28,444 research outputs found
Tensor-Structured Coupled Cluster Theory
We derive and implement a new way of solving coupled cluster equations with
lower computational scaling. Our method is based on decomposition of both
amplitudes and two electron integrals, using a combination of tensor
hypercontraction and canonical polyadic decomposition. While the original
theory scales as with respect to the number of basis functions, we
demonstrate numerically that we achieve sub-millihartree difference from the
original theory with scaling. This is accomplished by solving directly
for the factors that decompose the cluster operator. The proposed scheme is
quite general and can be easily extended to other many-body methods
A discrete-time approach to the steady-state and stability analysis of distributed nonlinear autonomous circuits
We present a direct method for the steady-state and stability
analysis of autonomous circuits with transmission lines and generic non-
linear elements. With the discretization of the equations that describe the
circuit in the time domain, we obtain a nonlinear algebraic formulation
where the unknowns to determine are the samples of the variables directly
in the steady state, along with the oscillation period, the main unknown in
autonomous circuits.An efficient scheme to buildtheJacobian matrix with
exact partial derivatives with respect to the oscillation period and with re-
spect to the samples of the unknowns is described. Without any modifica-
tion in the analysis method, the stability of the solution can be computed a
posteriori constructing an implicit map, where the last sample is viewed as
a function of the previous samples. The application of this technique to the
time-delayed Chua's circuit (TDCC) allows us to investigate the stability of
the periodic solutions and to locate the period-doubling bifurcations.Peer ReviewedPostprint (published version
An unconditionally energy stable finite difference scheme for a stochastic Cahn-Hilliard equation
In this work, the MMC-TDGL equation, a stochastic Cahn-Hilliard equation is
solved numerically by using the finite difference method in combination with a
convex splitting technique of the energy functional. For the non-stochastic
case, we develop an unconditionally energy stable difference scheme which is
proved to be uniquely solvable. For the stochastic case, by adopting the same
splitting of the energy functional, we construct a similar and uniquely
solvable difference scheme with the discretized stochastic term. The resulted
schemes are nonlinear and solved by Newton iteration. For the long time
simulation, an adaptive time stepping strategy is developed based on both
first- and second-order derivatives of the energy. Numerical experiments are
carried out to verify the energy stability, the efficiency of the adaptive time
stepping and the effect of the stochastic term.Comment: This paper has been accepted for publication in SCIENCE CHINA
Mathematic
Numerical solution of third order three-point boundary value problems of ordinary differential equations with variable coefficients using variational-composite hybrid fixed point iterative method
This paper explores variational–composite hybrid fixed point iterative scheme for the solution of third order three-point boundary value problems. The method shows a strong convergence which makes it an efficient and reliable technique for finding approximate analytical solutions for third order three-point boundary value problems of ordinary differential equations with variable coefficients. From the numerical experiments carried out, the accuracy of the method was confirmed through the order of convergence obtained
A direct solver with O(N) complexity for variable coefficient elliptic PDEs discretized via a high-order composite spectral collocation method
A numerical method for solving elliptic PDEs with variable coefficients on
two-dimensional domains is presented. The method is based on high-order
composite spectral approximations and is designed for problems with smooth
solutions. The resulting system of linear equations is solved using a direct
(as opposed to iterative) solver that has optimal O(N) complexity for all
stages of the computation when applied to problems with non-oscillatory
solutions such as the Laplace and the Stokes equations. Numerical examples
demonstrate that the scheme is capable of computing solutions with relative
accuracy of or better, even for challenging problems such as highly
oscillatory Helmholtz problems and convection-dominated convection diffusion
equations. In terms of speed, it is demonstrated that a problem with a
non-oscillatory solution that was discretized using nodes was solved
in 115 minutes on a personal work-station with two quad-core 3.3GHz CPUs. Since
the solver is direct, and the "solution operator" fits in RAM, any solves
beyond the first are very fast. In the example with unknowns, solves
require only 30 seconds.Comment: arXiv admin note: text overlap with arXiv:1302.599
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