# A comprehensive list of stationary solutions of shallow water models

Nina Aguillon, Emmanuel Audusse, Nelly Boulos, Martin Parisot

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Nina Aguillon, Emmanuel Audusse, Nelly Boulos, Martin Parisot. A comprehensive list of stationary solutions of shallow water models. 2025. hal-05025901

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<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
A comprehensive list of stationary solutions of shallow water
models

Nina Aguillon * Emmanuel Audusse † Nelly Boulos ‡ Martin Parisot §

April 8, 2025

Abstract

The shallow water equation of Barré de Saint-Venant describes the evolution of the water height and of the horizontal velocity of a column of water. It is valid when the horizontal length scale of the flow is much greater than its vertical length scale. This model is widely employed to model rivers and lake. It can be enriched to take into account a shear velocity along the vertical. In this work we classify all the stationary solutions of the shallow water equation with a shear velocity on a bounded domain. Boundary conditions are fixed at the inlet and the outlet, and their number depends on whether the regime is supercritical (Froude number larger than 1) or subcritical. The stationary solutions consist of a succession of regular parts where the momentum and hydraulic head are constant, separated by discontinuity (also called hydraulic jumps) through which the hydraulic head decreases but the momentum and momentum flux are constant. We exhibit the form of the stationary solution and its conditions of existence for any set of boundary conditions on monotonic, bumped shaped and hollow shaped topographies. We find that in some cases and for supercritical flow at the inlet, two different stationary solutions can fulfilled the same set of boundary conditions. The solutions with shocks on increasing topography are sparsely documented and seem to be instable. Eventually we consider a topography made of a succession of *N* decreasing bumps and prove in the subcritical case that up to 2<sup>*N*−1</sup> solutions with discontinuities on decreasing topography only may coexist.

**Keywords:** shallow water equation, stationary solution, boundary problem, hydraulic jumps

MSC classification: 35L40, 34K21, 34B60, 76L05

<footnote>*Sorbonne-Université, CNRS, Université de Paris, Laboratoire Jacques-Louis Lions (LJLL), F-75005 Paris,
France and Inria Paris</footnote>

<footnote>†LAGA, Institut Galilée, Université Sorbonne Paris Nord – CNRS UMR 7539 – 99 avenue Jean-Baptiste Clément, 93430 Villetaneuse.</footnote>

<footnote>‡LAGA, Institut Galilée, Université Sorbonne Paris Nord – CNRS UMR 7539 – 99 avenue Jean-Baptiste Clément, 93430 Villetaneuse.</footnote>

<footnote>§INRIA Bordeaux, team CARDAMOM, 200, avenue de la Vieille Tour, 33405 Talence CEDEX.</footnote>
# Introduction

This work is dedicated to the theoretical study of the steady state solutions of the 1D shallow water model with shear which reads

$$ 
\left\{ 
\begin{array}{l}
\partial_t h + \partial_x (h \bar{u}) = 0, \\
\partial_t (h \bar{u}) + \partial_x (h (\bar{u}^2 + \hat{u}^2) + \frac{g}{2} h^2) = -gh \partial_x Z, \\
\partial_t \hat{u} + \partial_x (\bar{u} \hat{u}) = 0.
\end{array}
\right. 
\quad (SW_2) \text{ SW2}
$$

with $g > 0$ the acceleration of gravity and $Z$ the $C^1$ regular topography. The unknowns of the system are $h(t, x) \in \mathbb{R}_+$ the water depth of the flow at time $t \ge 0$ and point $x \in \mathbb{R}$, $\bar{u}(t, x) \in \mathbb{R}$ the vertical-averaged of the horizontal velocity and $\hat{u}(t, x) \in \mathbb{R}$ the signed standard deviation of the horizontal velocity. This model encompasses the classical shallow water model of Barré de Saint-Venant when $\hat{u} = 0$.

System (SW₂) is related to two models that were recently introduced in the hyperbolic literature: the shear shallow water model presented in [9] and the two velocities version of the multiv velocities system introduced in [2]. More precisely, it is equivalent to these two systems in 1d and for continuous solutions. For discontinuous solutions, and since the other two models contain non-conservative source terms, system (SW₂) can be considered as a particular choice for defining these non-conservative products. In 2d, the models are no longer equivalent but we do not address this point in this work. Without bottom topography, i.e. when $Z(x) = 0$, system (SW₂) is a hyperbolic system of conservation laws and a complete study of the solution of the Riemann problem has been performed in [1].

Non-trivial analytical solutions of system of balance laws are of great interest because they can be used for at least two purposes: to validate models by comparing them to experimental data and to validate computational codes by comparing approximate solutions to exact ones. Among these solutions, the stationary solutions, if they are stable, are of particular interest because they are often easier to observe experimentally. The 1d stationary solutions of the classical Saint-Venant system, i.e. when the standard deviation $\hat{u}$ is identically zero, have been used for a long time and very widely to validate computational codes, see for example the Swashes library [5]. In particular, it is well known that these stationary solutions can belong to different flow regimes: subcritical (fluvial), supercritical (torrential) or transcritical (with one or more transitions from one regime to the other).

Surprisingly we have not found an exhaustive study of these stationary solutions in the following sense: given two boundary conditions, is it possible to construct a stationary solution? If so, how many? Answering those questions is the main objective of this work. We start with some definition and general results on stationary solutions of (SW₂) in Section 1. Section 2 gives a complete classification of the steady state solution depending on the boundary conditions. Results are stated for the model with shear (SW₂) and are in particular valid for the Saint-Venant when $\hat{u} = 0$. They permit to determine in advance which type of steady state solutions are obtained. When the regime is supercritical at the inlet we prove the coexistence of two steady state solutions verifying the same set of boundary condition which are not documented in the applied mathematical litterature, see for example [8] and the Swashes library [5], but appear in [3].
# 1 Definition of the steady state solutions

## 1.1 Some properties of the model

When the topography is not flat, the model (SW₂) is a hyperbolic system of balance laws that can be written in generic form

$$ \partial_t U + \partial_x F(U) = B(U, Z), $$

where

$$ U = \begin{pmatrix} h \\ h\bar{u} \\ \hat{u} \end{pmatrix}, \quad F(U) = \begin{pmatrix} h\bar{u} \\ h(\bar{u}^2 + \hat{u}^2) + \frac{g}{2}h^2 \\ \bar{u}\hat{u} \end{pmatrix} \quad \text{and} \quad B(U, Z) = \begin{pmatrix} 0 \\ -gh\partial_x Z \\ 0 \end{pmatrix}. $$

As shown in [1, 9], the Jacobian matrix DF(U) of the flux F of (SW₂) system has three real eigenvalues that reads

$$ \lambda_1 = \bar{u} - \sqrt{gh + 3\hat{u}^2}, \quad \lambda_2 = \bar{u} \quad \text{and} \quad \lambda_3 = \bar{u} + \sqrt{gh + 3\hat{u}^2}. \quad (1) \quad \text{eigenvalues} $$

As in the case of the classical shallow water model, we introduce the Froude number

$$ F_r := \frac{|\bar{u}|}{\sqrt{gh + 3\hat{u}^2}} \quad (2) \quad \text{Fr} $$

such that where $F_r < 1$, the smallest $\lambda_1$ and largest $\lambda_3$ eigenvalues are of opposite sign, hence information are going in both direction and the flow is called subcritical, where $F_r = 1$, the smallest $\lambda_1$ or largest $\lambda_3$ eigenvalues vanishes and the flow is called subcritical, where $F_r > 1$, the smallest $\lambda_1$ and largest $\lambda_3$ eigenvalues have the same sign, hence information are going in one direction and the flow is called supercritical.

As shown in [1], the smooth solutions of shallow water system (SW₂) verify the energy conservation law

$$ \partial_t E + \partial_x \left( \left( g(h+Z) + \frac{\bar{u}^2 + 3\hat{u}^2}{2} \right) h\bar{u} \right) = 0 \quad (3) \quad \text{entropequality} $$

where the energy is $E(U) = gh(Z + \frac{h}{2}) + \frac{h}{2}(\bar{u}^2 + \hat{u}^2)$.

Solutions of (SW₂) may develop discontinuities in finite time. The discontinuous solutions are weak solutions and they are characterized by the Rankine Hugoniot relations

$$ \sigma[U] = [F(U)], \quad (4) \quad \text{RHfull} $$

where $\sigma$ is the velocity of the discontinuity and $[X] = X_+ - X_-$ represents the jump of the discontinuity separating the solution's left trace $X_-$ and the solution's right trace $X_+$ respectively defined by

$$ X_{-}(t, x) = \lim_{\substack{\epsilon \to 0 \\ \epsilon > 0}} X(t, x - \epsilon) \quad \text{and} \quad X_{+}(t, x) = \lim_{\substack{\epsilon \to 0 \\ \epsilon > 0}} X(t, x + \epsilon). $$
Weak solutions are not always admissible. Following [7], we consider that the discontinuities in the solution must satisfy, in addition to the Rankine–Hugoniot relation (4), the entropy (or energy) inequality

$$ -\sigma[E(U)] + \left[ \left( g(h+Z) + \frac{\bar{u}^2 + 3\hat{u}^2}{2} \right) h\bar{u} \right] \le 0, \quad (5) \text{ entropy2} $$

This criterion is sufficient to recover the uniqueness of the solution, at least for the Riemann problem, see [1].

## 1.2 Generalities on steady state solutions

A steady state solution of $(SW_2)$ is a triplet of piecewise $C^1$-regular functions $x \mapsto (h(x), \bar{u}(x), \hat{u}(x))$ that do not depend on time. We first highlight that the mass conservation impose the conservation of the discharge in the domain.

**Proposition 1.** Let $(h, \bar{u}, \hat{u})$ be a steady state of $(SW_2)$, there exists a constant $M \in \mathbb{R}$ such that for any $x$ we have $h(x)\bar{u}(x) = M$.

**Proposition 2.** *Proof.* The first equation of $(SW_2)$ at steady state, i.e. with $\partial_t h = 0$, shows that $h\bar{u}$ is constant where the solution is regular. Similarly, the first Rankine-Hugoniot relation of (4) at steady state, i.e. with $\sigma = 0$, shows that the same hold across the discontinuity. $\square$

**Remark 1.** The case of zero vertical-averaged horizontal velocity $\bar{u} = M = 0$ is a special case. The fluxes of the evolution equations on the water height and on the signed standard deviation (first and third equation of $(SW_2)$) cancel out in this case. Moreover, an eigenvalue is also identically zero in this case. The problem of determining a stationary solution for given limit conditions is then clearly ill-posed in the sense that an infinity of solutions can coincide.

For the rest of the paper, we fix $M > 0$ without loss of generality. The case $M < 0$ follows by a change of reference frame $x \mapsto -x$, which imply $\bar{u} = -\bar{u}$ and $\hat{u} = -\hat{u}$. Let us start by highlighting some quantities of interest.

**Proposition 3.** Let $(h, \bar{u}, \hat{u})$ with $h\bar{u} = M > 0$ be a steady state of $(SW_2)$. There exists a constant $S \in \mathbb{R}$ such that for any $x$ we have $\frac{\hat{u}(x)}{h(x)} = S$.

