Abstract
The present work focuses on the numerical approximation of the weak solutions of the shallow water model over a non-flat topography. In particular, we pay close attention to steady solutions with nonzero velocity. The goal of this work is to derive a scheme that exactly preserves these stationary solutions, as well as the commonly preserved lake at rest steady solution. These moving steady states are solution to a nonlinear equation. We emphasize that the method proposed here never requires solving this nonlinear equation; instead, a suitable linearization is derived. To address this issue, we propose an extension of the well-known hydrostatic reconstruction. By appropriately defining the reconstructed states at the interfaces, any numerical flux function, combined with a relevant source term discretization, produces a well-balanced scheme that preserves both moving and non-moving steady solutions. This eliminates the need to construct specific numerical fluxes. Additionally, we prove that the resulting scheme is consistent with the homogeneous system on flat topographies, and that it reduces to the hydrostatic reconstruction when the velocity vanishes. To increase the accuracy of the simulations, we propose a well-balanced high-order procedure, which still does not require solving any nonlinear equation. Several numerical experiments demonstrate the effectiveness of the numerical scheme.
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funding C. Berthon acknowledges the support of ANR MUFFIN ANR-19-CE46-0004, funded by Agence Nationale de la Recherche. The SHARK-FV conference has greatly contributed to this work.
A Taylor expansions of 𝓗
The goal here is to provide a Taylor expansion of the function 𝓗 given by (4.5), in the case where ΔZ > 0, Δh > 0 and 1 – Fr2 > 0, when ΔZ approaches zero. The computations are performed below, where we have temporarily set 𝔽 = 1 – Fr2 in order to save some space.
B Taylor expansions of 𝓒
In this section, we give the Taylor expansions of the function 𝓒(hL, hR, qL, ΔZ), given by (4.7), when either hL or hR go to 0. The goal is to prove (𝓗-3), i.e., prove that 𝓒 is continuous when either hL tends to 0 and hR ≠ 0, and when hL tends to 0 and hR = hL. Recall that
Since qL = hLuL, with u the velocity, we also note that
First, we consider the case where hL goes to 0 and hR ≠ 0. According to assumptions (1.3), in this case, uL also goes to zero. To model this phenomenon, we assume that uL = u(hL), where the function u is such that u(0) = 0. In this case, we get, again using symbolic computation software,
In the above expression, for the sake of clarity, the ± symbols correspond to sgn(ΔZ). In any case, since u(0) = 0, this Taylor expansion shows that
which is what we had set out to prove.
Second, we have to prove that
However, recall from (4.6) that 𝓗 = (hR – hL) 𝓑, with 𝓑 a bounded function. The above result is established by arguing the boundedness (1.3) of the Froude number.
Therefore, property (𝓗-3) is satisfied by 𝓗, when 𝓗 is given by (4.5).
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