38,293 research outputs found
Expected exit time for time-periodic stochastic differential equations and applications to stochastic resonance
In this paper, we derive a parabolic partial differential equation for the expected exit time of non-autonomous time-periodic non-degenerate stochastic differential equations. This establishes a FeynmanâKac duality between expected exit time of time-periodic stochastic differential equations and time-periodic solutions of parabolic partial differential equations. Casting the time-periodic solution of the parabolic partial differential equation as a fixed point problem and a convex optimisation problem, we give sufficient conditions in which the partial differential equation is well-posed in a weak and classical sense. With no known closed formulae for the expected exit time, we show our method can be readily implemented by standard numerical schemes. With relatively weak conditions (e.g. locally Lipschitz coefficients), the method in this paper is applicable to wide range of physical systems including weakly dissipative systems. Particular applications towards stochastic resonance will be discussed
The Exact Ground State of the Frenkel-Kontorova Model with Repeated Parabolic Potential: II. Numerical Treatment
A procedure is described for efficiently finding the ground state energy and
configuration for a Frenkel-Kontorova model in a periodic potential, consisting
of N parabolic segments of identical curvature in each period, through a
numerical solution of the convex minimization problem described in the
preceding paper. The key elements are the use of subdifferentials to describe
the structure of the minimization problem; an intuitive picture of how to solve
it, based on motion of quasiparticles; and a fast linear optimization method
with a reduced memory requirement. The procedure has been tested for N up to
200.Comment: 9 RevTeX pages, using AMS-Fonts (amssym.tex,amssym.def), 3 Postscript
figures, accepted by Phys.Rev.B to be published together with
cond-mat/970722
MATHEMATICAL MODELING OF DIFFUSION BOUNDARY PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS USING MATLAB AND C++ FOR NUMERICAL AND ANALYTICAL SOLUTIONS
The article examines a second-order parabolic partial differential equation of a three-dimensional (3D) non-stationary boundary problem with constant diffusion coefficients and periodic boundary conditions in the x and y directions. The method for reducing the (3D) non-stationary boundary problem to the corresponding one-dimensional (1D) non-stationary boundary problem using periodic boundary conditions in the x and y directions is discussed. The stationary (analytical) solution of the obtained (1D) stationary boundary problem is also obtained. The numerical solutions of the 1D boundary problem are obtained using the Matlab package "pdepe" and the C++ programming language. As a practical application of the developed mathematical model, the article discusses calculating the concentration of heavy metal Ca in a peat layer based on the obtained experimental data (measurements)
Multiscale numerical methods for some types of parabolic equations
In this dissertation we study multiscale numerical methods for nonlinear parabolic
equations, turbulent diffusion problems, and high contrast parabolic equations. We
focus on designing and analysis of multiscale methods which can capture the effects
of the small scale locally.
At first, we study numerical homogenization of nonlinear parabolic equations
in periodic cases. We examine the convergence of the numerical homogenization
procedure formulated within the framework of the multiscale finite element method.
The goal of the second problem is to develop efficient multiscale numerical techniques
for solving turbulent diffusion equations governed by celluar flows. The solution near
the separatrices can be approximated by the solution of a system of one dimensional
heat equations on the graph. We study numerical implementation for this asymptotic
approach, and spectral methods and finite difference scheme on exponential grids are
used in solving coupled heat equations. The third problem we study is linear parabolic
equations in strongly channelized media. We concentrate on showing that the solution
depends on the steady state solution smoothly.
As for the first problem, we obtain quantitive estimates for the convergence of
the correctors and some parts of truncation error. These explicit estimates show us
the sources of the resonance errors. We perform numerical implementations for the
asymptotic approach in the second problem. We find that finite difference scheme with exponential grids are easy to implement and give us more accurate solutions
while spectral methods have difficulties finding the constant states without major
reformulation. Under some assumption, we justify rigorously the formal asymptotic
expansion using a special coordinate system and asymptotic analysis with respect to
high contrast for the third problem
Functional a posteriori error estimates for time-periodic parabolic optimal control problems
This paper is devoted to the a posteriori error analysis of multiharmonic
finite element approximations to distributed optimal control problems with
time-periodic state equations of parabolic type. We derive a posteriori
estimates of functional type, which are easily computable and provide
guaranteed upper bounds for the state and co-state errors as well as for the
cost functional. These theoretical results are confirmed by several numerical
tests that show high efficiency of the a posteriori error bounds
Exponential decay of the resonance error in numerical homogenization via parabolic and elliptic cell problems
This paper presents two new approaches for finding the homogenized
coefficients of multiscale elliptic PDEs. Standard approaches for computing the
homogenized coefficients suffer from the so-called resonance error, originating
from a mismatch between the true and the computational boundary conditions. Our
new methods, based on solutions of parabolic and elliptic cell-problems, result
in an exponential decay of the resonance error
Functional a posteriori error estimates for parabolic time-periodic boundary value problems
The paper is concerned with parabolic time-periodic boundary value problems
which are of theoretical interest and arise in different practical
applications. The multiharmonic finite element method is well adapted to this
class of parabolic problems. We study properties of multiharmonic
approximations and derive guaranteed and fully computable bounds of
approximation errors. For this purpose, we use the functional a posteriori
error estimation techniques earlier introduced by S. Repin. Numerical tests
confirm the efficiency of the a posteriori error bounds derived
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