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Title: A residual-based shock capturing scheme for the continuous/discontinuous spectral element solution of the 2D shallow water equations

Abstract

The high-order numerical solution of the non-linear shallow water equations is susceptible to Gibbs oscillations in the proximity of strong gradients. In this paper, we tackle this issue by presenting a shock capturing model based on the numerical residual of the solution. Via numerical tests, we demonstrate that the model removes the spurious oscillations in the proximity of strong wave fronts while preserving their strength. Furthermore, for coarse grids, it prevents energy from building up at small wave-numbers. When applied to the continuity equation to stabilize the water surface, the addition of the shock capturing scheme does not affect mass conservation. We found that our model improves the continuous and discontinuous Galerkin solutions alike in the proximity of sharp fronts propagating on wet surfaces. Furthermore, in the presence of wet/dry interfaces, however, the model needs to be enhanced with the addition of an inundation scheme which, however, we do not address in this paper.

Authors:
 [1]; ORCiD logo [2];  [3];  [1];  [4]
  1. Stanford Univ., Stanford, CA (United States)
  2. Univ. of California, Santa Cruz, CA (United States)
  3. Argonne National Lab. (ANL), Argonne, IL (United States); The Univ. of Chicago, Chicago, IL (United States)
  4. Naval Postgraduate School, Monterey, CA (United States)
Publication Date:
Research Org.:
Argonne National Lab. (ANL), Argonne, IL (United States)
Sponsoring Org.:
Air Force Research Laboratory (AFRL), Air Force Office of Scientific Research (AFOSR); USDOE
OSTI Identifier:
1463698
Grant/Contract Number:  
AC02-06CH11357
Resource Type:
Accepted Manuscript
Journal Name:
Advances in Water Resources
Additional Journal Information:
Journal Volume: 114; Journal Issue: C; Journal ID: ISSN 0309-1708
Publisher:
Elsevier
Country of Publication:
United States
Language:
English
Subject:
58 GEOSCIENCES; de-aliasing; dynamic artificial diffusion; element-based Galerkin methods; high-order methods; large eddy simulation; shallow water equations; unified continuous/discontinuous Galerkin

Citation Formats

Marras, Simone, Kopera, Michal A., Constantinescu, Emil M., Suckale, Jenny, and Giraldo, Francis X. A residual-based shock capturing scheme for the continuous/discontinuous spectral element solution of the 2D shallow water equations. United States: N. p., 2018. Web. doi:10.1016/j.advwatres.2018.02.003.
Marras, Simone, Kopera, Michal A., Constantinescu, Emil M., Suckale, Jenny, & Giraldo, Francis X. A residual-based shock capturing scheme for the continuous/discontinuous spectral element solution of the 2D shallow water equations. United States. doi:10.1016/j.advwatres.2018.02.003.
Marras, Simone, Kopera, Michal A., Constantinescu, Emil M., Suckale, Jenny, and Giraldo, Francis X. Thu . "A residual-based shock capturing scheme for the continuous/discontinuous spectral element solution of the 2D shallow water equations". United States. doi:10.1016/j.advwatres.2018.02.003. https://www.osti.gov/servlets/purl/1463698.
@article{osti_1463698,
title = {A residual-based shock capturing scheme for the continuous/discontinuous spectral element solution of the 2D shallow water equations},
author = {Marras, Simone and Kopera, Michal A. and Constantinescu, Emil M. and Suckale, Jenny and Giraldo, Francis X.},
abstractNote = {The high-order numerical solution of the non-linear shallow water equations is susceptible to Gibbs oscillations in the proximity of strong gradients. In this paper, we tackle this issue by presenting a shock capturing model based on the numerical residual of the solution. Via numerical tests, we demonstrate that the model removes the spurious oscillations in the proximity of strong wave fronts while preserving their strength. Furthermore, for coarse grids, it prevents energy from building up at small wave-numbers. When applied to the continuity equation to stabilize the water surface, the addition of the shock capturing scheme does not affect mass conservation. We found that our model improves the continuous and discontinuous Galerkin solutions alike in the proximity of sharp fronts propagating on wet surfaces. Furthermore, in the presence of wet/dry interfaces, however, the model needs to be enhanced with the addition of an inundation scheme which, however, we do not address in this paper.},
doi = {10.1016/j.advwatres.2018.02.003},
journal = {Advances in Water Resources},
number = C,
volume = 114,
place = {United States},
year = {2018},
month = {2}
}

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