Volume 10, Issue 2
Finite Volume Evolution Galerkin Methods for the Shallow Water Equations with Dry Beds

Andreas Bollermann, Sebastian Noelle & Maria Lukáčová-Medvid'ová

Commun. Comput. Phys., 10 (2011), pp. 371-404.

Published online: 2011-10

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  • Abstract

We present a new Finite Volume Evolution Galerkin (FVEG) scheme for the solution of the shallow water equations (SWE) with the bottom topography as a source term. Our new scheme will be based on the FVEG methods presented in (Noelle and Kraft, J. Comp. Phys., 221 (2007)), but adds the possibility to handle dry boundaries. The most important aspect is to preserve the positivity of the water height. We present a general approach to ensure this for arbitrary finite volume schemes. The main idea is to limit the outgoing fluxes of a cell whenever they would create negative water height. Physically, this corresponds to the absence of fluxes in the presence of vacuum. Well-balancing is then re-established by splitting gravitational and gravity driven parts of the flux. Moreover, a new entropy fix is introduced that improves the reproduction of sonic rarefaction waves.

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@Article{CiCP-10-371, author = {Andreas Bollermann , and Sebastian Noelle , and Maria Lukáčová-Medvid'ová , }, title = {Finite Volume Evolution Galerkin Methods for the Shallow Water Equations with Dry Beds}, journal = {Communications in Computational Physics}, year = {2011}, volume = {10}, number = {2}, pages = {371--404}, abstract = {

We present a new Finite Volume Evolution Galerkin (FVEG) scheme for the solution of the shallow water equations (SWE) with the bottom topography as a source term. Our new scheme will be based on the FVEG methods presented in (Noelle and Kraft, J. Comp. Phys., 221 (2007)), but adds the possibility to handle dry boundaries. The most important aspect is to preserve the positivity of the water height. We present a general approach to ensure this for arbitrary finite volume schemes. The main idea is to limit the outgoing fluxes of a cell whenever they would create negative water height. Physically, this corresponds to the absence of fluxes in the presence of vacuum. Well-balancing is then re-established by splitting gravitational and gravity driven parts of the flux. Moreover, a new entropy fix is introduced that improves the reproduction of sonic rarefaction waves.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.220210.020710a}, url = {http://global-sci.org/intro/article_detail/cicp/7446.html} }
TY - JOUR T1 - Finite Volume Evolution Galerkin Methods for the Shallow Water Equations with Dry Beds AU - Andreas Bollermann , AU - Sebastian Noelle , AU - Maria Lukáčová-Medvid'ová , JO - Communications in Computational Physics VL - 2 SP - 371 EP - 404 PY - 2011 DA - 2011/10 SN - 10 DO - http://dor.org/10.4208/cicp.220210.020710a UR - https://global-sci.org/intro/article_detail/cicp/7446.html KW - AB -

We present a new Finite Volume Evolution Galerkin (FVEG) scheme for the solution of the shallow water equations (SWE) with the bottom topography as a source term. Our new scheme will be based on the FVEG methods presented in (Noelle and Kraft, J. Comp. Phys., 221 (2007)), but adds the possibility to handle dry boundaries. The most important aspect is to preserve the positivity of the water height. We present a general approach to ensure this for arbitrary finite volume schemes. The main idea is to limit the outgoing fluxes of a cell whenever they would create negative water height. Physically, this corresponds to the absence of fluxes in the presence of vacuum. Well-balancing is then re-established by splitting gravitational and gravity driven parts of the flux. Moreover, a new entropy fix is introduced that improves the reproduction of sonic rarefaction waves.

Andreas Bollermann, Sebastian Noelle & Maria Lukáčová-Medvid'ová. (2020). Finite Volume Evolution Galerkin Methods for the Shallow Water Equations with Dry Beds. Communications in Computational Physics. 10 (2). 371-404. doi:10.4208/cicp.220210.020710a
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