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Volume 16, Issue 2
IMEX Large Time Step Finite Volume Methods for Low Froude Number Shallow Water Flows

Georgij Bispen, K. R. Arun, Mária Lukáčová-Medvid'ová & Sebastian Noelle

DOI: 10.4208/cicp.040413.160114a

Commun. Comput. Phys., 16 (2014), pp. 307-347.

Published online: 2014-08

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  • Abstract

We present new large time step methods for the shallow water flows in the low Froude number limit. In order to take into account multiscale phenomena that typically appear in geophysical flows nonlinear fluxes are split into a linear part governing the gravitational waves and the nonlinear advection. We propose to approximate fast linear waves implicitly in time and in space by means of a genuinely multidimensional evolution operator. On the other hand, we approximate nonlinear advection part explicitly in time and in space by means of the method of characteristics or some standard numerical flux function. Time integration is realized by the implicit-explicit (IMEX) method. We apply the IMEX Euler scheme, two step Runge Kutta Cranck Nicolson scheme, as well as the semi-implicit BDF scheme and prove their asymptotic preserving property in the low Froude number limit. Numerical experiments demonstrate stability, accuracy and robustness of these new large time step finite volume schemes with respect to small Froude number.

  • Keywords

Low Froude number flows asymptotic preserving schemes shallow water equations large time step semi-implicit approximation evolution Galerkin schemes.

  • AMS Subject Headings

35L65 76B15 65M08 65M06 35L45 35L65

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-16-307, author = {Georgij Bispen, K. R. Arun, Mária Lukáčová-Medvid'ová and Sebastian Noelle}, title = {IMEX Large Time Step Finite Volume Methods for Low Froude Number Shallow Water Flows}, journal = {Communications in Computational Physics}, year = {2014}, volume = {16}, number = {2}, pages = {307--347}, abstract = {

We present new large time step methods for the shallow water flows in the low Froude number limit. In order to take into account multiscale phenomena that typically appear in geophysical flows nonlinear fluxes are split into a linear part governing the gravitational waves and the nonlinear advection. We propose to approximate fast linear waves implicitly in time and in space by means of a genuinely multidimensional evolution operator. On the other hand, we approximate nonlinear advection part explicitly in time and in space by means of the method of characteristics or some standard numerical flux function. Time integration is realized by the implicit-explicit (IMEX) method. We apply the IMEX Euler scheme, two step Runge Kutta Cranck Nicolson scheme, as well as the semi-implicit BDF scheme and prove their asymptotic preserving property in the low Froude number limit. Numerical experiments demonstrate stability, accuracy and robustness of these new large time step finite volume schemes with respect to small Froude number.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.040413.160114a}, url = {http://global-sci.org/intro/article_detail/cicp/7044.html} }
TY - JOUR T1 - IMEX Large Time Step Finite Volume Methods for Low Froude Number Shallow Water Flows AU - Georgij Bispen, K. R. Arun, Mária Lukáčová-Medvid'ová & Sebastian Noelle JO - Communications in Computational Physics VL - 2 SP - 307 EP - 347 PY - 2014 DA - 2014/08 SN - 16 DO - http://doi.org/10.4208/cicp.040413.160114a UR - https://global-sci.org/intro/article_detail/cicp/7044.html KW - Low Froude number flows KW - asymptotic preserving schemes KW - shallow water equations KW - large time step KW - semi-implicit approximation KW - evolution Galerkin schemes. AB -

We present new large time step methods for the shallow water flows in the low Froude number limit. In order to take into account multiscale phenomena that typically appear in geophysical flows nonlinear fluxes are split into a linear part governing the gravitational waves and the nonlinear advection. We propose to approximate fast linear waves implicitly in time and in space by means of a genuinely multidimensional evolution operator. On the other hand, we approximate nonlinear advection part explicitly in time and in space by means of the method of characteristics or some standard numerical flux function. Time integration is realized by the implicit-explicit (IMEX) method. We apply the IMEX Euler scheme, two step Runge Kutta Cranck Nicolson scheme, as well as the semi-implicit BDF scheme and prove their asymptotic preserving property in the low Froude number limit. Numerical experiments demonstrate stability, accuracy and robustness of these new large time step finite volume schemes with respect to small Froude number.

Georgij Bispen, K. R. Arun, Mária Lukáčová-Medvid'ová and Sebastian Noelle. (2014). IMEX Large Time Step Finite Volume Methods for Low Froude Number Shallow Water Flows. Communications in Computational Physics. 16 (2). 307-347. doi:10.4208/cicp.040413.160114a
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