We derive equations for evaluating differential operators in transparent boundary conditions (TBCs) for a certain class of hyperbolic systems of second-order equations. This local part of TBCs can be used as approximate nonreflecting boundary conditions. We give examples of obtaining such truncated TBCs for 3D elasticity and Biot poroelasticity
The time-harmonic Maxwell equations at high wavenumber $k$ are discretized by edge elements of degree $p$ on a mesh of width $h$. For the case of a ball and exact, transparent boundary conditions, we show quasi-optimality of the Galerkin method under the $k$-explicit scale resolution condition that a) $kh/p$ is sufficient small and b) $p/\log k$ is bounded from below.
We consider the problem of constructing transparent boundary conditions for the time-dependent Schrödinger equation with a compactly supported binding potential and, if desired, a spatially uniform, time-dependent electromagnetic vector potential. Such conditions prevent nonphysical boundary effects from corrupting a numerical solution in a bounded computational domain. We use ideas from potential theory to build exact nonlocal conditions for arbitrary piecewise-smooth domains. These generalize the standard Dirichlet-to-Neumann and Neumann-to-Dirichlet maps known for the equation in one dimens…
The present work proposes a second-order time splitting scheme for a linear dispersive equation with a variable advection coefficient subject to transparent boundary conditions. For its spatial discretization, a dual Petrov--Galerkin method is considered which gives spectral accuracy. The main difficulty in constructing a second-order splitting scheme in such a situation lies in the compatibility condition at the boundaries of the sub-problems. In particular, the presence of an inflow boundary condition in the advection part results in order reduction. To overcome this issue a modified Strang …
The goal of the present work is to solve a linear dispersive equation with variable coefficient advection on an unbounded domain. In this setting, transparent boundary conditions are vital to allow waves to leave (or even re-enter) the, necessarily finite, computational domain. To obtain an efficient numerical scheme we discretize space using a spectral method. This allows us to drastically reduce the number of grid points required for a given accuracy. Applying a fully implicit time integrator, however, would require us to invert full matrices. This is addressed by performing an operator spli…
Local perturbations of an infinitely long rod go away to infinity. On the contrary, in the case of a finite length of the rod, the perturbations reach its boundary and are reflected from it. The boundary conditions constructed here for the implicit difference scheme imitate the Cauchy problem and provide almost no reflection. These boundary conditions are non-local in time, and their practical implementation requires additional calculations at every time step. To minimise them, a special rational approximation similar to the Hermite - Padé approximation is used. Numerical experiments confirm t…
We consider the problem of absence of backscattering in the transport of Manakov solitons on a line. The concept of transparent boundary conditions is used for modeling the reflectionless propagation of Manakov vector solitons in a one-dimensional domain. Artificial boundary conditions that ensure the absence of backscattering are derived and their numerical implementation is demonstrated.
We develop a general strategy in order to implement (approximate) discrete transparent boundary conditions for finite difference approximations of the two-dimensional transport equation. The computational domain is a rectangle equipped with a Cartesian grid. For the two-dimensional leapfrog scheme, we explain why our strategy provides with explicit numerical boundary conditions on the four sides of the rectangle and why it does not require prescribing any condition at the four corners of the computational domain. The stability of the numerical boundary condition on each side of the rectangle i…
Transparent boundary conditions based on the Pole Condition for time-dependent, two-dimensional problems
Institute of Computational Science, USI Lugano（卢加诺大学计算科学研究所）；Mathematisches Institut, Heinrich-Heine-Universität（海因里希·海涅大学数学研究所）；ZIB Berlin（柏林ZIB（德国数学家研究中心））
Discrete transparent boundary conditions for the linearized Green-Naghdi system of equations
Institut de Mathématiques de Toulouse（图卢兹数学研究所；UMR5219，图卢兹大学；法国国家科学研究中心，UPS IMT）；UMR5219, Université de Toulouse（图卢兹数学研究所；UMR5219，图卢兹大学；法国国家科学研究中心，INSA）；CNRS, UPS IMT；Institut de Mathématiques de Toulouse；UMR5219, Université de Toulouse；CNRS, INSA
The splitting in potential Crank-Nicolson scheme with discrete transparent boundary conditions for the Schrödinger equation on a semi-infinite strip
CEA, DAM, DIF（法国原子能委员会）；National Research University Higher School of Economics（国立研究型高等经济大学）；National Research University Moscow Power Engineering Institute（国立研究型莫斯科动力工程学院）
Transparent boundary conditions for the spatially discrete Schrödinger equation: Reflectionless quantum transport in 1D lattices
National University of Uzbekistan（乌兹别克斯坦国立大学）；Kimyo International University in Tashkent（塔什干化学国际大学）；Bergische Universität Wuppertal（伍珀塔尔大学）；Turin Polytechnic University in Tashkent（塔什干都灵理工大学）
A splitting higher order scheme with discrete transparent boundary conditions for the Schrödinger equation in a semi-infinite parallelepiped
CEA/DAM/DIF（法国原子能委员会 DAM DIF）；National Research University Higher School of Economics（国立研究型大学高等经济大学）；National Research University Moscow Power Engineering Institute（国立研究型大学莫斯科动力工程学院）
Maxwell's equations are considered with transparent boundary conditions, for initial conditions and inhomogeneity having support in a bounded, not necessarily convex three-dimensional domain or in a collection of such domains. The numerical method only involves the interior domain and its boundary. The transparent boundary conditions are imposed via a time-dependent boundary integral operator that is shown to satisfy a coercivity property. The stability of the numerical method relies on this coercivity and on an anti-symmetric structure of the discretized equations that is inherited from a wea…
Transparent Boundary Conditions for the Time-Fractional Heat Equation on Metric Graphs
University of Tunis El Manar（突尼斯埃尔马纳尔大学）；National Engineering School of Tunis（突尼斯国立工程学院）；ESPRIT School of Engineering（ESPRIT工程学院）；University of Wuppertal（伍珀塔尔大学）
Convergence of the PML solution for elastic wave scattering by biperiodic structures
Beijing University of Posts and Telecommunications（北京邮电大学）；Purdue University（普渡大学）；Jilin University（吉林大学）；Academy of Mathematics and System Sciences, CAS（中国科学院数学与系统科学研究院；中国科学院大学）；University of Chinese Academy of Sciences
This paper considers the valuation of a European call option under the Heston stochastic volatility model. We present the asymptotic solution to the option pricing problem in powers of the volatility of variance. Then we introduce the artificial boundary method for solving the problem on a truncated domain, and derive several artificial boundary conditions (ABCs) on the artificial boundary of the bounded computational domain. A typical finite difference scheme and quadrature rule are used for the numerical solution of the reduced problem. Numerical experiments show that the proposed ABCs are a…
A key issue in the solution of partial differential equations via integral equation methods is the evaluation of possibly singular integrals involving the Green's function and its derivatives multiplied by simple functions over discretized representations of the boundary. For the Helmholtz equation, while many authors use numerical quadrature to evaluate these boundary integrals, we present analytical expressions for such integrals over flat polygons in the form of infinite series. These can be efficiently truncated based on the accurate error bounds, which is key to their integration in metho…
For the Helmholtz equation posed in the exterior of a Dirichlet obstacle, we prove that if there exists a family of quasimodes (as is the case when the exterior of the obstacle has stable trapped rays), then there exist near-zero eigenvalues of the standard variational formulation of the exterior Dirichlet problem (recall that this formulation involves truncating the exterior domain and applying the exterior Dirichlet-to-Neumann map on the truncation boundary). Our motivation for proving this result is that a) the finite-element method for computing approximations to solutions of the Helmholtz…
