A Flexible Boundary Procedure for Hyperbolic Problems: Multiple Penalty Terms Applied in a Domain

Abstract

Abstract.A new weak boundary procedure for hyperbolic problems is presented. We consider high order finite difference operators of summation-by-parts form with weak boundary conditions and generalize that technique. The new boundary procedure is applied near boundaries in an extended domain where data is known. We show how to raise the order of accuracy of the scheme, how to modify the spectrum of the resulting operator and how to construct non-reflecting properties at the boundaries. The new boundary procedure is cheap, easy to implement and suitable for all numerical methods, not only finite difference methods, that employ weak boundary conditions. Numerical results that corroborate the analysis are presented.

A stable and accurate relaxation technique using multiple penalty terms in space and time

“…The MPT technique introduced in [13] has been extended such that it can be applied at limited time intervals, to model real life observations better. The resulting numerical scheme is proven to be unconditionally stable with appropriate choices of penalty parameters.…”

“…We denote these spatial grid points and time intervals by Ω s and the additional data by g(x, t). The additional data will be implemented using SAT's [13,6].…”

“…In this work, we introduce a similar but provable stable technique for data assimilation based on Summation-By-Parts (SBP) operators [8,11,16,10,14] and Simultaneous Approximation Terms (SAT) [2,3]. This new technique is an extension of the Multiple Penalty Technique (MPT) introduced in [13,6], where SAT's are implemented at grid points inside the computational domain. Besides being simple and easy to implement, the MPT always results in a provably stable scheme.…”

Stable, high order accurate adaptive schemes for long time, highly intermittent geophysics problems

“…We do this through the use of the so-called multiplepenalty technique, which will draw the velocity of the outgoing wave at the remote boundary towards the slow plate rate V p . This technique is described in detail in [35]. The semi-discrete form of the equations (1) with m additional penalty matrices is…”

“…The penalties β j are turned on when the wave hits the remote boundary, damping the outgoing wave (see [35] for more details).…”