*Proof.* The third equation of $(SW_2)$ at steady state, i.e. with $\partial_t \hat{u} = 0$, shows that $\bar{u}\hat{u}$ is constant where the solution is regular. Similarly, the third Rankine-Hugoniot relation of (4) at steady state, i.e. with $\sigma = 0$, shows that the same holds across the discontinuity. Using the fact that $h\bar{u} = M \ne 0$ is constant (Proposition 1), we conclude that $\frac{\bar{u}\hat{u}}{M} = \frac{\hat{u}}{h} = S$ is constant in the whole domain. $\square$

From Proposition 1 and Proposition 2 we conclude that the steady state solutions are entirely determined by $M > 0$, $S \in \mathbb{R}$ the water height function $h$. The mean velocity $\bar{u}$ and the standard deviation $\hat{u}$ are recovered with

$$ \forall x, \quad \bar{u}(x) = \frac{M}{h(x)} \quad \text{and} \quad \hat{u}(x) = Sh(x). $$
The water depth at steady state $x \mapsto h(x)$ is in fact defined from two functions accordingly to the following property.

### Proposition 3. 
Let $M > 0$, $S \in \mathbb{R}$ and $x \mapsto h(x)$ be a steady solution of (SW$_2$). The function $x \mapsto K(x) := K(h(x)) + Z(x)$ with

$$ \begin{array}{rcl} \mathcal{K} : & \mathbb{R}_+^* & \to \mathbb{R} \\ & h & \mapsto h + \frac{1}{2g} \left( \frac{M^2}{h^2} + 3h^2S^2 \right). \end{array} $$

is piecewise constant and decreasing. More precisely

* on intervals where the water depth $h$ is continuous, the function $K$ is constant;

* at any point of discontinuity $x$ of $h$, the function $K$ is discontinuous and decreasing and the momentum flux $x \mapsto F^{h\bar{u}}(h(x))$ is continuous, with

$$ F^{h\bar{u}} : \begin{array}{rcl} \mathbb{R}_+^* & \to & \mathbb{R}_+^* \\ h & \mapsto & \frac{M^2}{h} + h^3 S^2 + \frac{g}{2} h^2. \end{array} $$

This function $K$ is usually called the hydraulic head.

*Proof.* For regular steady states, the energy equality (3) at steady state, i.e. with $\partial_t E = 0$, implies that the flux of energy is constant. Dividing by the constant $M/g$, we conclude that the quantity $\mathcal{K} \circ h + Z$ is also constant. At a point of discontinuity the energy satisfies Inequality (5) with $\sigma = 0$ and the hydraulic head decreases. On the other hand the second Rankine-Hugoniot relation of (4) with $\sigma = 0$ implies that the momentum flux, rewritten as $F^{h\bar{u}}$ using the constant $M$ and $S$, is continuous through the discontinuity. $\square$

In order to determine the water level $x \mapsto h(x)$, the analysis of the functions $h \mapsto \mathcal{K}(h)$ and $h \mapsto F^{h\bar{u}}(h)$ are essential. The critical water depth $H_c$ corresponding to the Froude number $F_r = 1$ plays an important role.

### Proposition 4. 
Let $M > 0$ and $S \in \mathbb{R}$. Let us rewrite the Froude number (2) as

$$ F_r(h) = \frac{M}{h\sqrt{gh + 3h^2S^2}}. $$

There exists a unique water depth $H_c > 0$ such that $F_r(H_c) = 1$. The flow is supercritical if $h < H_c$, critical if $h = H_c$ and subcritical if $h > H_c$.

*Proof.* The function $h \mapsto F_r(h)$ is a strictly decreasing and continuous function on $]0, +\infty[$, that goes from $+\infty$ to $0$. It yields the existence and uniqueness of $H_c$. For smaller water depth $h < H_c$ the Froude number is larger and flow is supercritical, while for larger water depth $h > H_c$ the Froude number is smaller and the flow is subcritical. $\square$

We are now in position to analyse the functions $F^{h\bar{u}}$ and $\mathcal{K}$.

### Proposition 5. 
Let $M > 0$ and $S \in \mathbb{R}$. Both functions $h \mapsto F^{h\bar{u}}$ and $h \mapsto \mathcal{K}$ are strictly convex, tends to $+\infty$ as $h$ tends to 0 or to $+\infty$, and reach their unique global minimum at $h = H_c$.
<figure><img src="image_2.png" /></figure>

〈fig:tools〉

<caption>Figure 1: Illustration of mathematical tools $\mathcal{H}^{sup}$, $\mathcal{H}^{sub}$ and $\psi$.</caption>

*Proof.* We easily obtain

$$ \mathcal{K}'(h) = 1 + \frac{1}{2g} \left( \frac{-2M^2}{h^3} + 6hS^2 \right) \quad \text{and} \quad \mathcal{K}''(h) = \frac{3}{g} \left( \frac{M^2}{h^4} + S^2 \right) > 0. $$

Thus $\mathcal{K}$ is strictly convex and its limits $\lim_{h \to 0^+} \mathcal{K}(h) = \lim_{h \to +\infty} \mathcal{K}(h) = +\infty$ are clear. The function reaches its unique minimum in $\mathbb{R}_+^*$ when $3S^2h^4 + gh^3 - M^2 = 0$. This is nothing but the Equation $F_r^2 = 1$. Accordingly to Proposition 4, the unique solution is $h = H_c$. The momentum flux $F^{h\bar{u}}$ enjoys the same properties as

$$ (F^{h\bar{u}})' = \frac{-M^2}{h^2} + 3h^2S^2 + gh = gh\mathcal{K}' \quad \text{and} \quad (F^{h\bar{u}})'' = \frac{2M^2}{h^3} + 6hS^2 + g > 0. \quad (6) \text{ [Fhubarder]} $$

<figure><img src="image_3.png" /></figure>

For further use, we note by $\mathcal{K}^c := \mathcal{K}(H_c)$ the minimum value of function $\mathcal{K}$. As a consequence, we can defined the following mathematical objects, see the left of Figure 1 for an illustration of the functions.

### Corollary 1.
Let $M > 0$ and $S \in \mathbb{R}$. For any $\kappa \ge \mathcal{K}^c$ there exist a unique water depth $\mathcal{H}^{sup}(\kappa) \le H_c$ and a unique water depth $\mathcal{H}^{sub}(\kappa) \ge H_c$ such that $\mathcal{K}(\mathcal{H}^{sup}(\kappa)) = \mathcal{K}(\mathcal{H}^{sub}(\kappa)) = \kappa$. The two functions $\kappa \mapsto \mathcal{H}^{sup}(\kappa)$ and $\kappa \mapsto \mathcal{H}^{sub}(\kappa)$ are continuous on $[\mathcal{K}^c, +\infty[$. The function $\mathcal{H}^{sup}$ is decreasing and the function $\mathcal{H}^{sub}$ is increasing. No solution exists if $\kappa < \mathcal{K}^c$.

*Proof.* The proof immediately follows from Proposition 5.

<figure><img src="image_4.png" /></figure>

This corollary allows to construct regular steady state solutions of $(SW_2)$.

### Proposition 6.
Let us $M > 0$, $S \in \mathbb{R}$ and a continuous topography $x \mapsto Z(x)$ on an interval $[x_L, x_R]$. For any $K \ge \mathcal{K}^c + \max_{[x_L, x_R]} Z$,
* the function $h_K^{sub} : [x_L, x_R] \to \mathbb{R}_+^*$, $x \mapsto \mathcal{H}^{sub}(K - Z(x))$ is the unique subcritical steady state solution of $(SW_2)$ such that for all $x \in [x_L, x_R]$, the hydraulic head $\mathcal{K}(h_K^{sub}(x)) + Z(x)$ is equal to $K$;

* the function $h_K^{sup} : [x_L, x_R] \to \mathbb{R}_+^*$, $x \mapsto \mathcal{H}^{sup}(K - Z(x))$ is the unique subcritical steady state solution of $(SW_2)$ such that for all $x \in [x_L, x_R]$, the hydraulic head $\mathcal{K}(h_K^{sup}(x)) + Z(x)$ is equal to $K$.

*Proof.* Note that we restrict our attention to solutions $x \mapsto h(x)$ that are subcritical or supercritical on the entire interval $[x_L, x_R]$, thus the upcoming Proposition 7 ensures that they cannot contain any discontinuity. Then we know from Proposition Proposition 3 that the hydraulic head $\mathcal{K} \circ h + Z$ remains constant to some constant $K$. Thus for any $x \in [x_L, x_R]$ the solution is found in the subcritical domain (resp. supercritical domain) by solving the equation $\mathcal{K}(h_K^{sub}(x)) = K - Z(x)$ on $[H_c, +\infty[$ (resp. $\mathcal{K}(h_K^{sup}(x)) = K - Z(x)$ on $]0, H_c]$). Corollary 1 ensures that the solution is unique and given by $\mathcal{H}^{sub}(K - Z(x))$ (resp. $\mathcal{H}^{sup}(K - Z(x))$). Eventually at fixed $x$ these solutions exists if and only if $K - Z(x) \ge \mathcal{K}^c$ and the functions $h_K^{sub}$ and $h_K^{sup}$ are defined on the whole interval if and only if $K \ge \mathcal{K}^c + \max_{[x_L, x_R]} Z$. $\square$

Let now focus on the discontinuities. We introduce the jump operator $h \mapsto \psi(h)$ that links the water height $h_-$ on the left of the jump to $h_+$ the water height on its right in a way that the momentum flux $F^{h\bar{u}}$ remains constant as stated in the first bullet of Proposition 3.

**Corollary 2** (of Proposition 5). Let $M > 0$ and $S \in \mathbb{R}$. For any $h \ne H_c$, there exists a unique $\psi(h) > 0$, such that $\psi(h) \ne h$ and $F^{h\bar{u}}(h) = F^{h\bar{u}}(\psi(h))$. Extending the function by $\psi(H_c) = H_c$, the function $h \mapsto \psi(h)$ is strictly decreasing, continuous on $\mathbb{R}_+^*$ and $C^1$ on $]0, H_c[ \cup ]H_c, +\infty[$. For all $h \in \mathbb{R}_+^* - \{H_c\}$, $\psi(h)$ and $h$ are in different regimes, i.e. $(h - H_c)(\psi(h) - H_c) < 0$

*Proof.* The proof is a trivial consequence of Proposition 5 and the construction of $\psi$ is illustrated on the right of Figure 1. $\square$

We now turn to a full characterization of the admissible shocks, taking into account the decay of the hydraulic head $K$ of Proposition 3.

**Proposition 7.** An admissible discontinuity of a steady state solution $h$ in the sense of Proposition 3 only occurs during a transition from a supercritical regime to a subcritical regime. More precisely, the water depth $h_-$ on the left of the discontinuity and the water depth $h_+$ on the right of the discontinuity verify $h_+ = \psi(h_-)$ and $h_- < H_c < h_+$. Such discontinuities are also said to be compressible.

