An eigenvector‐based linear reconstruction approach for time stepping in discontinuous Galerkin scheme used to solve shallow water equations

Abstract

SUMMARY Discontinuous Galerkin (DG) methods have shown promising results for solving the two‐dimensional shallow water equations. In this paper, the classical Runge–Kutta (RK) time discretisation is replaced by the eigenvector‐based reconstruction (EVR) that allows the second‐order time accuracy to be achieved within a single time‐stepping procedure. Moreover, the EVRDG approach yields stable solutions near drying and wetting fronts, whereas the classical RKDG approach yields instabilities. The proposed EVRDG technique is compared with the original RKDG approach on various test cases with analytical solutions. The EVRDG solutions are shown to be as accurate as those obtained with the RKDG scheme. Besides, the EVRDG scheme is 1.6 times faster than the RKDG method. Simulating dambreaks involving dry beds confirms that EVRDG scheme gives correct solutions, whereas the RKDG method yields instabilities. Copyright © 2012 John Wiley & Sons, Ltd.

Discontinuous Galerkin flood model formulation: Luxury or necessity?

“…They are receiving increasing interest in water resource research in light of the applied advances adopted in finite volume methods [e.g., Bokhove , ; Ern et al ., ; Gourgue et al ., ; Kesserwani and Liang , ]. In addition to the factors discussed above under (ii) and (iii), these also include other characteristic features such as hp ‐adaptivity, local slope coefficient limiting, polynomial wet/dry front tracking, and the time stepping issue [e.g., Krivodonova et al ., ; Kubatko et al ., 2009; Kesserwani and Liang , ; Xing et al ., ; Araud et al ., ]. However, the DG formulation is seldom adopted for the simulation of flood propagation problems hindered, perhaps, by its excessive complexity.…”

Multiwavelet-based grid adaptation with discontinuous Galerkin schemes for shallow water equations

“…In particular, this requires more data transfer on distributed memory architectures and, thus, affects the efficiency of parallel implementations. Discontinuous Galerkin (DG) schemes have been recently applied to the shallow water equations [16,17,18,19,20,21]. The DG method generalizes the concept of the FV method, while relying on the Finite Element notion of projecting the solution onto a space of trial functions, but, without the restriction of keeping the functions continuous.…”