High-order Non-reflecting Boundary Conditions Research Articles

High-order non-reflecting boundary conditions for dispersive waves in polar coordinates using spectral elements

High-order non-reflecting boundary conditions are introduced to create a finite computational space and for the solution of dispersive waves using a spectral element formulation with high-order time integration. Numerical examples are used to demonstrate the synergy of using high-order spatial, time, and boundary discretization. We show that by balancing all numerical errors involved, high-order accuracy can be achieved for unbounded domain problems in polar coordinate systems.

Numerical comparison of high-order absorbing boundary conditions and perfectly matched layers for a dispersive one-dimensional medium

High-order absorbing boundary conditions (ABC) and perfectly matched layers (PML) are two powerful methods to numerically solve wave problems in unbounded domains. The aim of the proposed study is to analyze and compare the performance of these methods in the one-dimensional problem governed by the dispersive wave equation. The PMLs proposed in literature for time-harmonic dynamics are applied to time-dependent wave problems, and linear, quadratic and cubic polynomial stretching functions are considered. The resulting PMLs exhibit a double absorbing action: (i) they reduce the amplitude of the incident waves and (ii) they delay the wave propagation. Then the ABC proposed by Givoli and Neta and those proposed by Hagstrom, Mar-Or and Givoli are considered. The former are a reformulation of the Higdon high-order non-reflecting boundary conditions, the latter improve the Higdon conditions and extend them to take into account evanescent waves. The accuracy of the PMLs and ABCs, implemented in a finite element code, is first investigated with respect to the frequency of the incident wave, being it progressive or evanescent. Then the response to a wave train characterized by a broad frequency spectrum, resulting from an impulsive force, is studied. A detailed analysis is performed to detect the influence of the parameters of both the ABCs and the PMLs on the absorption of waves. The performances of PMLs and ABCs are compared, and merits and drawbacks of the two methods are pointed out.

Klein–Gordon equation with advection on unbounded domains using spectral elements and high-order non-reflecting boundary conditions

A reduced shallow water model under constant, non-zero advection in the infinite channel is considered. High-order (Givoli–Neta) non-reflecting boundary conditions are introduced in various configurations to create a finite computational space and solved using a spectral element formulation with high-order time integration. Numerical examples are used to demonstrate the synergy of using high-order spatial, time, and boundary discretization. We show that by balancing all numerical errors involved, high-order accuracy can be achieved for unbounded domain problems.

High-order non-reflecting boundary conditions for the linearized 2-D Euler equations: No mean flow case

Higdon-type non-reflecting boundary conditions (NRBCs) are developed for the 2-D linearized Euler equations with Coriolis forces. This implementation is applied to a simplified form of the equations, with the NRBCs applied to all four sides of the domain. We demonstrate the validity of the NRBCs to high order. We close with a list of areas for further research.

High‐order Higdon‐like boundary conditions for exterior transient wave problems

AbstractRecently developed non‐reflecting boundary conditions are applied for exterior time‐dependent wave problems in unbounded domains. The linear time‐dependent wave equation, with or without a dispersive term, is considered in an infinite domain. The infinite domain is truncated via an artificial boundary ℬ︁, and a high‐order non‐reflecting boundary condition (NRBC) is imposed on ℬ︁. Then the problem is solved numerically in the finite domain bounded by ℬ︁. The new boundary scheme is based on a reformulation of the sequence of NRBCs proposed by Higdon. We consider here two reformulations: one that involves high‐order derivatives with a special discretization scheme, and another that does not involve any high derivatives beyond second order. The latter formulation is made possible by introducing special auxiliary variables on ℬ︁. In both formulations the new NRBCs can easily be used up to any desired order. They can be incorporated in a finite element or a finite difference scheme; in the present paper the latter is used. In contrast to previous papers using similar formulations, here the method is applied to a fully exterior two‐dimensional problem, with a rectangular boundary. Numerical examples in infinite domains are used to demonstrate the performance and advantages of the new method. In the auxiliary‐variable formulation long‐time corner instability is observed, that requires special treatment of the corners (not addressed in this paper). No such difficulties arise in the high‐derivative formulation. Published in 2005 by John Wiley & Sons, Ltd.