*Proof.* This property is a particular case of [1, Proposition 4] at steady state. For readability we recall here the demonstration. At a point of discontinuity, the continuity of the momentum flux $F^{h\bar{u}}(h_+) = F^{h\bar{u}}(h_-)$ can be rewritten by definition as $h_+ = \psi(h_-)$. It also yields the following relation

$$M^2 = \left( S^2 (h_+^2 + h_+h_- + h_-^2) + \frac{g}{2} (h_+ + h_-) \right) h_-h_+ \quad (7) \quad \text{FluxContinuity}$$
On the other hand the hydraulic head decay leads to the following inequality

$$ \left( 1 + \frac{3S^2}{2g} (h_+ + h_-) - \frac{M^2}{2g} \frac{h_+ + h_-}{h_-^2 h_+^2} \right) [h] < 0. \qquad (8) \text{ChargeInequality} $$

Replacing $M^2$ in (8) by its expression (7) we get after some simplifications

$$ -\frac{g + 2S^2 (h_+ + h_-)}{4gh_-h_+} [h]^3 < 0 $$

which holds if and only if $h_+ - h_- = [h] > 0$, and we know that they are not in the same regime. □

**Remark 2.** Let $M > 0$, $S \in \mathbb{R}$. Since these quantities are constant through a steady dis-continuity, the fact that the water height increases through a discontinuity yields that the vertical-averaged of the horizontal velocity $\bar{u} = \frac{M}{h}$ decreases and the standard deviation of the horizontal velocity $|\hat{u}| = |S|h$ increases. This result will not directly be used in the fol- lowing, but implies that when shear is introduced in the model, it increases through steady discontinuities

We conclude this section by studying the continuous transition from one regime (sub- or supercritical) to the other one.

**Proposition 8.** Assume the topography $x \mapsto Z(x)$ is $C^1$-regular and let $M > 0$, $S \in \mathbb{R}$ and $x \mapsto h(x)$ is a steady state solution of (SW$_2$). Note that a direct consequence of Proposition 7 is that the transition from the subcritical regime to the supercritical regime is always contin-uous. A continuous transition from one regime (sub- or supercritical) to the other one can only occur at local maxima $\chi$ of the topography and satisfies $h(\chi) = H_c$.

*Proof.* Let us denote by $\chi$ a point where there is a continuous transition from one regime (sub- or supercritical) to the other one. First, since the solution is continuous, the Froude number is equal to one at position $\chi$ and the water height has to satisfy $h(\chi) = H_c$. But we also deduce from Proposition 3 that there exists $\epsilon > 0$ such that the hydraulic head $K \circ h + Z$ is constant on the interval $I = ]\chi - \epsilon, \chi + \epsilon[$. Thus for any $x \in I$, we have

$$ K(h(x)) + Z(x) = K(h(\chi)) + Z(\chi) = K^c + Z(\chi) \le K(h(x)) + Z(\chi). $$

Hence $Z(x) \le Z(\chi)$ which implies that $Z(\chi)$ is a local maximum of $Z$. □

We are now equipped with a family of regular solutions (Proposition 6) and a complete description of the discontinuous transition from the supercritical to the subcritical regime through a stationary shock (Proposition 7) and of the continuous transition from one regime to the other at local extrema of the topography (Proposition 8). We now turn to the construction of stationary solutions on a given topography that respect fixed boundary conditions by gluing together this building blocks.
## 2 Boundary conditions and moving steady states solutions

In this section, we address the existence and uniqueness of steady state solution of $(SW_2)$ on a bounded domain $[x_L, x_R]$ when some boundary conditions are imposed. Following [4, 6] the number of boundary conditions is given by the number eigenvalues impinging the domain. The fact that $M$ is strictly positive yields that two eigenvalues ($\lambda_2$ and $\lambda_3$, see (1)) are always positive on the whole domain. Thus we need at least two boundary condition at the inlet $x = x_L$, where we fix $M$ and $S$. The last eigenvalue $\lambda_1$ is negative when the regime at the inlet is subcritical in which case no other boundary condition is need. It is positive if the regime at the inlet is supercritical, in which case we additionally impose the water height $h(x_L) = H_L \le H_c$. Similarly if the regime at the outlet $x = x_R$ is subcritical, we impose the water height $h(x_R) = H_R \ge H_c$. Note every combination of regimes at inlet and the outlet are possible, and depending on the cases we have 2 or 3 boundary conditions at the inlet and 0 or 1 boundary condition at the outlet.

**Remark 3.** Let us give a more precise sense to "subcritical at the inlet" and in particular determine when a condition $h(x_L) = H_c$ is considered subcritical or supercritical. In the following we consider the case of topography $Z$ that are $C^1$ and composed of a finite succession of strictly monotone parts. We will prove that any solution contains at most one shock on every monotone part and we do not authorize solutions with a discontinuity located at the boundary. Thus a given solution is regular on a small interval $[x_L, x_L + \epsilon]$ and thus coincides for some $K \in \mathbb{R}$ with $h_K^{\text{sub}}$ or $h_K^{\text{sup}}$. The strict monotonicity of $Z$ yields that even though $h_K^{\text{sub}}(x_L) = H_c$, $h_K^{\text{sub}} > H_c$ in the vicinity of $x_L$. In other word if $h(x_L) = H_c$ is fixed, the limit $h(x)$ when $x$ tends to $x_L$ is either $H_c$ by excess and we say that the inlet is subcritical, or $H_c$ by default and we say that the inlet is supercritical.

**Remark 4.** The case of a topography only monotone with flat areas is easily deduced from the case with the same topography with the flat areas collapsed to zero size by placing it anywhere within the flat area. To overcome this somehow artificial multiplicity of solution we limit our study to a strictly monotone by piece topography.

The remaining of this section tackles different topography with increasing complexity. We start in §2.1 by the simplest case of strictly monotone topographies. The more complex cases are treated as a succession of topographies monotone by part. In section §2.2, we consider the case of a bump, i.e. a continuous topography that increases on a left subdomain and decreases on a right subdomain. This case is by far the most considered in the literature, in particular with subcritical boundary conditions, see for example [8] and the Swashes library [5]. In section §2.3 we consider the inverse case of a hollow. Eventually in section §2.4, we consider a continuous topography with several extrema and subcritical boundary conditions.

### 2.1 Steady states over a monotone topography

In the current section, we consider a strictly monotone $C^1$-regular topography $Z$ on a bounded domain $[x_L, x_R]$. We note $\hat{x} = \arg\max_{x \in [x_L, x_R]} Z(x)$. Here, either $\hat{x} = x_L$ (decreasing topography) or $\hat{x} = x_R$ (increasing topography). We consider the four possible cases (subcritical or supercritical regimes at the inlet and at the outlet) in four distinct propositions.
<figure><img src="image_5.png" /><figcaption>Figure 2: Illustration of the mathematical functions over monotone topographies and illustration of the transcritical solution (11).</figcaption></figure>

**Definition 1.** For all $x \in [x_L, x_R]$, we introduce the hydraulic head $K_x^c := K^c + Z(x) = K(H_c) + Z(x)$. If $h(x_L) = H_L$ or $h(x_R) = H_R$ are part of the boundary condition, we moreover introduce the corresponding hydraulic head $K_L := K(H_L) + Z(x_L)$ and $K_R := K(H_R) + Z(x_R)$.

**Proposition 9.** (Subcritical inlet / Supercritical outlet) Assume the topography Z is $C^1$-regular and strictly monotone. Let $M > 0$ and $S \in \mathbb{R}$ be set at the inlet with a subcritical inlet and a supercritical outlet. There is no steady state solution.

*Proof.* This property is a corollary of Proposition 8. This case requires a transition from the subcritical to the supercritical regime within the domain. This is not possible since it should occur at the maximum of the bathymetry. □

**Proposition 10.** (Subcritical inlet / Subcritical outlet) Assume the topography Z is $C^1$-regular and strictly monotone. Let $M > 0$ and $S \in \mathbb{R}$ be set at the inlet with a subcritical inlet and a subcritical outlet with $h(x_R) = H_R \ge H_c$. On increasing topography, there is always a steady state solution. On decreasing topography, there is a steady state solution if and only if $K_{xL}^c \le K_R$. When it exists, the solution is unique and is given by

$$h(x) = h_{K_R}^{sub}(x). \qquad (9) \text{ monotone_sub}$$

*Proof.* We know from Proposition 9 that there is no possible transition from subcritical to supercritical regime on a monotone topography. We conclude that if a stationary solution exists it is continuous on the whole domain and the flow is always subcritical. Thus by Proposition 3 the hydraulic head is constant on the whole domain, equals to the hydraulic head at the outlet where is it imposed by the boundary condition $K_R = K(H_R) + Z(x_R)$. The only possible solution is therefore $h_{K_R}^{sub}$. As mentioned in Proposition 6, this function is defined for all $x$ such that $Z(x) + K^c \le K_R$. This obviously hold if $Z$ increases since for all $x \in [x_L, x_R]$, $Z(x) + K^c \le Z(x_R) + K(H_R) = K_R$. If the topography decreases the domain of definition of $h_{K_R}^{sub}$ is $[x_L, x_R]$ if and only if $Z(x_L) + K^c = K_{xL}^c \le K_R$. □
### Proposition 11. (Supercritical inlet / Supercritical outlet)
Assume the topography Z is C¹-regular and strictly monotone. Let $M > 0$ and $S \in \mathbb{R}$ be set at the inlet with a supercritical inlet with $h(x_L) = H_L \le H_c$ and a supercritical outlet. On decreasing topography, there is always a steady state solution. On increasing topography, there is a steady state solution if and only if $K_L \ge K_{x_R}^c$. When it exists, the solution is unique and is given by

$$h(x) = \mathbf{h}_{K_L}^{sup}(x). \qquad (10) \text{ [monotone_sup]}$$

*Proof.* We prove this property with the same arguments as for Proposition 10. If there was a point $x$ where the solution is subcritical $h(x) < H_c$ it would be impossible to construct the solution on $[x, x_R]$ according to Proposition 9. We conclude that if a stationary solution exists the flow is supercritical in the whole domain with a constant hydraulic head determined by the inlet boundary condition $K_L$. In other words the solution is necessarily $\mathbf{h}_{K_L}^{sup}$. $\square$

Let us now consider the last case connecting a supercritical inlet to a subcritical outlet. It is the more complicated one since the solution contains a stationary shock. The existence of a solution is always subject to some conditions illustrated on Figure 2.

