Application of high‐order Higdon non‐reflecting boundary conditions to linear shallow water models

Abstract

SUMMARYA shallow water model with linear time-dependent dispersive waves in an unbounded domain is considered. The domain is truncated with artificial boundaries B where a sequence of high-order non-reflecting boundary conditions (NRBCs) proposed by Higdon are applied. Methods devised by Givoli and Neta that afford easy implementation of Higdon NRBCs are refined in order to reduce computational expenses. The new refinement makes the computational effort associated with the boundary treatment quadratic rather than exponential (as in the original scheme) with the order. This allows for implementation of NRBCs of higher orders than previously. A numerical example for a semi-infinite channel truncated on one side is presented. Finite difference schemes are used throughout.

Building generalized open boundary conditions for fluid dynamics problems

“…As pointed out in [5], the conditions that seem to lead to the best numerical performances rely more or less directly on the mathematical notion of transparent boundary conditions (TBCs). This is also consistent with several recent works that derive OBCs closely connected to TBCs BUILDING GENERALIZED OBC FOR FLUID DYNAMICS PROBLEMS 507 (e.g., [10][11][12][13][14]). The problematic of TBC consists in finding an adequate boundary condition, such that solving Lw loc D Q f in loc with this boundary condition leads to w loc D w j loc , where Lw D Q f in .…”

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

“…In [15] Givoli and Neta directly extended the Higdon scheme to high-order finite difference discretizations via an algorithm where the order of the NRBC was simply an input parameter. Long term stability using this formulation was demonstrated in [16]. They later extended this formulation to one that does not involve any high derivatives (hereafter referred to as the G-N formulation).…”

On the stability of the high-order Higdon Absorbing Boundary Conditions

“…The main advantage of the former approach over the latter is that it does not require special treatment of corners in the artificial boundary. On the other hand, the computational effort associated with the former scheme grows (at best) quadratically with the order [27], rather than linearly as in all available auxiliary-variable schemes. In addition, it requires a "deep" finite difference stencil in space and time, and cannot be easily adapted to finite element schemes.…”

“…In the first approach [12,31,27] a systematic way is devised to implement the Higdon condition directly (without auxiliary variables) up to any order, using high-order finite difference discretization. In the second approach [13,14], the high derivatives are eliminated by introducing auxiliary variables on the boundary, as in the Collino ABC.…”