A New Projection Method for the Zero Froude Number Shallow Water Equations

摘要: For non-zero Froude numbers the shallow water equations are a hyperbolic system of partial differential equations. In zero number limit, they mixed hyperbolic-elliptic type, and velocity field is subject to divergence constraint. A new semi-implicit projection method for number shallow water presented. This enforces divergence constraint on velocity field, in two steps. First, numerical fluxes an auxiliary computed with standard second order method. Then, these fluxes corrected by solving two Poisson-type These corrections guarantee that the new field satisfies discrete form above-mentioned divergence constraint. The main feature unified discretization equations, which rests on a Petrov-Galerkin finite element formulation piecewise bilinear ansatz functions unknown variable. discretization naturally leads piecewise linear momentum components. derived from semi-implicit finite volume Mach Euler which uses standard discretizations solution Poisson-type equations. The scheme can be formulated as approximate well as exact former case, not exactly satisfied. "approximateness" the method estimated asymptotic upper bound velocity divergence at time level, consistent method's second-order accuracy. method, piecewise linear components momentum employed computation of numerical level. In show stability step, primal-dual mixed derived, equivalent to scheme. Using abstract theory Nicola"ides (1982) generalized saddle point problems, existence uniqueness continuous problem proven. Furthermore, preliminary results regarding method are presented. The obtained significant accuracy improvements over version uses discretizations for L-two L-infinity norm, global error about four times smaller for smooth solutions. Simulating advection vortex discontinuous vorticity yields more accurate position of the center vortex.