A multilevel time integrator for computing longwave shallow water flows at low Froude numbers
Seminar Room 1, Newton Institute
A new multilevel semi-implicit scheme for the computation of low Froude number shallow water flows is presented. Motivated by the needs of atmospheric flow applications, it aims to minimize dispersion and amplitude errors in the computation of long wave gravity waves. While it correctly balances "slaved" dynamics of short-wave solution components induced by slow forcing, the method eliminates freely propagating compressible short-wave modes, which are under- resolved in time. This is achieved through the multilevel approach borrowing ideas from multigrid schemes for elliptic equations. The scheme is second order accurate and admits time steps depending only on the flow velocity. It incorporates a predictor step using a Godunov-type method for hyperbolic conservation laws and two elliptic corrections.
The multilevel method is initially derived for the one-dimensional linearized shallow water equations. Scale-wise decomposition of the data enables a scale- dependent blending of time integrators with different principal features. To guide the selection of these integrators, the discrete-dispersion relations of some standard second-order schemes are analyzed, and their response to high wave number low frequency source terms is discussed. The resulting method essentially consists of the solution of a Helmholtz problem on the original fine grid, where the operator and the right hand side incorporate the multiscale information of the discretization. The performance of the method in the linear case is illustrated on a test case with "multiscale" initial data and a problem with a slowly varying high wave number source term.
For the computation of fully nonlinear shallow water flows, a projection method for zero Froude number flows is generalized by incorporating the local time derivatives of the height. This semi-implicit method is combined with the multilevel method for the linearized equations to obtain the above mentioned properties of the scheme. Numerical tests address the scheme's ability to correctly cover the asymptotic flow regime of long-wave gravity waves passing over short-range topography and its balancing properties for the lake at rest.