### Proposition 12. (Supercritical inlet / Subcritical outlet)
Assume the topography Z is C¹-regular and strictly monotone. Let $M > 0$ and $S \in \mathbb{R}$ be set at the inlet with a supercritical inlet with $h(x_L) = H_L \le H_c$ and a subcritical outlet with $h(x_R) = H_R \ge H_c$. There is no such solution if $K_L \le K_R$. If $K_L > K_R$:

#### (casea)
(a) for increasing topography, there exists a steady state solution if and only if
$$H_L > \psi(\mathbf{h}_{K_R}^{sub}(x_L)) \text{ and } H_R < \psi(\mathbf{h}_{K_L}^{sup}(x_R));$$

#### (caseb)
(b) for decreasing topography, in the case $K_R > K_{x_L}^c$, there exists a steady state solution if and only if
$$H_L < \psi(\mathbf{h}_{K_R}^{sub}(x_L)) \text{ and } H_R > \psi(\mathbf{h}_{K_L}^{sup}(x_R));$$

#### (casec)
(c) for decreasing topography, in the case $K_R \le K_{x_L}^c$, there exists a steady state solution if and only if
$$H_R > \psi(\mathbf{h}_{K_L}^{sup}(x_R)).$$

When it exists, the solution is unique and contains a single stationary shock. We denote by $\chi_{x_L, x_R}^{K_L, K_R}$ the position of the discontinuity. The solution is

$$h(x) = \begin{cases} \mathbf{h}_{K_L}^{sup}(x), & \text{if } x < \chi_{x_L, x_R}^{K_L, K_R} \\ \mathbf{h}_{K_R}^{sub}(x), & \text{if } x > \chi_{x_L, x_R}^{K_L, K_R} \end{cases} \qquad (11) \text{ [monotone_choc]}$$

*Proof.* The condition $K_L \ge K_R$ was proven in Proposition 3. The case $K_L = K_R$ would imply that the transition from supercritical to subcritical regime is continuous, which is not possible since it should occur at a local maximum of the topography, as explained in Proposition 9. So one has to consider the case $K_L > K_R$ and the solution has to contain a stationary discontinuity, see Proposition 7. After the shock the regime is subcritical, thus the
solution cannot contain a second discontinuity nor a continuous transition to the supercritical regime in the absence of local maxima of topography. Thus the solution necessarily has the form (11) since the boundary conditions $h(x_L) = H_L$ and $h(x_R) = H_R$ impose the hydraulic heads $K_L$ at the inlet and $K_R$ at the outlet.

The other conditions of the statement ensure that the functions $h_{K_L}^{sup}$ and $h_{K_R}^{sub}$ can be connected by a stationary discontinuity in the interval $]x_L, x_R[$. From Proposition 3, the discontinuity takes place at a point $\chi$ that is a solution of the equation $f(\chi) = 0$ with

$$f(x) = F^{h\bar{u}}(h_{K_R}^{sub}(x)) - F^{h\bar{u}}(h_{K_L}^{sup}(x)).$$

By derivation and using (6), we have

$$ \begin{aligned} f'(x) &= (F^{h\bar{u}})'(h_{K_R}^{sub}(x)) \times (h_{K_R}^{sub})'(x) - (F^{h\bar{u}})'(h_{K_L}^{sup}(x)) \times (h_{K_L}^{sup})'(x) \\ &= g h_{K_R}^{sub}(x) \mathcal{K}'(h_{K_R}^{sub}(x)) \times (h_{K_R}^{sub})'(x) - g h_{K_L}^{sup}(x) \mathcal{K}'(h_{K_L}^{sup}(x)) \times (h_{K_L}^{sup})'(x) \\ &= g h_{K_R}^{sub}(x) \partial_x (\mathcal{K}(h_{K_R}^{sub}(x))) - g h_{K_L}^{sup}(x) \partial_x (\mathcal{K}(h_{K_L}^{sup}(x))). \end{aligned} $$

Since $h_{K_R}^{sub}(x)$ and $h_{K_L}^{sup}(x)$ are both solutions of $\partial_x(\mathcal{K}(h) + Z(x)) = 0$, we get

$$f'(x) = -gZ'(x)(h_{K_R}^{sub}(x) - h_{K_L}^{sup}(x)).$$

On the part of the domain where $h_{K_L}^{sup}$ and $h_{K_R}^{sub}$ are both well defined, they satisfy $h_{K_R}^{sub}(x) > h_{K_L}^{sup}(x)$ (a subcritical water height is always greater than a supercritical one). Hence the function $f$ is strictly monotone since the topography is strictly monotone. More precisely the function $f$ is decreasing (resp. increasing) when the topography is increasing (resp. decreasing). It follows that, if it exists, the zero of this function is unique and then corresponds to the unique possible position of the discontinuity. To determine if the function $f$ actually has a zero in the interval $]x_L, x_R[$, we must determine its domain of definition and study whether its sign is different at the two bounds of this interval.

Let us consider the case of an increasing topography. The hypothesis $H_R \ge H_c$ implies that $K_R \ge K_{x_R}^c$ and since the maximum of the topography is obtained at the outlet, the function $h_{K_R}^{sub}$ is defined in the whole domain. The hypothesis $K_L > K_R$ ensures that $K_L > K_{x_R}^c$ and then that $h_{K_L}^{sup}$ is defined in the whole domain. Hence the function $f$ is defined in the whole domain. It is strictly decreasing since the topography is strictly increasing. Then it only remains to find the conditions under which $f(x_L) > 0 > f(x_R)$. Using the notation $\psi$ introduced in Corollary 2, we have

$$f(x_L) = F^{h\bar{u}}(h_{K_R}^{sub}(x_L)) - F^{h\bar{u}}(H_L) = F^{h\bar{u}}(\psi(h_{K_R}^{sub}(x_L))) - F^{h\bar{u}}(H_L).$$

Since $F^{h\bar{u}}(h)$ is decreasing for $h \le H_c$, we conclude that $f(x_L) > 0$ if and only if $H_L > \psi(h_{K_R}^{sub}(x_L))$. We prove similarly that $f(x_R) < 0$ if and only if $H_R < \psi(h_{K_L}^{sup}(x_R))$ and we conclude the proof in this case.
We now consider the case of a decreasing topography. The hypothesis $H_L \ge H_c$ implies that $K_L \ge K_{x_L}^c$ and since the maximum of the topography is obtained at the inlet, the function $h_{K_L}^{sup}$ is defined in the whole domain. However the assumption on the hydraulic head $K_L > K_R$ no longer ensures that the function $h_{K_R}^{sub}$ is defined on the entire domain. Let us first assume that $K_R > K_{x_L}^c$. In that case the functions $h_{K_R}^{sub}$ and $f$ are defined over the entire domain. The computations are similar to those carried out in the previous case except that the inequalities are reversed on the now decreasing topography. We obtain the two conditions $H_L < \psi(h_{K_R}^{sub}(x_L))$ and $H_R > \psi(h_{K_L}^{sup}(x_R))$. Let us now assume that $K_R < K_{x_L}^c$. In that case the functions $h_{K_R}^{sub}$ and $f$ are only defined on the interval $[x^*, x_R]$ where $x^* > x_L$ is such that $Z(x^*) + K^c = K_R$, see Proposition 6. One has to find the conditions under which $f(x^*) \le 0 < f(x_R)$. Since $h_{K_R}^{sub}(x^*) = H_c$ and $F^{h\bar{u}}$ reaches its minimum at $H_c$, we have

$$f(x^*) = F^{h\bar{u}}(h_{K_R}^{sub}(x^*)) - F^{h\bar{u}}(h_{K_L}^{sup}(x^*)) = F^{h\bar{u}}(H_c) - F^{h\bar{u}}(h_{K_L}^{sup}(x^*)) \le 0$$

Hence, one only has to check the second inequality which is true under the same condition as in the previous case $H_R > \psi(h_{K_L}^{sup}(x_R))$. In the last case $K_R = K_{x_L}^c$, the functions $h_{K_R}^{sub}$ and $f$ are defined on the whole domain. Moreover $K_L > K_{x_L}^c$ and then $H_L < H_c$. It follows that

$$f(x_L) = F^{h\bar{u}}(h_{K_R}^{sub}(x_L)) - F^{h\bar{u}}(H_L) = F^{h\bar{u}}(h_{K_{x_L}^c}^{sub}(x_L)) - F^{h\bar{u}}(H_L) = F^{h\bar{u}}(H_c) - F^{h\bar{u}}(H_L) < 0$$

and one also only has to ensure that $H_R > \psi(h_{K_L}^{sup}(x_R))$. $\square$

**<rmk:baines>** **Remark 5.** As far as we know, steady solutions (11) with a jump on increasing topography are not well known in the literature, even in the case of classical shallow water $S = 0$. A few points can be made to argue that this jump is not stable to small perturbations [3], as well as numerical evidences, see Figure (3). However, a full proof of instability is left as an open question for this work.

Before tackling the construction of stationary solutions on piecewise monotone topographies, we give on Figure 4 a graphical representation in the $(h(x_R), h(x_L))$-plane of Proposition 10, Proposition 11 and Proposition 12 by indicating which pairs of water heights can be linked by a stationary solution. To do this, we must first write in terms of water height the conditions which, in the previous statements, were written on the hydraulic heads $K_L$ and $K_R$. This writing is less natural, which explains our choice in the statements, but it is essential for this graphical representation. We therefore give here all the correspondences, detailing only the first one.

* Proposition 10. Let us prove that the condition on the hydraulic heads $K_{x_L}^c \le K_R$ is equivalent to the condition on the water heights $h_{K_{x_L}^c}^{sub}(x_R) \le H_R$. We first recall that in the case considered here, the function $h_{K_{x_L}^c}^{sub}$ is defined over the entire interval. The hydraulic head being constant through regular solution, we then deduce

$$K_{x_L}^c = \mathcal{K}(h_{K_{x_L}^c}^{sub}(x_L)) + Z(x_L) = \mathcal{K}(h_{K_{x_L}^c}^{sub}(x_R)) + Z(x_R)$$

The definition of $K_R = \mathcal{K}(H_R) + Z(x_R)$ and the increase of the function $\mathcal{K}$ on the subcritical domain $[H_c, +\infty[$ yields the result.
<figure><img src="image_6.png" /><figcaption>Figure 3: Small perturbation of Solution (11): the shock moves further away on increasing topography, while it stabilizes close by on decreasing topography. Numerical simulation obtained with a HLL scheme with a centered treatment of the source term.

$\langle F:stability \rangle$</figcaption></figure>

* Proposition 11. A similar argument yields that the condition on the hydraulic heads $K_L \ge K_{x_R}^c$ is equivalent to the condition on the water heights $H_L \le h_{K_L}^{sup}(x_L)$.

* Proposition 12. The condition on the hydraulic heads $K_L > K_R$ is necessary to construct the solution but is eventually always less restrictive than another condition that is mentioned in the proposition, hence it does not appear in the graphical representation. For increasing topography, the condition $H_R < \psi(h_{K_L}^{sup}(x_R))$ implies $K_L > K_R$: we use two properties of the function $\mathcal{K}$ that is increasing on the subcritical domain and decreasing through a discontinuity from supercritical to subcritical regime to establish

$$ K_R = \mathcal{K}(H_R) + Z(x_R) < \mathcal{K} \left( \psi(h_{K_L}^{sup}(x_R)) \right) + Z(x_R) < \mathcal{K} \left( h_{K_L}^{sup}(x_R) \right) + Z(x_R) = \mathcal{K} \left( h_{K_L}^{sup}(x_L) \right) + Z(x_L) = K_L $$

On decreasing topography either the condition $H_L < \psi(h_{K_R}^{sub}(x_L))$ (for similar reason) or $K_R \le K_{x_L}^c$ (because $K_{x_L}^c \le K_L$ is obvious and both equalities are mutually exclusive due to the presence of the discontinuity) also implies $K_L > K_R$. The translation of the condition $K_R \le K_{x_L}^c$ (or the opposite) on the water heights has already been explained for a previous proposition and remains the same.

