2,584,931 research outputs found
High order amplitude equation for steps on creep curve
We consider a model proposed by one of the authors for a type of plastic
instability found in creep experiments which reproduces a number of
experimentally observed features. The model consists of three coupled
non-linear differential equations describing the evolution of three types of
dislocations. The transition to the instability has been shown to be via Hopf
bifurcation leading to limit cycle solutions with respect to physically
relevant drive parameters. Here we use reductive perturbative method to extract
an amplitude equation of up to seventh order to obtain an approximate analytic
expression for the order parameter. The analysis also enables us to obtain the
bifurcation (phase) diagram of the instability. We find that while
supercritical bifurcation dominates the major part of the instability region,
subcritical bifurcation gradually takes over at one end of the region. These
results are compared with the known experimental results. Approximate analytic
expressions for the limit cycles for different types of bifurcations are shown
to agree with their corresponding numerical solutions of the equations
describing the model. The analysis also shows that high order nonlinearities
are important in the problem. This approach further allows us to map the
theoretical parameters to the experimentally observed macroscopic quantities.Comment: LaTex file and eps figures; Communicated to Phys. Rev.
On transparent boundary conditions for the high--order heat equation
In this paper we develop an artificial initial boundary value problem for the
high-order heat equation in a bounded domain . It is found an unique
classical solution of this problem in an explicit form and shown that the
solution of the artificial initial boundary value problem is equal to the
solution of the infinite problem (Cauchy problem) in .Comment: 9 page
High Order Asymptotic Preserving DG-IMEX Schemes for Discrete-Velocity Kinetic Equations in a Diffusive Scaling
In this paper, we develop a family of high order asymptotic preserving
schemes for some discrete-velocity kinetic equations under a diffusive scaling,
that in the asymptotic limit lead to macroscopic models such as the heat
equation, the porous media equation, the advection-diffusion equation, and the
viscous Burgers equation. Our approach is based on the micro-macro
reformulation of the kinetic equation which involves a natural decomposition of
the equation to the equilibrium and non-equilibrium parts. To achieve high
order accuracy and uniform stability as well as to capture the correct
asymptotic limit, two new ingredients are employed in the proposed methods:
discontinuous Galerkin spatial discretization of arbitrary order of accuracy
with suitable numerical fluxes; high order globally stiffly accurate
implicit-explicit Runge-Kutta scheme in time equipped with a properly chosen
implicit-explicit strategy. Formal asymptotic analysis shows that the proposed
scheme in the limit of epsilon -> 0 is an explicit, consistent and high order
discretization for the limiting equation. Numerical results are presented to
demonstrate the stability and high order accuracy of the proposed schemes
together with their performance in the limit
Stabilization of high-order solutions of the cubic Nonlinear Schrodinger Equation
In this paper we consider the stabilization of non-fundamental unstable
stationary solutions of the cubic nonlinear Schrodinger equation. Specifically
we study the stabilization of radially symmetric solutions with nodes and
asymmetric complex stationary solutions. For the first ones we find partial
stabilization similar to that recently found for vortex solutions while for the
later ones stabilization does not seem possible
High-order Compact Difference Schemes for the Modified Anomalous Subdiffusion Equation
In this paper, two kinds of high-order compact finite difference schemes for
second-order derivative are developed. Then a second-order numerical scheme for
Riemann-Liouvile derivative is established based on fractional center
difference operator. We apply these methods to fractional anomalous
subdiffusion equation to construct two kinds of novel numerical schemes. The
solvability, stability and convergence analysis of these difference schemes are
studied by Fourier method in details. The convergence orders of these numerical
schemes are and ,
respectively. Finally, numerical experiments are displayed which are in line
with the theoretical analysis.Comment:
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