A Comparative Study of Iterative Riemann Solvers for the Shallow Water and Euler Equations

dc.contributor.authorMoncayo, Carlos Muñoz
dc.contributor.authorQuezada de Luna, Manuel
dc.contributor.authorKetcheson, David I.
dc.contributor.departmentApplied Mathematics and Computational Science Program
dc.contributor.departmentComputer, Electrical and Mathematical Science and Engineering (CEMSE) Division
dc.contributor.departmentExtreme Computing Research Center
dc.contributor.departmentNumerical Mathematics Group
dc.date.accessioned2022-09-28T08:48:01Z
dc.date.available2022-09-28T08:48:01Z
dc.date.issued2022-09-25
dc.description.abstractThe Riemann problem for first-order hyperbolic systems of partial differential equations is of fundamental importance for both theoretical and numerical purposes. Many approximate solvers have been developed for such systems; exact solution algorithms have received less attention because computation of the exact solution typically requires iterative solution of algebraic equations. Iterative algorithms may be less computationally efficient or might fail to converge in some cases. We investigate the achievable efficiency of robust iterative Riemann solvers for relatively simple systems, focusing on the shallow water and Euler equations. We consider a range of initial guesses and iterative schemes applied to an ensemble of test Riemann problems. For the shallow water equations, we find that Newton's method with a simple modification converges quickly and reliably. For the Euler equations we obtain similar results; however, when the required precision is high, a combination of Ostrowski and Newton iterations converges faster. These solvers are slower than standard approximate solvers like Roe and HLLE, but come within a factor of two in speed. We also provide a preliminary comparison of the accuracy of a finite volume discretization using an exact solver versus standard approximate solvers.
dc.eprint.versionPre-print
dc.identifier.arxivid2209.12235
dc.identifier.urihttps://repository.kaust.edu.sa/handle/10754/681722.1
dc.publisherarXiv
dc.relation.urlhttps://arxiv.org/pdf/2209.12235.pdf
dc.rightsThis is a preprint version of a paper and has not been peer reviewed. Archived with thanks to arXiv.
dc.titleA Comparative Study of Iterative Riemann Solvers for the Shallow Water and Euler Equations
dc.typePreprint
display.details.left<span><h5>Type</h5>Preprint<br><br><h5>Authors</h5><a href="https://repository.kaust.edu.sa/search?spc.sf=dc.date.issued&spc.sd=DESC&f.author=Moncayo, Carlos Muñoz,equals">Moncayo, Carlos Muñoz</a><br><a href="https://repository.kaust.edu.sa/search?spc.sf=dc.date.issued&spc.sd=DESC&f.author=Quezada de Luna, Manuel,equals">Quezada de Luna, Manuel</a><br><a href="https://repository.kaust.edu.sa/search?query=orcid.id:0000-0002-1212-126X&spc.sf=dc.date.issued&spc.sd=DESC">Ketcheson, David I.</a> <a href="https://orcid.org/0000-0002-1212-126X" target="_blank"><img src="https://repository.kaust.edu.sa/server/api/core/bitstreams/82a625b4-ed4b-40c8-865a-d6a5225a26a4/content" width="16" height="16"/></a><br><br><h5>KAUST Department</h5><a href="https://repository.kaust.edu.sa/search?spc.sf=dc.date.issued&spc.sd=DESC&f.department=Applied Mathematics and Computational Science Program,equals">Applied Mathematics and Computational Science Program</a><br><a href="https://repository.kaust.edu.sa/search?spc.sf=dc.date.issued&spc.sd=DESC&f.department=Computer, Electrical and Mathematical Science and Engineering (CEMSE) Division,equals">Computer, Electrical and Mathematical Science and Engineering (CEMSE) Division</a><br><a href="https://repository.kaust.edu.sa/search?spc.sf=dc.date.issued&spc.sd=DESC&f.department=Extreme Computing Research Center,equals">Extreme Computing Research Center</a><br><a href="https://repository.kaust.edu.sa/search?spc.sf=dc.date.issued&spc.sd=DESC&f.department=Numerical Mathematics Group,equals">Numerical Mathematics Group</a><br><br><h5>Date</h5>2022-09-25</span>
display.details.right<span><h5>Abstract</h5>The Riemann problem for first-order hyperbolic systems of partial differential equations is of fundamental importance for both theoretical and numerical purposes. Many approximate solvers have been developed for such systems; exact solution algorithms have received less attention because computation of the exact solution typically requires iterative solution of algebraic equations. Iterative algorithms may be less computationally efficient or might fail to converge in some cases. We investigate the achievable efficiency of robust iterative Riemann solvers for relatively simple systems, focusing on the shallow water and Euler equations. We consider a range of initial guesses and iterative schemes applied to an ensemble of test Riemann problems. For the shallow water equations, we find that Newton's method with a simple modification converges quickly and reliably. For the Euler equations we obtain similar results; however, when the required precision is high, a combination of Ostrowski and Newton iterations converges faster. These solvers are slower than standard approximate solvers like Roe and HLLE, but come within a factor of two in speed. We also provide a preliminary comparison of the accuracy of a finite volume discretization using an exact solver versus standard approximate solvers.<br><br><h5>Publisher</h5><a href="https://repository.kaust.edu.sa/search?spc.sf=dc.date.issued&spc.sd=DESC&f.publisher=arXiv,equals">arXiv</a><br><br><h5>arXiv</h5><a href="https://arxiv.org/abs/2209.12235">2209.12235</a><br><br><h5>Additional Links</h5>https://arxiv.org/pdf/2209.12235.pdf</span>
kaust.personMoncayo, Carlos Muñoz
kaust.personQuezada de Luna, Manuel
kaust.personKetcheson, David I.
orcid.authorMoncayo, Carlos Muñoz
orcid.authorQuezada de Luna, Manuel
orcid.authorKetcheson, David I.::0000-0002-1212-126X
orcid.id0000-0002-1212-126X
refterms.dateFOA2022-09-28T08:48:44Z
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