## 2.2 Steady states over a bump

$\langle \text{sec:bump} \rangle$ Let us now consider the classical case of a $C^1$-regular topography that increases on $[x_L, \hat{x}]$ and decreases on $[\hat{x}, x_R]$. We denote by $K_{\hat{x}}^c = \mathcal{K}(H_c) + Z(\hat{x})$ the critical hydraulic head at the top of the topography. As in Section 2.1, we consider the four possible cases (subcritical or supercritical regimes at the inlet and at the outlet) in four distinct propositions. In each case, the solutions on the whole domain will be obtained by connecting the solutions on increasing and decreasing parts described in Section 2.1.
<figure><img src="image_7.png" /><figcaption>Figure 4: Pairs of boundary values $(h(x_R), h(x_L))$ for which it exists a stationary solution on increasing topography (top) or decreasing topography (bottom). Black curves are solutions without jump, red curves are solutions with one jump on increasing topography, blue curves are solutions with one jump on decreasing topography.</figcaption></figure>
### Proposition 13. (Subcritical inlet / Supercritical outlet)
Assume the topography Z is $C^1$-regular, increasing on $[x_L, \hat{x}]$ and decreasing on $[\hat{x}, x_R]$. Let $M > 0$ and $S \in \mathbb{R}$ be set at the inlet with a subcritical inlet and a supercritical outlet.
There exists a unique steady state solution given by

$$h(x) = \begin{cases} \mathbf{h}_{K_{\hat{x}}^c}^{\text{sub}}(x), & \text{if } x \le \hat{x} \\ \mathbf{h}_{K_{\hat{x}}^c}^{\text{sup}}(x), & \text{if } x \ge \hat{x} \end{cases} \qquad (12) \text{bumpsol:critical}$$

*Proof.* On the increasing slope $[x_L, \hat{x}]$ with a subcritical inlet, Propositions 9 and Proposition 10 allow us to conclude that there is always a unique solution given by (9) : $\mathbf{h}_{K(\hat{x})}^{\text{sub}}$.

On the decreasing slope $[\hat{x}, x_R]$ with a supercritical outlet, Propositions 9 and Proposition 11 allow us to conclude that there is always a unique solution given by (10) : $\mathbf{h}_{K(\hat{x})}^{\text{sup}}$.

Moreover Proposition 8 tells us that the transition from a subcritical to a supercritical regime is continuous and can only take place at a local maximum of the topography.

From these three ingredients, we conclude that there exists a unique solution on the whole interval. This solution is continuous, subcritical on the left of the bump, supercritical on the right of the bump and is such that $h(\hat{x}) = H_c$. Then the hydraulic head is constant on the whole interval and equal to $K_{\hat{x}}^c$ and the solution is given by (12). □

### Proposition 14. (Subcritical inlet / Subcritical outlet)
Assume the topography Z is $C^1$-regular, increasing on $[x_L, \hat{x}]$ and decreasing on $[\hat{x}, x_R]$. Let $M > 0$ and $S \in \mathbb{R}$ be set at the inlet with a subcritical inlet and a subcritical outlet with $h(x_R) = H_R \ge H_c$.

There exists a steady state solution if and only if $\psi(\mathbf{h}_{K_{\hat{x}}^c}^{\text{sup}}(x_R)) \le H_R$. If it exists the solution is unique and moreover:

#### *subsubcase b)* If $K_R \ge K_{\hat{x}}^c$, the solution is $\mathbf{h}_{K_R}^{\text{sub}}$.

#### *subsubcase a)* If $\psi(\mathbf{h}_{K_{\hat{x}}^c}^{\text{sup}}(x_R)) < H_R$ and $K_R < K_{\hat{x}}^c$, the solution is given by

$$h(x) = \begin{cases} \mathbf{h}_{K_{\hat{x}}^c}^{\text{sub}}(x) & \text{if } x \le \hat{x}, \\ \mathbf{h}_{K_{\hat{x}}^c}^{\text{sup}}(x) & \text{if } \hat{x} \le x < \chi_{[\hat{x},x_R]}^{K_{\hat{x}}^c,K_R}, \\ \mathbf{h}_{K_R}^{\text{sub}}(x) & \text{if } x > \chi_{[\hat{x},x_R]}^{K_{\hat{x}}^c,K_R} \end{cases}, \qquad (13) \text{bumpsol_trans}$$

**Remark 6.** The condition $\psi(\mathbf{h}_{K_{\hat{x}}^c}^{\text{sup}}(x_R)) \le H_R$ is equivalent to the fact that the hydraulic head $K_R$ is larger than the one on the right of a stationary shock with left state $\mathbf{h}_{K_{\hat{x}}^c}^{\text{sup}}$, which is smaller than $K_{\hat{x}}^c$. Thus this condition is automatically fulfilled in case i).

*Proof.* On the increasing slope $[x_L, \hat{x}]$ with a subcritical inlet, Propositions 9 and 10 allow us to conclude that there is always a unique solution that is subcritical on $[x_L, \hat{x}]$ and given by (9) : $\mathbf{h}_{K(\hat{x})}^{\text{sub}}$.

On the decreasing slope $[\hat{x}, x_R]$ with a subcritical outlet, several solutions can take place, as stated in Propositions 10 and 12, depending on whether the regime on the right neighborhood of the top of the bump is sub- or supercritical.
Let us first consider the case of a subcritical right neighborhood of the top of the bump, as described in Proposition 10. It is stated in this proposition that this situation can happen if and only if $K_{\hat{x}}^c \le K_R$. The solution is then given by (9) : $h_{K_R}^{sub}$. The solution on the whole domain $[x_L, x_R]$ is then obtained by connecting the two subcritical solutions $h_{K(\hat{x})}^{sub}$ on $[x_L, \hat{x}]$ and $h_{K_R}^{sub}$ on $[\hat{x}, x_R]$. This connection has to be continuous since a discontinuity can only connect a supercritical and a subcritical part, see Proposition 7. Then the hydraulic head is constant on the whole domain and equal to the hydraulic head $K_R$ imposed at the right boundary. The solution is described in case i).

From Proposition 12, there is another possibility which consists on a solution that is subcritical at the right boundary $x_R$ but supercritical on the right neighborhood of the top of the bump $\hat{x}$. In that case, since the flow is subcritical on the left of the bump and as stated in Proposition 13, the transition from the subcritical part and the supercritical part has to take place at the top of the bump $\hat{x}$ and is continuous. It follows that, as in Proposition 13, the top of the bump is a sonic point with $h(\hat{x}) = H_c$ with a corresponding hydraulic head $K_{\hat{x}}^c$. A first consequence is that the subcritical solution on the left of the bump is given by $h_{K_{\hat{x}}^c}^{sub}$. On the right of the bump, as stated in Proposition 12, the solution has to contain a discontinuity in the interval $[\hat{x}, x_R]$ to connect the supercritical (on the right of $\hat{x}$) and the subcritical (on the left of $x_R$) parts. Since the hydraulic head is equal to $K_{\hat{x}}^c$ on the left of this discontinuity and to $K_R$ on its right, the flow has to satisfy $K_R < K_{\hat{x}}^c$ and we are in case Proposition 12-(c) (the slope is decreasing on the right of the bump). The hypothesis of case ii) and the form of the solution (13) immediately follow. $\square$

**$\langle \text{bump\_supsup} \rangle$** **Proposition 15. (Supercritical inlet / Supercritical outlet)** Assume the topography Z is $C^1$-regular, increasing on $[x_L, \hat{x}]$ and decreasing on $[\hat{x}, x_R]$. Let $M > 0$ and $S \in \mathbb{R}$ be set at the inlet with a supercritical inlet with $h(x_L) = H_L \le H_c$ and a supercritical outlet. There exist steady state solutions if and only if $K_L \ge K_{\hat{x}}^c$. More precisely:

**$\langle \text{supsupcasea} \rangle$** i) $h_{K_L}^{sup}$ is always a solution;

**$\langle \text{supsupcaseb} \rangle$?** ii) if $\psi(h_{K_{\hat{x}}^c}^{sub}(x)) < H_L$ and $K_L > K_{\hat{x}}^c$, there is a second solution given by

$$h(x) = \begin{cases} h_{K_L}^{sup}(x) & \text{if } x < \chi_{[x_L, \hat{x}]}^{K_{\hat{x}}^c, K_R}, \\ h_{K_{\hat{x}}^c}^{sub}(x) & \text{if } \chi_{[x_L, \hat{x}]}^{K_{\hat{x}}^c, K_R} < x \le \hat{x}, \\ h_{K_{\hat{x}}^c}^{supp}(x) & \text{if } x \ge \hat{x}. \end{cases} \quad (14) \text{ ?bumpsol:transsupsup}$$

**Remark 7.** The proof will be quite similar to the previous one but note that in this case the conditions are not mutually exclusive, meaning that two solutions can exist for some data.

*Proof.* On the increasing slope $[x_L, \hat{x}]$ with a supercritical inlet, several solutions can take place, as stated in Propositions 11 and 12, depending on whether the regime on the left neighborhood of the top of the bump is sub- or supercritical.

On the decreasing slope $[\hat{x}, x_R]$ with a supercritical outlet, Propositions 9 and Proposition 11 allow us to conclude that there is always a unique solution given by (10) : $h_{K(\hat{x})}^{sup}$.

Let us first consider the case of a supercritical left neighborhood of the top of the bump, as
described in Proposition 11. It is stated in this proposition that this situation can happen if and only if $K_L \ge K_{\hat{x}}^c$. The solution is then given by (10) : $h_{K_L}^{sup}$. The solution on the whole domain $[x_L, x_R]$ is then obtained by connecting the two supercritical solutions $h_{K_L}^{sup}$ on $[x_L, \hat{x}]$ and $h_{K(\hat{x})}^{sup}$ on $[\hat{x}, x_R]$. This connection has to be continuous since a discontinuity can only connect a supercritical and a subcritical part, see Proposition 7. Then the hydraulic head is constant on the whole domain and equal to the hydraulic head $K_L$ imposed at the left boundary. The solution is described in case i).

From Proposition 12, there is another possibility which consists on a solution that is supercritical at the left boundary $x_L$ but subcritical on the left neighborhood of the top of the bump $\hat{x}$. In that case, since the flow is supercritical on the right of the bump and as stated in Proposition 13, the transition from the subcritical part and the supercritical part has to take place at the top of the bump $\hat{x}$ and is continuous. It follows that the top of the bump is a sonic point with $h(\hat{x}) = H_c$ with a corresponding hydraulic head $K_{\hat{x}}^c$. A first consequence is that the supercritical solution on the right of the bump is given by $h_{K_{\hat{x}}^c}^{sup}$. On the left of the bump, as stated in Proposition 12, the solution has to contain a discontinuity in the interval $]x_L, \hat{x}[$ to connect the supercritical (on the right of $x_L$) and the subcritical (on the left of $\hat{x}$) parts. Since the slope is increasing on the left of the bump, we are in case Proposition 12-(a) and the conditions of existence are $K_L > K_{\hat{x}}^c$, $H_L > \psi(h_{K_{\hat{x}}^c}^{sub}(x_L))$ and $H_c < \psi(h_{K_L}^{sup}(\hat{x}))$. The last condition is automatically fullfilled, which concludes the proof. $\square$

#### ?<bump_supsub>?
**Proposition 16.** (Supercritical inlet / Subcritical outlet) Assume the topography Z is $C^1$-regular, increasing on $[x_L, \hat{x}]$ and decreasing on $[\hat{x}, x_R]$. Let $M > 0$ and $S \in \mathbb{R}$ be set at the inlet with a supercritical inlet with $h(x_L) = H_L \le H_c$ and a subcritical outlet with $h(x_R) = H_R \ge H_c$.

