A Lagrangian method for the shallow water equations based on a Voronoi mesh - flows on a rotating sphere.

*(English)*Zbl 0591.76020
The free-Lagrange method, Proc. 1st Int. Conf., Hilton Head Island/S.C. 1985, Lect. Notes Phys. 238, 54-86 (1985).

[For the entire collection see Zbl 0573.00014.]

We have presented in this work and in a previous work [e.g.: the author, J. Comput. Phys. 53, 240-265 (1984; Zbl 0538.76028)] a new Lagrangian method for the shallow water equations. The advantages of this method may be summarized as follows. Firstly, there is no dependence on coordinate systems, and thus we can treat flows on a sphere without worrying about pole singularities. Secondly, unlike for other Lagrangian schemes, there is no restriction that points have to retain their initial neighbors. On the contrary, at each time step the particles find their natural neighbors, and the derivatives are computed using these neighbors. Thus the method allows for large deformations. Thirdly, a novel feature of the method is that it handles shocks in a very natural way. We consider a shock as one fluid particle overtaking another and colliding with it. This procedure not only makes it possible to handle shocks (which are unimportant in atmospheric flow calculations since the rotation of the earth causes waves to be dispersive) but it also guarantees the stability of the scheme by enforcing the Courant-Friedrichs-Lewy conditions in the neighborhood of each fluid marker. Fourthly, since the markers can be placed anywhere, the fluid markers may be placed, initially, at points where satellite or other measured data are available. The main problem with free Lagrangian methods however, is that the discrete operators seem to be of low order accuracy.

We have presented in this work and in a previous work [e.g.: the author, J. Comput. Phys. 53, 240-265 (1984; Zbl 0538.76028)] a new Lagrangian method for the shallow water equations. The advantages of this method may be summarized as follows. Firstly, there is no dependence on coordinate systems, and thus we can treat flows on a sphere without worrying about pole singularities. Secondly, unlike for other Lagrangian schemes, there is no restriction that points have to retain their initial neighbors. On the contrary, at each time step the particles find their natural neighbors, and the derivatives are computed using these neighbors. Thus the method allows for large deformations. Thirdly, a novel feature of the method is that it handles shocks in a very natural way. We consider a shock as one fluid particle overtaking another and colliding with it. This procedure not only makes it possible to handle shocks (which are unimportant in atmospheric flow calculations since the rotation of the earth causes waves to be dispersive) but it also guarantees the stability of the scheme by enforcing the Courant-Friedrichs-Lewy conditions in the neighborhood of each fluid marker. Fourthly, since the markers can be placed anywhere, the fluid markers may be placed, initially, at points where satellite or other measured data are available. The main problem with free Lagrangian methods however, is that the discrete operators seem to be of low order accuracy.

##### MSC:

76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |

76U05 | General theory of rotating fluids |

76H05 | Transonic flows |

76M99 | Basic methods in fluid mechanics |