A New Projection Method for the Zero Froude Number Shallow Water Equations
S. Vater (June 2005)
For non-zero Froude numbers the shallow water equations are a hyperbolic system
of partial differential equations. In the zero Froude number limit, they are of mixed
hyperbolic-elliptic type, and the velocity field is subject to a divergence constraint.
A new semi-implicit projection method for the zero Froude number shallow water
equations is presented. This method enforces the divergence constraint on the
velocity field, in two steps. First, the numerical fluxes of an auxiliary hyperbolic
system are computed with a standard second order method. Then, these fluxes are
corrected by solving two Poisson-type equations. These corrections guarantee that
the new velocity field satisfies a discrete form of the above-mentioned divergence constraint.
The main feature of the new method is a unified discretization of the two
Poisson-type equations, which rests on a Petrov-Galerkin finite element formulation
with piecewise bilinear ansatz functions for the unknown variable. This discretization
naturally leads to piecewise linear ansatz functions for the momentum components.
The projection method is derived from a semi-implicit finite volume method for the
zero Mach number Euler equations, which uses standard discretizations for the solution
of the Poisson-type equations.
The new scheme can be formulated as an approximate as well as an exact projection
method. In the former case, the divergence constraint is not exactly satisfied.
The “approximateness” of the method can be estimated with an asymptotic upper
bound of the velocity divergence at the new time level, which is consistent with the
method’s second-order accuracy. In the exact projection method, the piecewise linear
components of the momentum are employed for the computation of the numerical
fluxes of the auxiliary system at the new time level.
In order to show the stability of the new projection step, a primal-dual mixed finite
element formulation is derived, which is equivalent to the Poisson-type equations of
the new scheme. Using the abstract theory of Nicolaïdes for generalized saddle
point problems, existence and uniqueness of the continuous problem are proven. Furthermore,
preliminary results regarding the stability of the discrete method are presented.
The numerical results obtained with the new exact method show significant accuracy
improvements over the version that uses standard discretizations for the solution
of the Poisson-type equations. In the L2 as well as the L¥ norm, the global error is
about four times smaller for smooth solutions. Simulating the advection of a vortex
with discontinuous vorticity field, the new method yields a more accurate position of
the center of the vortex.