One or two solutions exist if and only if $K_L \ge K_R$, $K_L \ge K_{\hat{x}}^c$ and one or two of the following sets of conditions holds:

i) if $K_L = K_R = K_{\hat{x}}^c$, one solution is

$$h(x) = \begin{cases} \mathbf{h}_{K_{\hat{x}}^c}^{sup}(x) & \text{if } x \le \hat{x}, \\ \mathbf{h}_{K_{\hat{x}}^c}^{sub}(x) & \text{if } x > \hat{x}; \end{cases} \qquad (15) \text{ bump_noshock}$$

ii) if $K_L > K_R \ge K_{\hat{x}}^c$, $h_{K_R}^{sub}(\hat{x}) < \psi(h_{K_L}^{sup}(\hat{x}))$ and $\psi(h_{K_R}^{sub}(x_L)) < H_L$, one solution is

$$h(x) = \begin{cases} \mathbf{h}_{K_L}^{sup}(x) & \text{if } x < \chi_{[x_L, \hat{x}]}^{K_L, K_R}, \\ \mathbf{h}_{K_R}^{sub}(x) & \text{if } x > \chi_{[x_L, \hat{x}]}^{K_L, K_R}, \end{cases} \qquad (16) \text{ bump_ishock1}$$

#### ?<supsubcasec>?

iii) if $K_L > K_R$, $H_R > \psi(h_{K_L}^{sup}(x_R))$ and either $(K_R > K_{\hat{x}}^c$ and $h_{K_L}^{sup}(\hat{x}) \le \psi(h_{K_R}^{sub}(\hat{x}))$ or $K_R \le K_{\hat{x}}^c$, one solution is

$$h(x) = \begin{cases} \mathbf{h}_{K_L}^{sup}(x) & \text{if } x < \chi_{[\hat{x}, x_R]}^{K_L, K_R}, \\ \mathbf{h}_{K_R}^{sub}(x) & \text{if } x > \chi_{[\hat{x}, x_R]}^{K_L, K_R}, \end{cases} \qquad (17) \text{ bump_ishockr}$$
iv) if $K_R < K_{\hat{x}}^c < K_L$, $H_R > \psi(\mathbf{h}_{K_{\hat{x}}^c}^{sup}(x_R))$ and $H_L > \psi(\mathbf{h}_{K_{\hat{x}}^c}^{sub}(x_L))$ one solution is given by

$$h(x) = \begin{cases} \mathbf{h}_{K_L}^{sup}(x) & \text{if } x < \chi_{[x_L, \hat{x}]}^{K_L, K_{\hat{x}}^c}, \\ \mathbf{h}_{K_{\hat{x}}^c}^{sub}(x) & \text{if } \chi_{[x_L, \hat{x}]}^{K_L, K_{\hat{x}}^c} < x \le \hat{x}, \\ \mathbf{h}_{K_{\hat{x}}^c}^{sup}(x) & \text{if } \hat{x} \le x < \chi_{[\hat{x}, x_R]}^{K_{\hat{x}}^c, K_R}, \\ \mathbf{h}_{K_R}^{sub}(x) & \text{if } x > \chi_{[\hat{x}, x_R]}^{K_{\hat{x}}^c, K_R}. \end{cases} \quad (18) \text{ bump_2shocks}$$

The coexistence of solutions (16) and (18) on one side and of solutions (17) and (16) on the other side is illustrated on Figure 5 and Figure 6 respectively. See also the bottom right quadrant of Figure 7 for a representation in the $(H_L, H_R)$-plan. Solutions (16) and (18) are mutually exclusive.

<figure><img src="image_8.png" /><figcaption>Figure 5: Coexistence of Solution (17) in blue and Solution (18) in green in the case $K_L > K_{\hat{x}}^c > K_R$. The conditions of existence are represented by stationary shock of the corresponding color at the boundaries.
$\langle F: \text{coex2R} \rangle$</figcaption></figure>

**Remark 8.** The coexistence of the solutions with only one shock is easily explained. Imagine a solution of the form (16) with a shock on the left exists. Then as soon as there exists a point on the right with the same topography, it is also possible to place the shock at that point, yielding a solution of the form (17). The reverse holds true as long as $\mathbf{h}_{K_L}^{sub}$ exists on the whole domain, which is ensured by the hypothesis $K_R \ge K_{\hat{x}}^c$.

**Remark 9.** Note that when the left topography or the right topography tend to the maximum topography, we recover the results of §2.1. In particular, when $Z(x_L) \to Z(\hat{x})$, the solutions (16) and (18) are no longer possible, and the conditions of solution (17) coincides with those of solution (11). Similarly, when $Z(x_R) \to Z(\hat{x})$, the solutions (17) and (18) are no longer possible, and the conditions of solution (16) coincides with those of solution (11).
<figure><img src="image_9.png" /><figcaption>Figure 6: Coexistence of Solution (17) in blue and Solution (16) in red in the case $K_L > K_R > K_{\hat{x}}^c$. The conditions of existence are represented by stationary shock of the corresponding color at the boundaries.</figcaption></figure>

⟨F:coexLR⟩
*Proof.* On the increasing slope $[x_L, \hat{x}]$ with a supercritical inlet, several solutions can take place, as stated in Propositions 11 and 12, depending on whether the regime on the left neighborhood of the top of the bump is sub- or supercritical.

On the decreasing slope $[\hat{x}, x_R]$ with a subcritical outlet, several solutions can take place, as stated in Propositions 10 and 12, depending on whether the regime on the right neighborhood of the top of the bump is sub- or supercritical.

Hence we have four cases to investigate.

Let us start with the case we connect a solution given by Proposition 12 on the increasing slope on the left of the bump and by Proposition 10 on the decreasing slope on the right of the bump. Both solutions are subcritical in the vicinity of the top of the bump $\hat{x}$. Then the solution has to be continuous at that point. If it exists, the subcritical solution on the right of the bump, given by Proposition 10, is $h_{K_R}^{sub}$. Hence the water depth at the top of the bump has to be equal to $h_{K_R}^{sub}(\hat{x})$ and the form of the solution on the left of the bump is given in Proposition 12 by (11). We now are in position to write the form of the solution on the whole domain $[x_L, x_R]$, it is given by (16). The conditions for the existence of such a solution are $K_R \ge K_\hat{x}^c$ (from Proposition 10) and $K_L > K_R$, $H_L > \psi(h_{K_R}^{sub}(x_L))$ and $h_{K_R}^{sub}(\hat{x}) < \psi(h_{K_L}^{sup}(\hat{x}))$ (from Proposition 12-(a), increasing slope).

Let us now consider the case we connect a solution given by Proposition 11 on the increasing slope on the left of the bump and by Proposition 12 on the decreasing slope on the right of the bump. Both solutions are supercritical in the vicinity of the top of the bump $\hat{x}$. Then the solution has to be continuous at that point. If it exists, the supercritical solution on the left of the bump, given by Proposition 11, is $h_{K_L}^{sup}$. Hence the water depth at the top of the bump has to be equal to $h_{K_L}^{sup}(\hat{x})$ and the form of the solution on the right of the bump is given by (11), see Proposition 12. We now are in position to write the form of the solution on the whole domain $[x_L, x_R]$, it is given by (17). The conditions for the existence of such a solution are $K_L \ge K_\hat{x}^c$ (from Proposition 11) and those corresponding to cases Proposition 12-(b) or Proposition 12-(c), depending on the relative order of the quantities $K_R$ and $K_\hat{x}^c$, i.e. $K_L > K_R$, $H_R > \psi(h_{K_L}^{sup}(x_R))$ and either $K_R \le K_\hat{x}^c$ or $K_R > K_\hat{x}^c$ and $h_{K_L}^{sup}(\hat{x}) < \psi(h_{K_R}^{sub}(\hat{x}))$.

Let us now consider the case we connect a solution given by Proposition 11 on the increasing slope on the left of the bump and by Proposition 10 on the decreasing slope on the right of the bump. In that case, the solution is $h_{K_L}^{sup}$ on the left of the bump and $h_{K_R}^{sub}$ on its right. The conditions for the existence of these solutions are $K_L \ge K_\hat{x}^c$ (from Proposition 11) and $K_R \ge K_\hat{x}^c$ (from Proposition 10). We must now study under what conditions we can connect these two solutions. Since the regime is supercritical on the left of the bump and subcritical on the right of the bump, the transition can be continuous or discontinuous, see Propositions 7 and 8. If it continuous, the top of the bump is a sonic point with $h(\hat{x}) = H_c$ with a corresponding hydraulic head $K_\hat{x}^c$. In that case, the whole solution is continuous, the hydraulic head is constant, equal to $K_\hat{x}^c$. The condition of existence of such a solution is then $K_L = K_R = K_\hat{x}^c$ and the solution is given by (15). If the transition is discontinuous, the continuity of the flux implies $h_{K_R}^{sub}(\hat{x}) = \psi(h_{K_L}^{sup}(\hat{x}))$, see Proposition 7. This relation immediately implies that $H_R > \psi(h_{K_L}^{sup}(x_R))$: the function $f$ introduced in Proposition 12
that characterizes the position of the bump is strictly monotone and vanishes at $\hat{x}$ in this case, then it has to be nonzero at $x_R$. Moreover, it also implies that $K_R > K_{\hat{x}}^c$ since in the equality case, the water height would tends to $H_c$ when $x$ tends to $\hat{x}^+$ but it is not possible to connect $H_c$ to another water height through a stationary shock. This case is then included in case iii) and the solution is given by (17) with $\chi_{[\hat{x},x_R]}^{K_L,K_R} = \hat{x}$.

Let us now consider the last case where we connect two solutions given by Proposition 12 on each side of the bump. In that case, the solution is subcritical on the left of the bump and supercritical on its right. The transition has to be continuous and the top of the bump is a sonic point with $h(\hat{x}) = H_c$ with a corresponding hydraulic head $K_{\hat{x}}^c$, see Proposition 13. It immediately follows that the solution is given by (18) and that the conditions for the existence of such a solution are, for the increasing slope part, $K_L > K_{\hat{x}}^c$ and $H_L > \psi(h_{K_{\hat{x}}^c}^{sub}(x_L))$ (from Proposition 12-(a), see also the proof of Proposition 15 where this case is already considered) and, for the decreasing slope part, $K_{\hat{x}}^c > K_R$ and $H_R > \psi(h_{K_{\hat{x}}^c}^{sup}(x_R))$ (from Proposition 12-(c), see also the proof of Proposition 14 where this case is already considered). $\square$

As in Section 2.1, the conditions on the hydraulic heads can be translated into conditions on the water heights and we can give on Figure 7 a graphical representation in the $(h(x_R), h(x_L))$-plane of the four propositions of this section by indicating which pairs of water heights can be linked by a stationary solution. Here the two figures correspond to the case where the minimum of the topography is located at the left boundary (figure on the left) or at the right boundary (figure on the right). Note that only the right bottom part of the figure is modified. It corresponds to the zone where it is possible to impose the water height at the inlet and at the outlet (supercritical inlet and subcritical outlet). Moreover, it is in this case that, for some boundary data, there is no unicity of the steady state solution: it is characterized in the figure by the fact that the intersection of the three zones that corresponds to three different kinds of solutions (respectively one shock on the left of the bump, one shock on the right or two shocks) is clearly not empty.