High-order Non-reflecting Boundary Conditions for Dispersive Waves in Cartesian, Cylindrical and Spherical Coordinate Systems

Among the many areas of research that Professor Kawahara has been active in is the subject of open boundaries in which linear time-dependent dispersive waves are considered in an unbounded domain. The infinite domain is truncated via an artificial boundary B on which an open boundary condition (OBC) is imposed. In this paper, Higdon OBCs and Hagstrom–Hariharan (HH) OBCs are considered. Higdon-type conditions, originally implemented as low-order OBCs, are made accessible for any desired order via a new scheme. The higher-order Higdon OBC is then reformulated using auxiliary variables and made compatible for use with finite element (FE) methods. Methodologies for selecting Higdon parameters are also proposed. The performances of these schemes are demonstrated in two numerical examples. This is followed by a discussion of the HH OBC, which is applicable to non-dispersive media on cylindrical and spherical geometries. The paper extends this OBC to the “slightly dispersive” case.

High-order nonreflecting boundary conditions for the dispersive shallow water equations

Time-dependent dispersive shallow water waves in an unbounded domain are considered. The infinite domain is truncated via an artificial boundary B , and a high-order non-reflecting boundary condition (NRBC) is imposed on B . Then the problem is solved by a finite difference scheme in the finite domain bounded by B . The sequence of NRBCs proposed by Higdon is used. However, in contrast to the original low-order implementation of the Higdon conditions, a new scheme is devised which allows the easy use of a Higdon-type NRBC of any desired order. In addition, a procedure for the automatic choice of the parameters appearing in the NRBC is proposed. The performance of the scheme is demonstrated via a numerical example.

High-order non-reflecting boundary scheme for time-dependent waves

A new non-reflecting boundary scheme is proposed for time-dependent wave problems in unbounded domains. The linear time-dependent wave equation, with or without a dispersive term, is considered in a semi-infinite wave guide. The infinite domain is truncated via an artificial boundary B , and a high-order non-reflecting boundary condition (NRBC) is imposed on B . Then the problem is solved numerically in the finite domain bounded by B . The new boundary scheme is based on a reformulation of the sequence of NRBCs proposed by Higdon. In contrast to the original formulation of the Higdon conditions, the scheme constructed here does not involve any high derivatives beyond second order. This is made possible by introducing special auxiliary variables on B . As a result, the new NRBCs can easily be used up to any desired order. They can be incorporated in a finite element or a finite difference scheme; in the present paper the latter is used. The parameters appearing in the NRBC are chosen automatically via a special procedure. Numerical examples concerning a semi-infinite wave guide are used to demonstrate the performance of the new method.

High-order non-reflecting boundary conditions for dispersive waves

Problems of linear time-dependent dispersive waves in an unbounded domain are considered. The infinite domain is truncated via an artificial boundary B , and a high-order non-reflecting boundary condition (NRBC) is imposed on B . Then the problem is solved by a finite difference (FD) scheme in the finite domain bounded by B . The sequence of NRBCs proposed by Higdon is used. However, in contrast to the original low-order implementation of the Higdon conditions, a new scheme is devised which allows the easy use of a Higdon-type NRBC of any desired order. In addition, a procedure for the automatic choice of the parameters appearing in the NRBC is proposed. The performance of the scheme is demonstrated via numerical examples.

An optimal high‐order non‐reflecting finite element scheme for wave scattering problems

AbstractA new finite element scheme is proposed for the numerical solution of time‐harmonic wave scattering problems in unbounded domains. The infinite domain in truncated via an artificial boundary ℬ︁ which encloses a finite computational domain Ω. On ℬ︁ a local high‐order non‐reflecting boundary condition (NRBC) is applied which is constructed to be optimal in a certain sense. This NRBC is implemented in a special way, by using auxiliary variables along the boundary ℬ︁, so that it involves no high‐order derivatives regardless of its order. The order of the scheme is simply an input parameter, and it may be arbitrarily high. This leads to a symmetric finite element formulation where standard C0 finite elements are used in Ω. The performance of the method is demonstrated via numerical examples, and it is compared to other NRBC‐based schemes. The method is shown to be highly accurate and stable, and to lead to a well‐conditioned matrix problem. Copyright © 2002 John Wiley & Sons, Ltd.

High-order Non-reflecting Boundary Conditions For Dispersive WaveProblems In Stratified Media

High-order Non-reflecting Boundary Conditions For Dispersive WaveProblems In Stratified Media