On stability of the Crank-Nicolson scheme with approximate transparent boundary conditions for the Schrödinger equation. I

Ducomet, Bernard; Zlotnik, A.

doi:10.4310/cms.2006.v4.n4.a4

Cited by 23 publications

(41 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The derivation of transparent numerical boundary conditions for (67) was performed in [EA01] so we shall not reproduce it here. Approximate (namely, absorbing) numerical boundary conditions for (67) are proposed and studied in [EA01, AES03, AAB + 08, DZ06]. We also refer to [Sze04, AAB + 08] and references therein for the construction of absorbing boundary conditions for the nonlinear Schrödinger equation.…”

Section: Numerical Schemes For Dispersive Equationsmentioning

confidence: 99%

Transparent numerical boundary conditions for evolution equations: Derivation and stability analysis

Coulombel¹

2019

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

View full text Add to dashboard Cite

The aim of this article is to propose a systematic study of transparent boundary conditions for finite difference approximations of evolution equations. We try to keep the discussion at the highest level of generality in order to apply the theory to the broadest class of problems.We deal with two main issues. We first derive transparent numerical boundary conditions, that is, we exhibit the relations satisfied by the solution to the pure Cauchy problem when the initial condition vanishes outside of some domain. Our derivation encompasses discretized transport, diffusion and dispersive equations with arbitrarily wide stencils. The second issue is to prove sharp stability estimates for the initial boundary value problem obtained by enforcing the boundary conditions derived in the first step. We focus here on discretized transport equations. Under the assumption that the numerical boundary is non-characteristic, our main result characterizes the class of numerical schemes for which the corresponding transparent boundary conditions satisfy the so-called Uniform Kreiss-Lopatinskii Condition introduced in [GKS72]. Adapting some previous works to the non-local boundary conditions considered here, our analysis culminates in the derivation of trace and semigroup estimates for such transparent numerical boundary conditions. Several examples and possible extensions are given.

show abstract

Section: Numerical Schemes For Dispersive Equationsmentioning

confidence: 99%

Transparent numerical boundary conditions for evolution equations: Derivation and stability analysis

Coulombel¹

2019

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

View full text Add to dashboard Cite

show abstract

“…Since the formulation of the boundary condition is independent of the cross section's shape, a cylindrical waveguide is depicted just for illustration. (42) where denotes at . The time is discretized in length of several picoseconds (as in a normal gyrotron cavity simulation).…”

Section: Numerical Validationmentioning

confidence: 99%

“…The reasons for the observed instability of high-order polynomials should be numerical, most probably due to the used finite-difference process and improper space-time discretization [40]- [42]. Exact reasons are still under investigation.…”

Section: Numerical Validationmentioning

confidence: 99%

“…2) An analytical current excitation has been considered instead of a realistic electron beam, whereas in gyrotron cavity simulations the most time-consuming part is the motion of a great amount of electrons (particle pushing). In this situation, the time cost for solving (42) and (37) is negligible. Hence, the performance of the proposed improved broadband boundary condition is fully acceptable; especially in gyrotron simulations, its time cost can even be ignored.…”

Section: Requirement Of Computational Resourcesmentioning

confidence: 99%

See 1 more Smart Citation

An Improved Broadband Boundary Condition for the RF Field in Gyrotron Interaction Modeling

Avramidis

Thumm

et al. 2015

IEEE Trans. Microwave Theory Techn.

View full text Add to dashboard Cite

Gyrotrons are the only RF sources providing significantly high output powers at continuous-wave operation in the sub-terahertz frequency range. A prerequisite for a proper study of gyrotron behavior is the accurate time-dependent self-consistent simulation of the interaction between the electron beam and the RF wave inside the interaction region (cavity). That requires an accurate description of the RF fields at the boundaries of the interaction region. In this work, an improved broadband boundary condition is proposed. Based on polynomial series expansion of the load impedance and of the wavenumber, this boundary condition not only features a significantly improved broadband matching, but also allows a customizable frequency-dependent reflection coefficient at the boundaries. The boundary condition has a general formulation, i.e., it does not rely on a special numerical method. For the case of perfect matching, the new formulation is validated through comparing the numerically calculated reflection coefficients with those obtained by the state-of-the-art matched gyrotron broadband boundary condition, whereas the validation in the case of a given frequency-dependent reflection is done by comparing the numerically calculated reflection with the prescribed one. Although this concept is initially intended for the simulation of gyrotron cavities, it can be easily extended to any other open-cavity resonators, for which proper definition of the boundary conditions is required.

show abstract

“…In Part I of this study (see [10]), a new form of the approximate TBCs has been suggested, with a non-local operator S governing properties of the schemes. The uniform-in-time stability bounds in L 2 have been proved under suitable condition (inequality) on S, for non-uniform meshes in space and time.…”

Section: Introductionmentioning

confidence: 99%

On stability of the Crank-Nicolson scheme with approximate transparent boundary conditions for the Schrödinger equation. Part II

Ducomet

Злотник²

2007

Communications in Mathematical Sciences

Self Cite

View full text Add to dashboard Cite

Abstract. We continue to consider initial-boundary value problems for a generalized timedependent Schrödinger equation in 1D on the semi-axis and in 2D on a semi-bounded strip. The Crank-Nicolson finite-difference schemes with general approximate transparent boundary conditions (TBCs), including the discrete TBCs, are investigated. We prove unconditional stability in L 2 and in the energy norm with respect to initial data and free terms in the equation and the approximate TBC, in general non-uniform in time, under new suitable conditions (inequalities) on a non-local operator S of the approximate TBC. IntroductionWe continue to consider initial-boundary value problems for a generalized timedependent Schrödinger equation in 1D on the semi-axis and in 2D on a semi-bounded strip closely related to a microscopic description of low-energy nuclear fission dynamics [7,13]. We investigate the Crank-Nicolson finite-difference schemes with general approximate transparent boundary conditions (TBCs) including the discrete TBCs [4,5,12].The importance of the stability problem is well-known for mesh methods with the approximate TBCs of solving time-dependent Schrödinger-like equations in unbounded domains see . In Part I of this study (see [10]), a new form of the approximate TBCs has been suggested, with a non-local operator S governing properties of the schemes. The uniform-in-time stability bounds in L 2 have been proved under suitable condition (inequality) on S, for non-uniform meshes in space and time.In the present Part II, we establish essential further results in this direction. We prove unconditional stability in L 2 and in the energy norm with respect to initial data and free terms in the equation and the approximate TBC, in general non-uniform in time in order to cover broader applications. To this end, we introduce new suitable conditions (inequalities) on S. These inequalities are valid for the operators S ref of the discrete TBCs (ensuring the uniform-in-time stability); we clarify them by considering the corresponding schemes on infinite space meshes. We suggest a trick reducing the derivation of general non-uniform in time bounds to the derivation of simpler uniform

show abstract

On stability of the Crank-Nicolson scheme with approximate transparent boundary conditions for the Schrödinger equation. I

Cited by 23 publications

References 18 publications

Transparent numerical boundary conditions for evolution equations: Derivation and stability analysis

Transparent numerical boundary conditions for evolution equations: Derivation and stability analysis

An Improved Broadband Boundary Condition for the RF Field in Gyrotron Interaction Modeling

On stability of the Crank-Nicolson scheme with approximate transparent boundary conditions for the Schrödinger equation. Part II

Contact Info

Product

Resources

About