## 2.3 Steady states over a hollow

Let us now consider the case of a $C^1$-regular topography that decreases on $[x_L, \tilde{x}]$ and increases on $[\tilde{x}, x_R]$. As in the previous section §2.2, we split the domain into two subdomains $[x_L, \tilde{x}]$ and $[\tilde{x}, x_R]$ where the topography is monotone, we deduce from the four propositions stated in Section 2.1 which solutions can take place on these monotone parts and under which conditions and we study when it is possible to connect such solutions. Note that the situation is much more simple here than in the bump case since, as the point $\tilde{x}$ corresponds to a minimum of the topography, it is no more possible to connect a subcritical solution on its left to a supercritical solution on its right, see Proposition 8. It follows that the number of global solution is drastically reduced.

**Proposition 17. (Subcritical inlet / Supercritical outlet)** Assume the topography Z is $C^1$-regular and decreasing on $[x_L, \tilde{x}]$ and increasing on $[\tilde{x}, x_R]$. Let $M > 0$ and $S \in \mathbb{R}$ be set at the inlet with a subcritical inlet and a supercritical outlet. There is no steady state solution.

*Proof.* Proposition 9 applied on the decreasing part $[x_L, \tilde{x}]$ implies that the steady solution, if it exists, has to be subcritical on this subdomain. But the same proposition applied on the
<figure><img src="image_10.png" /><figcaption>Figure 7: Pairs of boundary values $(h(x_R), h(x_L))$ for which it exists a stationary solution on bump shaped topography with $Z(x_L) < Z(x_R)$ (top) and $Z(x_L) > Z(x_R)$ (bottom). Black curves are solutions without jump, red curves are solutions with one jump on increasing topography, blue curves are solutions with one jump on decreasing topography, green curves are solutions with two jumps.</figcaption></figure>

〈fig:Bump:SolPlan〉
increasing part [x̃, x_R] implies that the steady solution, if it exists, has to be supercritical on this part. Since it is not possible to connect a subcritical solution to a supercritical one at a minimum of the topography, there is no global steady solution. □

?hollow:subsub>? **Proposition 18. (Subcritical inlet / Subcritical outlet)** Assume the topography Z is $C^1$-regular and decreasing on $[x_L, \tilde{x}]$ and increasing on $[\tilde{x}, x_R]$. Let $M > 0$ and $S \in \mathbb{R}$ be set at the inlet with a subcritical inlet and a subcritical outlet with $h(x_R) = H_R \ge H_c$. The unique possible steady state solution is $h_{K_R}^{sup}$. It always exists if $Z(x_L) \le Z(x_R)$ and exists if only if $K_{x_L}^c \le K_R$ otherwise.

*Proof.* Here again, Proposition 9 applied on the decreasing part $[x_L, \tilde{x}]$ implies that the steady solution, if it exists, has to be subcritical on this whole subdomain. Such a solution exists if and only if $K_{x_L}^c \le K(\tilde{x})$ and, in that case, is given by $h_{K(\tilde{x})}^{sup}$, see Proposition 10. On the increasing part $[\tilde{x}, x_R]$, two kinds of steady solutions can take place, see Propositions 10 and 12. But the ones issued from Proposition 12 are supercritical on the right vicinity of the minimum of the topography $\tilde{x}$ and then it is not possible to connect them to a subcritical solution on the left of this point. The other one, given by Proposition 10, is subcritical on $[\tilde{x}, x_R]$, equal to $h_{K_R}^{sup}$ and no condition is required for its existence. The connection at the minimum of the topography $\tilde{x}$ between these two solutions has to be continuous since the regime does not change. It follows that the hydraulic head is constant on the whole domain and then equal to the value $K_R$ imposed by the right boundary condition. Hence the solution on the whole domain is equal to $h_{K_R}^{sup}$. Since $K(\tilde{x}) = K_R$, the condition of existence on the decreasing part now reads $K_{x_L}^c \le K_R$ and that condition is always fullfilled if $Z(x_L) \le Z(x_R)$ since in that case

$$K_R = K(H_R) + Z(x_R) \ge K(H_c) + Z(x_L) = K_{x_L}^c.$$

?hollow:supsup>? **Proposition 19. (Supercritical inlet / Supercritical outlet)** Assume the topography Z is $C^1$-regular and decreasing on $[x_L, \tilde{x}]$ and increasing on $[\tilde{x}, x_R]$. Let $M > 0$ and $S \in \mathbb{R}$ be set at the inlet with a supercritical inlet with $h(x_L) = H_L \le H_c$ and a supercritical outlet. The unique possible steady state solution is $h_{K_L}^{sup}$. It always exists if $Z(x_L) \ge Z(x_R)$ and exists if and only if $K_L \ge K_{x_R}^c$ otherwise.

*Proof.* The proof is quite similar to the previous one. From Proposition 9 applied on the increasing part $[\tilde{x}, x_R]$, the steady solution, if it exists, has to be supercritical on this subdomain. According to Proposition 11, such a solution is given by $h_{K(\tilde{x})}^{sup}$ and the condition for existence is $K(\tilde{x}) \ge K_{x_R}^c$. On the decreasing part $[x_L, \tilde{x}]$, two kinds of steady solutions can take place, see Propositions 11 and 12. But here again, the ones issued from Proposition 12 are subcritical on the left vicinity of the minimum of the topography $\tilde{x}$ and then it is not possible to connect them to a supercritical solution on the right of this point. The other one, given by Proposition 11, is supercritical, equal to $h_{K_L}^{sup}$ and there is no condition for its existence. Since we connect two supercritical solutions at $\tilde{x}$, the connection has to be continuous, the hydraulic head is constant and then equal to the value $K_L$ imposed at the left boundary. The solution is given by $h_{K_L}^{sup}$ on the whole domain and since $K(\tilde{x}) = K_L$, the condition for existence reads $K_L \ge K_{x_R}^c$, condition that is always fullfilled if $Z(x_L) \ge Z(x_R)$ since

$$K_L = K(H_L) + Z(x_L) \ge K(H_c) + Z(x_R) = K_{x_R}^c.$$
**Proposition 20.** (Supercritical inlet / Subcritical outlet) Assume the topography Z is $C^1$-regular and decreasing on $[x_L, \tilde{x}]$ and increasing on $[\tilde{x}, x_R]$. Let $M > 0$ and $S \in \mathbb{R}$ be set at the inlet with a supercritical inlet with $h(x_L) = H_L \le H_c$ and a subcritical outlet with $h(x_R) = H_R \ge H_c$.

There is at most two steady state solutions, which exist if and only if $K_L > K_R$, $h_{K_R}^{sub}(\tilde{x}) \ge \psi(h_{K_L}^{sup}(\tilde{x}))$ and one or two of the following sets of conditions are verified:

* if $(K_R > K_{x_L}^c$ and $H_L < \psi(h_{K_R}^{sub}(x_L)))$ or if $K_R \le K_{x_L}^c$, one solution is

$$h(x) = \begin{cases} \mathbf{h}_{K_L}^{sup}(x) & \text{if } x < \chi_{[x_L, \tilde{x}]}^{K_L, K_R}, \\ \mathbf{h}_{K_R}^{sub}(x) & \text{if } x > \chi_{[x_L, \tilde{x}]}^{K_L, K_R}. \end{cases} \qquad (19) \text{ hollow:1shockl}$$

* if $H_R < \psi(h_{K_L}^{sup}(x_R))$ one solution is

$$h(x) = \begin{cases} \mathbf{h}_{K_L}^{sup}(x) & \text{if } x < \chi_{[\tilde{x}, x_R]}^{K_L, K_R}, \\ \mathbf{h}_{K_R}^{sub}(x) & \text{if } x > \chi_{[\tilde{x}, x_R]}^{K_L, K_R}. \end{cases} \qquad (20) \text{ hollow:1shockr}$$

**Remark 10.** Note that if $Z(x_L) \le Z(x_R)$, $K_R \ge K_{x_R}^c \ge K_{x_L}^c$ and only the first subcase arises in the first point of the proposition.

**Remark 11.** When $Z(x_L) \to Z(\tilde{x})$, the solution (20) is no longer possible, and the conditions of solution (19) coincides with the condition for the solution on decreasing topography (11), cases (b) and (c). Similarly, when $Z(x_R) \to Z(\tilde{x})$, the solution (19) is no longer possible, and the conditions of solution (20) coincides with those of the solution on increasing topography (11), case (a).

*Proof.* On the decreasing part $[x_L, \tilde{x}]$, two kinds of steady solutions can take place, see Propositions 11 and 12. On the increasing part $[\tilde{x}, x_R]$, two kinds of steady solutions can take place, see Propositions 10 and 12. But since it is not possible to connect a subcritical solution on the left of $\tilde{x}$ to a supercritical solution on the right of $\tilde{x}$, there are only three possibilities to study.

The first one is to connect a solution issued from Proposition 11 on the decreasing part to a solution issued from Proposition 12 on the increasing part. In that case the solution is supercritical on both sides of $\tilde{x}$ and then the connection has to be continuous. On the decreasing part the solution is supercritical, equal to $h_{K_L}^{sup}$ and there is no condition for existence. The continuity at $\tilde{x}$ implies that $K(\tilde{x}) = K_L$ and $h(\tilde{x}) = h_{K_L}^{sup}(\tilde{x})$. On the increasing part, the solution is given by Proposition 12, case (a) and, taking into account the previous equalities, the conditions for existence read $h_{K_L}^{sup}(\tilde{x}) > \psi(h_{K_R}^{sub}(\tilde{x}))$ and $H_R < \psi(h_{K_L}^{sup}(x_R))$, this second condition having the consequence $K_L > K_R$. The solution on the whole domain is given by (20).

