65M12 Stability and convergence of numerical methods
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- convergence (3)
- time discretization (3)
- discrete convolution (2)
- error estimates (2)
- phase field model (2)
- Cahn--Hilliard equation (1)
- Cahn-Hilliard equation (1)
- Schrödinger equation (1)
- advection--diffusion equation (1)
- air pollution (1)
In this paper we propose a time discretization of a system of two parabolic equations describing diffusion-driven atom rearrangement in crystalline matter. The equations express the balances of microforces and microenergy; the two phase fields are the order parameter and the chemical potential. The initial and boundary-value problem for the evolutionary system is known to be well posed. Convergence of the discrete scheme to the solution of the continuous problem is proved by a careful development of uniform estimates, by weak compactness and a suitable treatment of
nonlinearities. Moreover, for the difference of discrete
and continuous solutions we prove an error estimate of
order one with respect to the time step.
We investigate the convergence of an implicit Voronoi finite volume
method for reaction-diffusion problems including nonlinear diffusion
in two space dimensions. The model allows to handle heterogeneous
materials and uses the chemical potentials of the involved species as
primary variables. The numerical scheme uses boundary conforming Delaunay
meshes and preserves positivity and the dissipative property of the
continuous system. Starting from a result on the global stability of
the scheme (uniform, mesh-independent global upper and lower bounds),
we prove strong convergence of the chemical activities and their gradients
to a weak solution of the continuous problem. In order to illustrate the
preservation of qualitative properties by the numerical scheme, we present
a long-term simulation of the Michaelis-Menten-Henri system. Especially,
we investigate the decay properties of the relative free energy and the
evolution of the dissipation rate over several magnitudes of time, and
obtain experimental orders of convergence for these quantities.
In this paper we propose a time discretization of a system of two parabolic equations describing diffusion-driven atom rearrangement in crystalline matter. The equations express the balances of microforces and microenergy; the two phase fields are the order parameter and the chemical potential. The initial and boundary-value problem for the evolutionary system is known to be well posed. Convergence of the discrete scheme to the solution of the continuous problem is proved by a careful development of uniform estimates, by weak compactness and a suitable treatment of
nonlinearities. Moreover, for the difference of discrete
and continuous solutions we prove an error estimate of
order one with respect to the time step.
We consider anisotropic Allen--Cahn equations with interfacial energy
induced by an anisotropic surface energy density $\gamma$.
Assuming that $\gamma$
is positive, positively homogeneous of degree one,
strictly convex in tangential directions to the unit sphere,
and sufficiently smooth, we show stability of
various time discretizations. In particular,
we consider a fully implicit and a linearized time discretization
of the interfacial energy combined with implicit
and semi-implicit time discretizations
of the double-well potential. In the semi-implicit variant,
concave terms are taken explicitly.
The arising discrete spatial problems are solved by
globally convergent truncated nonsmooth Newton multigrid methods.
Numerical experiments show the accuracy of the different
discretizations.
We also illustrate that pinch-off under anisotropic
mean curvature flow is no longer frame invariant,
but depends on the orientation of the initial configuration.
In this review article we discuss different techniques to solve numerically the
time-dependent Schrödinger equation on unbounded domains.
We present in detail the most recent approaches and describe briefly alternative ideas pointing out the relations between these works.
We conclude with several numerical examples from
different application areas to compare the presented techniques. We mainly focus on the one-dimensional problem but also touch upon the situation in two space dimensions and the cubic nonlinear case.
This paper is concerned with transparent boundary
conditions (TBCs) for the time-dependent Schrödinger equation
on a circular domain.
Discrete TBCs are introduced in the
numerical simulations of problems on unbounded domains in order to reduce
the computational domain to a finite region in order to make this problem feasible for numerical simulations.
The main focus of this article is on the
appropriate discretization of such
TBCs for the two-dimensional Schrödinger equation
in conjunction with a conservative Crank-Nicolson-type finite difference discretization.
The presented discrete TBCs yield an unconditionally stable
numerical scheme and are completely reflection-free at the boundary.
Furthermore we prove concisely the stability of the recurrence formulas used to
obtain the convolution coefficients of the new discrete TBC
for a spatially dependent potential.
In this work we deal with the numerical solution of some problems of air pollution.
Since the problems are posed on unbounded domains we have to introduce
artificial boundaries to confine the computational region.
We construct and analyse (discrete) transparent boundary conditions
for an implicit difference scheme.
We discuss the concepts of positivity and monotonicity of
difference schemes and briefly consider these
properties of difference schemes for advection-diffusion equations
arising in problems of air (and water) pollution.
The efficiency and accuracy of our method is illustrated by an example.