuracy. Moreover, a simple Richardson
extrapolation is used to cheaply get a fourth order approximate solution on the
entire fine grid. Test results are reported to show that ECMG has much
better efficiency compared to the classical MG methods. Since the initial guess
for the iterative solution is a quite good approximation to the FE solution,
numerical results show that only few number of iterations are required on the
finest grid for ECMG with an appropriate tolerance of the relative
residual to achieve full second order accuracy, which is particularly important
when solving large systems of equations and can greatly reduce the
computational cost. It should be pointed out that when the tolerance becomes
smaller, ECMG still needs only few iterations to obtain fourth order
extrapolated solution on each grid, except on the finest grid. Finally, we
present the reason why our ECMG algorithms are so highly efficient for solving
such problems.Comment: 20 pages, 4 figures, 10 tables; abbreviated abstrac
An extrapolation cascadic multigrid method combined with a fourth order compact scheme for 3D poisson equation
In this paper, we develop an EXCMG method to solve the three-dimensional
Poisson equation on rectangular domains by using the compact finite difference
(FD) method with unequal meshsizes in different coordinate directions. The
resulting linear system from compact FD discretization is solved by the
conjugate gradient (CG) method with a relative residual stopping criterion. By
combining the Richardson extrapolation and tri-quartic Lagrange interpolation
for the numerical solutions from two-level of grids (current and previous
grids), we are able to produce an extremely accurate approximation of the
actual numerical solution on the next finer grid, which can greatly reduce the
number of relaxation sweeps needed. Additionally, a simple method based on the
midpoint extrapolation formula is used for the fourth-order FD solutions on
two-level of grids to achieve sixth-order accuracy on the entire fine grid
cheaply and directly. The gradient of the numerical solution can also be easily
obtained through solving a series of tridiagonal linear systems resulting from
the fourth-order compact FD discretizations. Numerical results show that our
EXCMG method is much more efficient than the classical V-cycle and W-cycle
multigrid methods. Moreover, only few CG iterations are required on the finest
grid to achieve full fourth-order accuracy in both the -norm and
-norm for the solution and its gradient when the exact solution
belongs to . Finally, numerical result shows that our EXCMG method is
still effective when the exact solution has a lower regularity, which widens
the scope of applicability of our EXCMG method.Comment: Accepted for publication in Journal of Scientific Computing. arXiv
admin note: text overlap with arXiv:1506.0298
A fourth-order maximum principle preserving operator splitting scheme for three-dimensional fractional Allen-Cahn equations
In this paper, by using Strang's second-order splitting method, the numerical
procedure for the three-dimensional (3D) space fractional Allen-Cahn equation
can be divided into three steps. The first and third steps involve an ordinary
differential equation, which can be solved analytically. The intermediate step
involves a 3D linear fractional diffusion equation, which is solved by the
Crank-Nicolson alternating directional implicit (ADI) method. The ADI technique
can convert the multidimensional problem into a series of one-dimensional
problems, which greatly reduces the computational cost. A fourth-order
difference scheme is adopted for discretization of the space fractional
derivatives. Finally, Richardson extrapolation is exploited to increase the
temporal accuracy. The proposed method is shown to be unconditionally stable by
Fourier analysis. Another contribution of this paper is to show that the
numerical solutions satisfy the discrete maximum principle under reasonable
time step constraint. For fabricated smooth solutions, numerical results show
that the proposed method is unconditionally stable and fourth-order accurate in
both time and space variables. In addition, the discrete maximum principle is
also numerically verified.Comment: 24 pages, 7 figures, 10 table
Quadratic Decomposable Submodular Function Minimization: Theory and Practice (Computation and Analysis of PageRank over Hypergraphs)
We introduce a new convex optimization problem, termed quadratic decomposable
submodular function minimization (QDSFM), which allows to model a number of
learning tasks on graphs and hypergraphs. The problem exhibits close ties to
decomposable submodular function minimization (DSFM), yet is much more
challenging to solve. We approach the problem via a new dual strategy and
formulate an objective that can be optimized through a number of double-loop
algorithms. The outer-loop uses either random coordinate descent (RCD) or
alternative projection (AP) methods, for both of which we prove linear
convergence rates. The inner-loop computes projections onto cones generated by
base polytopes of the submodular functions, via the modified min-norm-point or
Frank-Wolfe algorithm. We also describe two new applications of QDSFM:
hypergraph-adapted PageRank and semi-supervised learning. The proposed
hypergraph-based PageRank algorithm can be used for local hypergraph
partitioning, and comes with provable performance guarantees. For
hypergraph-adapted semi-supervised learning, we provide numerical experiments
demonstrating the efficiency of our QDSFM solvers and their significant
improvements on prediction accuracy when compared to state-of-the-art methods.Comment: A part of the work appeared in NeurIPS 2018. The current version is
to appear in JMLR. arXiv admin note: substantial text overlap with
arXiv:1806.0984
Electronic structures and Raman features of a carbon nanobud
By employing the first-principles calculations, we investigate electronic
properties of a novel carbon nanostructure called a carbon nanobud, in which a
molecule covalently attaches or embeds in an armchair carbon nanotube.
We find that the carbon nanobud exhibits either semiconducting or metallic
behavior, depending on the size of the nanotube, as well as the combination
mode. Moreover, with respect to the case of the corresponding pristine
nanotubes, some new electronic states appear at 0.3-0.8 eV above the Fermi
level for the carbon nanobuds with the attaching mode, which agrees well with
the experimental reports. In addition, the vibrational properties of the carbon
nanobuds are explored. The characteristic Raman active modes for both
and the corresponding pristine nanotube present in Raman spectra of the carbon
nanobuds with attaching modes, consistent with the observations of a recent
experiment. In contrast, such situation does not appear for the case of the
carbon nanobud with the embedding mode. This indicates that the synthesized
carbon nanobuds are probably of the attaching configuration rather than the
embedding configuration.Comment: 5 pages, 4 figures. accepted by J. Phys. Chem.
- β¦