Mirco Ciallella

Davide Torlo

Mario Ricchiuto

Arbitrary High Order WENO Finite Volume Scheme with Flux Globalization for Moving Equilibria Preservation

Keywords: Flux Globalization, WENO, well-balanced, moving equilibria, shallow water

. This approach consists in including the integral of the source term in the global flux and reconstructing the new global flux rather than the conservative variables. The resulting scheme is able to preserve a large family of smooth and discontinuous steady state moving equilibria. In this work, we focus on an arbitrary high order WENO Finite Volume (FV) generalization of the global flux approach. The most delicate aspect of the algorithm is the appropriate definition of the source flux (integral of the source term) and the quadrature strategy used to match it with the WENO reconstruction of the hyperbolic flux. When this construction is correctly done, one can show that the resulting WENO FV scheme admits exact discrete steady states characterized by constant global fluxes. We also show that, by an appropriate quadrature strategy for the source, we can embed exactly some particular steady states, e.g. the lake at rest for the shallow water equations. It can be shown that an exact approximation of global fluxes leads to a scheme with better convergence properties and improved solutions. The novel method has been tested and validated on classical cases: subcritical, supercritical and transcritical flows.

Introduction

Water waves can be modeled with various systems of equations. The Saint-Venant system [START_REF] De Saint | Theorie du mouvement non-permanent des eaux avec application aux crues des rivers et a l'introduntion des marees dans leur lit[END_REF] of shallow water equations is widely used to simulate water waves in regimes where the wave lengths are larger than the water height. These equations are a system of balance laws that models the conservation of mass and momentum under the force of gravity, taking also care of the bottom topography effects through a source term. Their applications range between coastal waves, rivers, seas, lakes, tsunamis, and so on. In this work we will consider the Saint-Venant equations with a friction term. Analytical solutions of these equations are not always available and it is necessary to recur to numerical simulations in order to approximate the exact solutions. In the past, many techniques have been used to discretize these equations: finite volume methods [START_REF] Audusse | A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows[END_REF][START_REF] Noelle | High-order well-balanced finite volume WENO schemes for shallow water equation with moving water[END_REF][START_REF] Kurganov | Finite-volume schemes for shallow-water equations[END_REF][START_REF] Michel-Dansac | A two-dimensional high-order well-balanced scheme for the shallow water equations with topography and manning friction[END_REF][START_REF] Ciallella | An arbitrary high order and positivity preserving method for the shallow water equations[END_REF], finite element methods [START_REF] Ricchiuto | Stabilized residual distribution for shallow water simulations[END_REF][START_REF] Ricchiuto | Application of conservative residual distribution schemes to the solution of the shallow water equations on unstructured meshes[END_REF][START_REF] Ricchiuto | An explicit residual based approach for shallow water flows[END_REF][START_REF] Ricchiuto | On the C-property and generalized C-property of residual distribution for the shallow water equations[END_REF][START_REF] Arpaia | r-adaptation for shallow water flows: conservation, well balancedness, efficiency[END_REF], discontinuous Galerkin [START_REF] Mantri | Well-balanced discontinuous Galerkin scheme for 2× 2 hyperbolic balance law[END_REF][START_REF] Arpaia | An efficient covariant frame for the spherical shallow water equations: Well balanced DG approximation and application to tsunami and storm surge[END_REF][START_REF] Meister | On unconditionally positive implicit time integration for the DG scheme applied to shallow water flows[END_REF] and so on. The goal of such methods is to accurately approximate the solutions with as little as possible computational costs. To achieve this, many strategies have been adopted. High order methods are a clear example of a technique to obtain more accurate results within the same dimension of the semidiscretized system. In particular, in this work we will consider the WENO high order reconstruction finite volume method to obtain (arbitrarily) high order spatial discretizations [START_REF] Shu | Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws[END_REF][START_REF] Shu | Efficient implementation of essentially non-oscillatory shock-capturing schemes[END_REF], while for the temporal discretization we will use the Deferred Correction [START_REF] Daniel | Iterated deferred corrections for initial value problems[END_REF][START_REF] Dutt | Spectral Deferred Correction Methods for Ordinary Differential Equations[END_REF][START_REF] Minion | Semi-implicit spectral deferred correction methods for ordinary differential equations[END_REF][START_REF] Abgrall | High order schemes for hyperbolic problems using globally continuous approximation and avoiding mass matrices[END_REF][START_REF] Veiga | DeC and ADER: Similarities, Differences and a Unified Framework[END_REF], an (arbitrarily) high order time integration method.

