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

Abstract

Dedicated to Professor Mutsuto Kawahara on the occasion of his 60th birthdayAmong 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. Higdontype 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.

A Study of Lateral Boundary Conditions for the NRL's Coupled Ocean/Atmosphere Mesoscale Prediction System (COAMPS)

“…In the latter case, auxiliary variables are required. We have generalized our idea to the linearized SWE with zero mean flow [10,12,13]. We also allowed NRBC on all 4 sides of the lateral boundaries [15].…”

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

“…The question now arises, can this formulation be extended to include dispersive effects such as Coriolis? In [22], van Joolen et al presented a method to extend the HH formulation for the standard wave equation under mild dispersion. While this formulation was well grounded mathematically it was never implemented.…”

“…Transforming these equations back into the time domain results in the final HH boundary formulation for the KGE: It should be noted that van Joolen et al [22] show how m(t) and n(t) can be calculated in each time-step to keep the boundary condition local in time without having to store and operate on the history of the solution. For this analysis, a simple trapezoidal approximation was used to approximate the integral.…”

“…The reason is that only in the latter the coefficients are constant. To overcome this, van Joolen et al [22] extended the Hagstrom-Hariharan (HH) conditions to dispersive media. Here we show how to implement this idea.…”

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