A stable and accurate Davies-like relaxation procedure using multiple penalty terms for lateral boundary conditions

Abstract

A lateral boundary treatment using summation-by-parts operators and simultaneous approximation terms is introduced. The method is similar to Davies relaxation technique used in the weather prediction community and have similar areas of application, but is also provably stable. In this paper, it is shown how this technique can be applied to the shallow water equations, and that it reduces the errors in the computational domain.

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

“…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.…”

On the Theoretical Foundation of Overset Grid Methods for Hyperbolic Problems: Well-Posedness and Conservation

“…To get energy bounded problems without assuming simultaneous diagonalization of the coefficient matrices, we use multiple interior penalty functions to eliminate terms not bounded by the data. Multiple penalty procedures have been used previously to increase the accuracy and stability of numerical solutions [9,21], and here we use them as part of the original IBVP to enforce energy boundedness. By adding penalty terms to the original equations we can remove the parasitic terms and obtain a suitable energy bound.…”