The second possibility is to connect a solution issued from Proposition 12 on the decreasing
<figure><img src="image_11.png" /><figcaption>Figure 8: Pairs of boundary values $(h(x_R), h(x_L))$ for which it exists a stationary solution on hollow shaped topography with $Z(x_L) < Z(x_R)$ (top) and $Z(x_L) > Z(x_R)$ (bottom). Black curves are solutions without jump, red curves are solutions with one jump on increasing topography, blue curves are solutions with one jump on decreasing topography, green curves are solutions with two jumps.</figcaption></figure>
part to a solution issued from Proposition 10 on the increasing part. In that case the solution is subcritical on both sides of $\tilde{x}$ and then the connection has to be continuous. On the increasing part the solution is subcritical, equal to $h_{K_R}^{sub}$ and there is no condition for existence. The continuity at $\tilde{x}$ implies that $K(\tilde{x}) = K_R$ and $h(\tilde{x}) = h_{K_R}^{sub}(\tilde{x})$. On the decreasing part, the solution is given by Proposition 12, case (b) or (c), and taking into account the previous equalities, the conditions for existence read $h_{K_R}^{sub}(\tilde{x}) > \psi(h_{K_L}^{sup}(\tilde{x}))$ and, if $K_R > K_{x_L}^c$, $H_L < \psi(h_{K_R}^{sub}(x_L))$, this last condition having the consequence $K_L > K_R$. The solution on the whole domain is given by (19). Note that $\psi$ is a decreasing function, see Corollary 2, and so, the condition $h_{K_L}^{sup}(\tilde{x}) > \psi(h_{K_R}^{sub}(\tilde{x}))$ encountered in the previous case is equivalent to the condition $h_{K_R}^{sub}(\tilde{x}) > \psi(h_{K_L}^{sup}(\tilde{x}))$ issued from this case.

The last possibility is to connect a solution issued from Proposition 11 on the decreasing part to a solution issued from Proposition 10 on the increasing part. In that case, the solution is equal to $h_{K_L}^{sup}$ on the decreasing part and to $h_{K_R}^{sub}$ on the increasing part and it has to contain a discontinuity at $\tilde{x}$ to connect the two parts since a continuous transition from a regime to another one is not possible at a minimum of the topography. Both parts of the solution exist without any condition but the discontinuity can take place at $\tilde{x}$ if and only if $h_{K_R}^{sub}(\tilde{x}) = \psi(h_{K_L}^{sup}(\tilde{x}))$. This condition implies $K_L > K_R$. Let us show that it also implies $H_R < \psi(h_{K_L}^{sup}(x_R))$. First the functions $h_{K_L}^{sup}$ and $h_{K_R}^{sub}$ are both well defined on the increasing part $[\tilde{x}, x_R]$ : it is obvious for $h_{K_R}^{sub}$ since $x_R$ is a maximum of the topography on this domain and it is also true for $h_{K_L}^{sup}$ since $K_L > K_R \ge K_{x_R}^c$. So the function $f$ introduced in Proposition 12 is well-defined on this interval. It is strictly decreasing since the topography is strictly increasing and it vanishes at $\tilde{x}$ since the shock takes place at this location. So it is strictly negative at $x_R$, which implies $H_R < \psi(h_{K_L}^{sup}(x_R))$ since the function $F^{h\tilde{u}}$ is increasing for subcritical water heights. It follows that this last case is a particular case of the second point of the proposition. $\square$

## 2.4 Steady states over a general regular topography

In this section we construct several solutions for the same subcritical boundary conditions. We restrict our attention to solutions that only contain shocks on decreasing slope, see Remark 5. The topography satisfies the following assumption.

**HypZ2** **Hypothesis 1.** We assume that $Z : [x_L, x_R] \to \mathbb{R}$ is a $C^1$-regular function describing a succession of $N \ge 1$ smaller and smaller bumps. More precisely, suppose that there exist

$$ \tilde{x}_0 = x_L < \hat{x}_1 < \tilde{x}_1 < \dots < \tilde{x}_{j-1} < \hat{x}_j < \tilde{x}_j < \dots < \hat{x}_N < \tilde{x}_N = x_R $$

such that for all $j \in \{1, \dots, N\}$, $Z$ is strictly increasing on $[\tilde{x}_{j-1}, \hat{x}_j]$ and strictly decreasing on $[\hat{x}_j, \tilde{x}_j]$ and the maximum of topography decrease

$$ Z(\hat{x}_1) > Z(\hat{x}_2) > \dots > Z(\hat{x}_N). $$

We now describe all the possible solution in the case of a subcritical inlet and a subcritical outlet with a small hydraulic head. There is a most $2^{N-1}$ solutions. They all contain a shock
on the far right decreasing slope. Then under some condition of existence it is possible to place a shock on each decreasing slope. On Figure 9 we represented all the solution with one or two jumps in the case $N = 5$.

**Proposition 21.** Let $Z$ verify Hypothesis 1 and denote by $K_i^c := K^c + Z(\hat{x}_i)$. Suppose that $h(x_L) > H_c$ (but not fixed), $M > 0$ and $S \in \mathbb{R}$ fixed at the inlet and $h(x_R) = H_R > H_c$ fixed at the outlet satisfy $H_R < h_{K_N^c}^{sub}(x_R)$. Then there exist at most $2^{N-1}$ solutions of $(SW_2)$ without jump on increasing topography and satisfying the boundary conditions. The possible solutions can be constructed as follow. Pick $n \in \{1, \dots, N\}$ the total number of jumps and $(i_1, \dots, i_{n-1})$ one of the $\binom{N-1}{n-1}$ strictly increasing sequence of length $n-1$ in $\{1, \dots, N-1\}$. Set $i_N = N$. In the class of solution with only shocks on decreasing topography, there exists at most one solution that contains exactly one shock on each interval $(\hat{x}_{i_k}, \tilde{x}_{i_k})$, $1 \le k \le n$. This solution exists if and only if

$$ \forall k \in \{1, \dots, n-1\}, \quad h_{K_{i_k}^c}^{sub}(\tilde{x}_{i_k}) \ge \psi \left( h_{K_{i_{k-1}}^c}^{sup}(\tilde{x}_{i_k}) \right) \quad \text{and} \quad h_{K_R}^{sub}(x_R) \ge \psi \left( h_{K_{i_{n-1}}^c}^{sup}(x_R) \right). $$

The solution goes from the subcritical regime to the supercritical regimes at the top of the first, $i_1 + 1$-th, ..., $i_{n-1} + 1$-th bumps.

**Proof.** The uniqueness is not an issue since we restrict our attention to solution with shocks on decreasing slopes only and there cannot be more than one shock on each monotone part of the topography.

The solution necessarily contains one shock in the interval $[\hat{x}_N, x_R]$ since the function $h_{K_R}^{sub}$ is not defined up to $\hat{x}_N$. The solution is subcritical at the inlet $\tilde{x}_0$ and thus on $[\tilde{x}_0, \hat{x}_1]$. The hydraulic head $K_0$ on this interval is necessarily larger than $K_1^c$ to ensure the existence of $h_{K_0}^{sub}$. But if $K_0 > K_1^c$, then at the first maxima $h_{K_0}^{sub}(\hat{x}_1) > H_c$ and the regime is subcritical at the inlet of the interval $[\hat{x}_1, \hat{x}_2]$ and thus on the whole interval (Proposition 17). The solution would still be $h_{K_0}^{sub}$ on this interval. As $Z(\hat{x}_1) \ge Z(\hat{x}_2)$ the function $h_{K_0}^{sub}$ would be defined on the whole interval and $h_{K_L}^{sub}(\hat{x}_2) > H_c$. We apply the same reasoning recursively and obtain that the solution would be $h_{K_0}^{sub}$ everywhere, which contradicts the boundary condition $K_R < K_N^c < K_1^c$ at the outlet. As a conclusion the first hydraulic head is necessarily $K_1^c$ and the transition to the supercritical regime can only occur at $\hat{x}_1$.

Suppose now that the solution contains a shock on $[\hat{x}_{i_k}, \tilde{x}_{i_k}]$ at position $\chi$. The regime is subcritical on the right of the shock and remains so up to the next local maxima $\hat{x}_{i_{k+1}}$. The corresponding hydraulic K head should be larger than $K_{i_{k+1}}^c$ so that $h_K^{sub}$ is defined at point $\hat{x}_{i_{k+1}}$. The same reasoning as before shows that if it is not equal to $K_{i_{k+1}}^c$ the regime would stay subcritical on $[\chi, x_L]$ with the constant hydraulic $K > K_{i_{k+1}}^c$. It once again contradicts the boundary condition at $x_R$. Thus there is always a transcritical transition on the local extrema following a shock. This fixes the hydraulic head on $[x_i, \hat{x}_{i_{k+1}}]$ to $K_{i_{k+1}}^c$. At $\hat{x}_{i_{k+1}}$ there is a transcritical transition to the supercritical regime and we apply the same reasoning recursively up to the last shock on $[\hat{x}_N, x_R]$.

Alltogether we proved that the solution has the following form, where we set $i_0 = 0$ and
$K_{n+1}^c = K_R:$

$$
h(x) =
\begin{cases}
\mathbf{h}_{K_1^c}^{\text{sub}} & \text{on } [x_L, \hat{x}_1] \\
\mathbf{h}_{K_1^c}^{\text{sup}} & \text{on } [\hat{x}_1, \chi]_{\hat{x}_{i_1}, \tilde{x}_{i_1}}^{K_1^c, K_{i_1+1}^c} \\
\vdots & \\
\mathbf{h}_{K_{i_k+1}^c}^{\text{sub}} & \text{on } [\chi]_{\hat{x}_{i_k}, \tilde{x}_{i_k}}^{K_{i_k-1+1}^c, K_{i_k+1}^c}, \hat{x}_{i_k+1}, 1 \le k \le n-1 \\
\mathbf{h}_{K_{i_k+1}^c}^{\text{sup}} & \text{on } [\hat{x}_{i_k+1}, \chi]_{\hat{x}_{i_k+1}, \tilde{x}_{i_k+1}}^{K_{i_k+1}^c, K_{i_k+1}^c}, 1 \le k \le n-1 \\
\vdots & \\
\mathbf{h}_{K_R^c}^{\text{sub}} & \text{on } [\chi]_{\hat{x}_N, \tilde{x}_{N+1}}^{K_{i_{n-1}+1}^c, K_R}, x_R
\end{cases}
$$

We now turn to the conditions of existence of the shocks. The hydraulic head is decreasing as it should be: Hypothesis (1) yields $K_1^c > K_{i_1+1}^c > \dots > K_{i_{n-1}+1}^c \ge K_N^c$ and $H_R < \mathbf{h}_{K_N^c}^{\text{sub}}(x_R)$ is equivalent to $K_N^c > K_R$. The hydraulic head on the left of the k-th shock is $K_{i_{k-1}+1}^c$ which is larger than the one on its right $K_{\tilde{x}_{i_k}+1}^c$. The condition of the statement follows from Proposition 12, case (c), applied on $[\hat{x}_{i_k}, \tilde{x}_{i_k}]$ with $H_L = \mathbf{h}_{K_{i_{k-1}+1}^c}^{\text{sup}}(\hat{x}_{i_k})$ and $H_R = \mathbf{h}_{K_{i_k+1}^c}^{\text{sub}}(\tilde{x}_{i_k})$. The same condition applies for the last shock with right hydraulic head $K_R < K_N^c$. $\square$

Acknowledgment

This research was funded, in whole or in part, by the Agence Nationale de la Recherche (ANR), grant ANR-23-CE56-0005-01. A CC BY license is applied to the AAM arising from this submission, in accordance with the grant's open access conditions.
<figure><img src="image_12.png" /><figcaption>Figure 9: Solutions one or two shocks on a topography with 5 bumps.</figcaption></figure>

<Fig:NonUnicSol>
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