A multigrid-based multiscale numerical scheme for shallow water flows at low froude number
Stefan Vater · Universitätsbibliothek der FU Berlin Hochschulschriftenstelle u. Dokumentenserver · 2013
A new multiscale 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 a multilevel approach borrowing ideas from multigrid schemes for elliptic equations. The scheme is second- order accurate and admits time steps depending essentially on the flow velocity. First, a multilevel method is 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 differencing operator and the right hand side incorporate the multiscale information of the discretization. The performance of the new multilevel method is illustrated on a test case with "multiscale" initial data and a problem with a slowly varying high-wave- number source term. The scheme for simulating fully nonlinear shallow water flows is a generalization of a projection method for the zero Froude number equations. Therefore, the method described in Vater and Klein (Numer. Math. 113, pp. 123-161, 2009) is extended to account for time dependent bottom topography. Numerical simulations show that the method is well-balanced in that it can reproduce the steady state of a lake at rest with non-trivial bottom topography. Further results verify the correct representation of time dependent bottom topography. Finally, the multiscale semi-implicit method for the nonlinear shallow water equations at low Froude numbers is derived. This incorporates two different extensions of the aforementioned projection method by incorporating the local time derivatives of the height. The two variants are combined in the multiscale method, which incorporates the solution of a multilevel Helmholtz problem, similar to the linear case. The method is implemented in one space dimension. The convergence of the method is analyzed by means of different test cases. Moreover, the method's balancing properties are addressed for the lake at rest, and the asymptotic regime of fast gravity waves traveling over short-range topography is considered. The numerical results of the proposed method suggest that the scheme correctly reproduces this regime, and can be therefore considered as a so-called asymptotically adaptive numerical method.