Volume 28, Issue 3
Analysis and Numerical Simulation of Hyperbolic Shallow Water Moment Equations

Julian KoellermeierMarvin Rominger

Commun. Comput. Phys., 28 (2020), pp. 1038-1084.

Published online: 2020-07

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  • Abstract

Shallow Water Moment Equations allow for vertical changes in the horizontal velocity, so that complex shallow flows can be described accurately. However, we show that these models lack global hyperbolicity and the loss of hyperbolicity already occurs for small deviations from equilibrium. This leads to instabilities in a numerical test case. We then derive new Hyperbolic Shallow Water Moment Equations based on a modification of the system matrix. The model can be written in analytical form and hyperbolicity can be proven for a large number of equations. A second variant of this model is obtained by generalizing the modification with the help of additional parameters. Numerical tests of a smooth periodic problem and a dam break problem using the new models yield accurate and fast solutions while guaranteeing hyperbolicity.

  • Keywords

Shallow water equation, moment method, hyperbolicity.

  • AMS Subject Headings

76D05, 35L65, 65M08

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-28-1038, author = {Koellermeier , Julian and Rominger , Marvin}, title = {Analysis and Numerical Simulation of Hyperbolic Shallow Water Moment Equations}, journal = {Communications in Computational Physics}, year = {2020}, volume = {28}, number = {3}, pages = {1038--1084}, abstract = {

Shallow Water Moment Equations allow for vertical changes in the horizontal velocity, so that complex shallow flows can be described accurately. However, we show that these models lack global hyperbolicity and the loss of hyperbolicity already occurs for small deviations from equilibrium. This leads to instabilities in a numerical test case. We then derive new Hyperbolic Shallow Water Moment Equations based on a modification of the system matrix. The model can be written in analytical form and hyperbolicity can be proven for a large number of equations. A second variant of this model is obtained by generalizing the modification with the help of additional parameters. Numerical tests of a smooth periodic problem and a dam break problem using the new models yield accurate and fast solutions while guaranteeing hyperbolicity.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2019-0065}, url = {http://global-sci.org/intro/article_detail/cicp/17675.html} }
TY - JOUR T1 - Analysis and Numerical Simulation of Hyperbolic Shallow Water Moment Equations AU - Koellermeier , Julian AU - Rominger , Marvin JO - Communications in Computational Physics VL - 3 SP - 1038 EP - 1084 PY - 2020 DA - 2020/07 SN - 28 DO - http://doi.org/10.4208/cicp.OA-2019-0065 UR - https://global-sci.org/intro/article_detail/cicp/17675.html KW - Shallow water equation, moment method, hyperbolicity. AB -

Shallow Water Moment Equations allow for vertical changes in the horizontal velocity, so that complex shallow flows can be described accurately. However, we show that these models lack global hyperbolicity and the loss of hyperbolicity already occurs for small deviations from equilibrium. This leads to instabilities in a numerical test case. We then derive new Hyperbolic Shallow Water Moment Equations based on a modification of the system matrix. The model can be written in analytical form and hyperbolicity can be proven for a large number of equations. A second variant of this model is obtained by generalizing the modification with the help of additional parameters. Numerical tests of a smooth periodic problem and a dam break problem using the new models yield accurate and fast solutions while guaranteeing hyperbolicity.

Julian Koellermeier & Marvin Rominger. (2020). Analysis and Numerical Simulation of Hyperbolic Shallow Water Moment Equations. Communications in Computational Physics. 28 (3). 1038-1084. doi:10.4208/cicp.OA-2019-0065
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