HR: 08:00h
AN: NG51A-04    [Abstracts]
TI: Conservative fully discrete schemes for the shallow-water model
AU: * Skiba, Y N
EM: skiba@servidor.unam.mx
AF: Centro de Ciencias de la Atmosfera, Universidad Nacional Autonoma de Mexico, Av. Universidad # 3000, Ciudad Universitaria, Coyoacan, Mexico, DF 04510, Mexico
AU: Filatov, D M
EM: denisfilatov@gmail.com
AF: Centro de Ciencias de la Atmosfera, Universidad Nacional Autonoma de Mexico, Av. Universidad # 3000, Ciudad Universitaria, Coyoacan, Mexico, DF 04510, Mexico
AB: It is considered the classical nonlinear shallow-water model (SWM) of an ideal fluid. It is well known that the model conserves several integral characteristics such as the mass, total energy and potential enstrophy. It is extremely desirable to conserve the same characteristics in a fully discrete SWM (discrete both in space and time), because they guarantee the stability of calculations and correct description of the energy cascades in the discrete model. However, the full discretization usually destroys some, if not all, of the conservation laws, and therefore, the construction of conservative fully discrete SWMs is a non-trivial and actual scientific problem. For the last forty years there have been suggested various semi-discrete SWMs (discrete in space, but still continuous in time), which conserve one or all of the three above-mentioned integral characteristics. In particular, the model by Ringler and Randall (2002) uses rather complicated geodesic grids on a sphere, while that by Salmon (2004) applies a sophisticated stencil (containing 25 nodes) in a doubly periodic domain on the f-plane. Nevertheless, explicit time discretization used in both works resulted in the loss of all the conservation laws except the mass conservation. In this work, new fully discrete SWMs are suggested, which exactly conserve the mass and total energy. The splitting of the SWM operator in geometric coordinates provides substantial benefits in the computational cost of the solution, as well as in the applicability to a doubly periodic domain on the plane, in a periodic channel on a rotating sphere, and on the whole sphere. Each split one-dimensional fully discrete system conserves the mass and total energy, too. In fact, a family of finite-difference schemes of different approximation order is suggested, either linear or nonlinear, depending on the choice of certain parameters. Note that on a sphere and in a doubly periodic domain, our approach allows constructing various linear conservative schemes of arbitrary approximation order in space. Results of numerical experiments are discussed.
DE: 0545 Modeling (4255)
DE: 0550 Model verification and validation
DE: 0560 Numerical solutions (4255)
DE: 0774 Dynamics
DE: 4445 Nonlinear differential equations
SC: Nonlinear Geophysics [NG]
MN: 2007 Joint Assembly