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Register a new userOn the advantage of well-balanced schemes for moving-water equilibria of the shallow water equations
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This note aims at demonstrating the advantage of moving-water well-balanced schemes over still-water well-balanced schemes for the shallow water equations. We concentrate on numerical examples with solutions near a moving-water equilibrium. For such examples, still-water well-balanced methods are not capable of capturing the small perturbations of the moving-water equilibrium and may generate significant spurious oscillations, unless an extremely refined mesh is used. On the other hand, moving- water well-balanced methods perform well in these tests. The numerical examples in this note clearly demonstrate the importance of utilizing moving-water well-balanced methods for solutions near a moving-water equilibrium.
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In this paper, we develop a well-balanced oscillation-free discontinuous Galerkin (OFDG) method for solving the shallow water equations with a non-flat bottom topography. One notable feature of the constructed scheme is the well-balanced property, which preserves exactly the hydrostatic equilibrium solutions up to machine error. Another feature is the non-oscillatory property, which is very important in the numerical simulation when there exist some shock discontinuities. To control the spurious oscillations, we construct an OFDG method with an extra damping term to the existing well-balanced DG schemes proposed in [Y. Xing and C.-W. Shu, CICP, 1(2006), 100-134.]. With a careful construction of the damping term, the proposed method achieves both the well-balanced property and non-oscillatory property simultaneously without compromising any order of accuracy. We also present a detailed procedure for the construction and a theoretical analysis for the preservation of the well-balancedness property. Extensive numerical experiments including one- and two-dimensional space demonstrate that the proposed methods possess the desired properties without sacrificing any order of accuracy.
In this paper, high order semi-implicit well-balanced and asymptotic preserving finite difference WENO schemes are proposed for the shallow water equations with a non-flat bottom topography. We consider the Froude number ranging from O(1) to 0, which in the zero Froude limit becomes the lake equations for balanced flow without gravity waves. We apply a well-balanced finite difference WENO reconstruction, coupled with a stiffly accurate implicit-explicit (IMEX) Runge-Kutta time discretization. The resulting semi-implicit scheme can be shown to be well-balanced, asymptotic preserving (AP) and asymptotically accurate (AA) at the same time. Both one- and two-dimensional numerical results are provided to demonstrate the high order accuracy, AP property and good performance of the proposed methods in capturing small perturbations of steady state solutions.
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We describe a compatible finite element discretisation for the shallow water equations on the rotating sphere, concentrating on integrating consistent upwind stabilisation into the framework. Although the prognostic variables are velocity and layer depth, the discretisation has a diagnostic potential vorticity that satisfies a stable upwinded advection equation through a Taylor-Galerkin scheme; this provides a mechanism for dissipating enstrophy at the gridscale whilst retaining optimal order consistency. We also use upwind discontinuous Galerkin schemes for the transport of layer depth. These transport schemes are incorporated into a semi-implicit formulation that is facilitated by a hybridisation method for solving the resulting mixed Helmholtz equation. We illustrate our discretisation with some standard rotating sphere test problems.
Because of their capability to preserve steady-states, well-balanced schemes for Shallow Water equations are becoming popular. Among them, the hydrostatic reconstruction proposed in Audusse et al. (2004), coupled with a positive numerical flux, allows to verify important mathematical and physical properties like the positivity of the water height and, thus, to avoid unstabilities when dealing with dry zones. In this note, we prove that this method exhibits an abnormal behavior for some combinations of slope, mesh size and water height.
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