A multigrid-based multiscale numerical scheme for shallow water flows at low froude number

摘要: A new multiscale semi-implicit scheme for the computation of low Froude number shallow water flows is presented. Motivated by needs atmospheric flow applications, it aims to minimize dispersion and amplitude errors in long-wave gravity waves. While correctly balances “slaved” dynamics shortwave solution components induced slow forcing, method eliminates freely propagating compressible short-wave modes, which are under-resolved time. This achieved through a multilevel approach borrowing ideas from multigrid schemes elliptic equations. The second-order accurate admits time steps depending essentially on velocity. It incorporates predictor step using Godunov-type hyperbolic conservation laws two corrections numerical fluxes. First, derived one-dimensional linearized Scale-wise decomposition data enables scale-dependent blending integrators with different principal features. To guide selection these integrators, discrete-dispersion relations some standard analyzed, their response high-wave-number low-frequency source terms discussed. In particular, implicit trapezoidal rule backward differentiation formula (BDF) considered. resulting consists Helmholtz problem original fine grid, where differencing operator right hand side incorporate information discretization. performance illustrated test case “multiscale” initial slowly varying high-wavenumber term. simulating fully nonlinear generalization projection zero Therefore, described Vater Klein (Numer. Math. 113, pp. 123–161, 2009) extended account dependent bottom topography. Numerical simulations show that well-balanced can reproduce steady state lake at rest non-trivial Furthermore, results other cases verify correct representation Finally, equations numbers derived. extensions aforementioned incorporating local derivatives