22,465 research outputs found
The role of numerical boundary procedures in the stability of perfectly matched layers
In this paper we address the temporal energy growth associated with numerical
approximations of the perfectly matched layer (PML) for Maxwell's equations in
first order form. In the literature, several studies have shown that a
numerical method which is stable in the absence of the PML can become unstable
when the PML is introduced. We demonstrate in this paper that this instability
can be directly related to numerical treatment of boundary conditions in the
PML. First, at the continuous level, we establish the stability of the constant
coefficient initial boundary value problem for the PML. To enable the
construction of stable numerical boundary procedures, we derive energy
estimates for the variable coefficient PML. Second, we develop a high order
accurate and stable numerical approximation for the PML using
summation--by--parts finite difference operators to approximate spatial
derivatives and weak enforcement of boundary conditions using penalties. By
constructing analogous discrete energy estimates we show discrete stability and
convergence of the numerical method. Numerical experiments verify the
theoretical result
Nonlinear Boundary Conditions for Initial Boundary Value Problems with Applications in Computational Fluid Dynamics
We derive new boundary conditions and implementation procedures for nonlinear
initial boundary value problems (IBVPs) with non-zero boundary data that lead
to bounded solutions. The new boundary procedure is applied to nonlinear IBVPs
on skew-symmetric form, including dissipative terms. The complete procedure has
two main ingredients. In the first part (published in [1, 2]), the energy and
entropy rate in terms of a surface integral with boundary terms was produced
for problems with first derivatives.
In this second part we complement it by adding second derivative dissipative
terms and bound the boundary terms. We develop a new nonlinear boundary
procedure which generalise the characteristic boundary procedure for linear
problems. Both strong and weak imposition of the nonlinear boundary conditions
with non-zero boundary data are considered, and we prove that the solution is
bounded. The boundary procedure is applied to four important IBVPs in
computational fluid dynamics: the incompressible Euler and Navier-Stokes, the
shallow water and the compressible Euler equations. Finally we show that stable
discrete approximations follow by using summation-by-parts operators combined
with weak boundary conditions.Comment: arXiv admin note: substantial text overlap with arXiv:2301.0456
Review of Summation-by-parts schemes for initial-boundary-value problems
High-order finite difference methods are efficient, easy to program, scales
well in multiple dimensions and can be modified locally for various reasons
(such as shock treatment for example). The main drawback have been the
complicated and sometimes even mysterious stability treatment at boundaries and
interfaces required for a stable scheme. The research on summation-by-parts
operators and weak boundary conditions during the last 20 years have removed
this drawback and now reached a mature state. It is now possible to construct
stable and high order accurate multi-block finite difference schemes in a
systematic building-block-like manner. In this paper we will review this
development, point out the main contributions and speculate about the next
lines of research in this area
Structure constants of operators on the Wilson loop from integrability
We study structure constants of local operators inserted on the Wilson loop
in super Yang-Mills theory. We compute the structure constants in
the SU(2) sector at tree level using the correspondence between operators on
the Wilson loop and the open spin chain. The results are interpreted as the
summation over all possible ways of changing the signs of magnon momenta in the
hexagon form factors. This is consistent with a holographic description of the
correlator as the cubic open string vertex, which consists of one hexagonal
patch and three boundaries. We then conjecture that a similar expression should
hold also at finite coupling.Comment: 38 pages; v3: JHEP published versio
High-order accurate difference schemes for the Hodgkin-Huxley equations
A novel approach for simulating potential propagation in neuronal branches
with high accuracy is developed. The method relies on high-order accurate
difference schemes using the Summation-By-Parts operators with weak boundary
and interface conditions applied to the Hodgkin-Huxley equations. This work is
the first demonstrating high accuracy for that equation. Several boundary
conditions are considered including the non-standard one accounting for the
soma presence, which is characterized by its own partial differential equation.
Well-posedness for the continuous problem as well as stability of the discrete
approximation is proved for all the boundary conditions. Gains in terms of CPU
times are observed when high-order operators are used, demonstrating the
advantage of the high-order schemes for simulating potential propagation in
large neuronal trees
Trembling cavities in the canonical approach
We present a canonical formalism facilitating investigations of the dynamical
Casimir effect by means of a response theory approach. We consider a massless
scalar field confined inside of an arbitaray domain , which undergoes
small displacements for a certain period of time. Under rather general
conditions a formula for the number of created particles per mode is derived.
The pertubative approach reveals the occurance of two generic processes
contributing to the particle production: the squeezing of the vacuum by
changing the shape and an acceleration effect due to motion af the boundaries.
The method is applied to the configuration of moving mirror(s). Some properties
as well as the relation to local Green function methods are discussed.
PACS-numbers: 12.20; 42.50; 03.70.+k; 42.65.Vh Keywords: Dynamical Casimir
effect; Moving mirrors; Cavity quantum field theory; Vibrating boundary
One-loop Noncommutative U(1) Gauge Theory from Bosonic Worldline Approach
We develop a method to compute the one-loop effective action of
noncommutative U(1) gauge theory based on the bosonic worldline formalism, and
derive compact expressions for N-point 1PI amplitudes. The method, resembling
perturbative string computations, shows that open Wilson lines emerge as a
gauge invariant completion of certain terms in the effective action. The terms
involving open Wilson lines are of the form reminiscent of closed string
exchanges between the states living on the two boundaries of a cylinder. They
are also consistent with recent matrix theory analysis and the results from
noncommutative scalar field theories with cubic interactions.Comment: 1+35 pages, Latex, address info adde
- …