Another way to obtain accurate results with low computational costs is to design methods which are capable of preserving some physical properties. The shallow water equations, for example, are verified by some steady state solutions that are very easily described. The simplest equilibrium is the lake at rest steady state where the water level is constant and velocity is zero. The preservation of this equilibrium at the numerical level is not automatic and numerical schemes must be specifically designed in order to balance the discretization of the flux and the source in this situation. Such schemes are often denoted as well-balanced or they are said to enjoy the C-property [START_REF] Audusse | A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows[END_REF][START_REF] Bollermann | A well-balanced reconstruction of wet/dry fronts for the shallow water equations[END_REF][START_REF] Díaz | High order exactly well-balanced numerical methods for shallow water systems[END_REF][START_REF] Gallardo | On a well-balanced high-order finite volume scheme for shallow water equations with topography and dry areas[END_REF][START_REF] Kurganov | A second-order well-balanced positivity preserving central-upwind scheme for the saint-venant system[END_REF][START_REF] Arpaia | r-adaptation for shallow water flows: conservation, well balancedness, efficiency[END_REF][START_REF] Michel-Dansac | A well-balanced scheme for the shallow-water equations with topography[END_REF][START_REF] Michel-Dansac | A well-balanced scheme for the shallow-water equations with topography or manning friction[END_REF][START_REF] Michel-Dansac | A two-dimensional high-order well-balanced scheme for the shallow water equations with topography and manning friction[END_REF]. These methods increase the accuracy of the solutions when close to these equilibria, but not only. It is possible to see great improvement in their errors also for more general tests. There are also other types of steady states that solve the shallow water equations. A generalized form of the lake at rest is obtained when the discharge is not zero, but constant in space. In this situation, many equilibria can occur. Again, an analytical solution is not always available, but some implicit forms of the solution [START_REF] Delestre | Swashes: a compilation of shallow water analytic solutions for hydraulic and environmental studies[END_REF] are known. It is then interesting to build schemes that are able to preserve also these equilibria at the discrete level. A class of schemes has been built in the recent years with the aim of preserving such equilibria [START_REF] Chertock | Well-balanced schemes for the euler equations with gravitation: Conservative formulation using global fluxes[END_REF][START_REF] Cheng | A new approach for designing movingwater equilibria preserving schemes for the shallow water equations[END_REF][START_REF] Chertock | Well-balancing via flux globalization: Applications to shallow water equations with wet/dry fronts[END_REF]. They rely on the idea of defining a global flux that incorporates both the flux and the source term thanks to a discretization of the integral of the source term. It is possible to define a numerical scheme that depends only on this global flux and, hence, when reaching the equilibrium the scheme guarantees to preserve the constant global flux at machine precision, including the constant discharge. These techniques not only guarantee the preservation of these equilibria, but also improve the general accuracy of the solution. In this work, we aim at maximizing the accuracy of simulations by combining the flux globalization technique and the high order accuracy of finite volume methods with WENO reconstructions. We study how a high order reconstruction can be included in the definition of the global flux. A special care is needed in the choice of the discretization of the source term, in particular in the used quadrature formula. Moreover, to exactly preserve the lake-at-rest steady state, a further modification must be done in the definition of the discretized source integral, in order to guarantee the well-balancedness property. The procedure requires a special discretization of the source term and an upwinding technique to compute the numerical flux. The work is organized as follows. In Section 2 we introduce the shallow water equations and the steady state equilibria that we want to preserve. In Section 3 we introduce the spatial discretization including the finite volume discretization, the global flux definition, the lake-at-rest well-balanced modification and the high order WENO reconstruction. In Section 4 we describe the Deferred Correction method, an arbitrarily high order time integration method. In Section 5 we perform several tests showing the superior capability of the global flux high order method with respect to other high order methods. Finally, in Section 6 we draw some conclusions and we describe some perspective future works.

Shallow Water Equations

Model

The shallow water equations (SWE) model the behaviour of shallow free surface flows under the action of gravity. They are used to simulate the flows in rivers and coastal areas, and can be applied to predict tides, storm surge levels and coastline changes from hurricanes and ocean currents. They are also used in atmospheric flows, debris flows, and certain hydraulic structures like open channels and sedimentation tanks. SWEs take the form of non-homogeneous hyperbolic conservation laws with source terms modeling the effects of bathymetry and viscous friction. In this paper, we will consider the effect of the bathymetry and the friction as source terms. If the bottom topography is assumed to be constant with respect to time, the SWEs can be recast in balance law form as: with conserved variables, flux and source term given by

∂u ∂t + ∂F (u) ∂x = S(u, x) on Ω T = Ω × [0, T ] ⊂ R × R + (1) x b(x) h η u
u = h q , F (u) = q q 2 h + g h 2 2 , S(u, x) = 0 S(u, x) = -gh 0 ∂b(x) ∂x -gq 0 n 2 h 7/3 |q| (2)
where h represents the relative water height, q is its discharge (equal to hu, where u is the vertically averaged velocity), g is the gravitational acceleration, b(x) is the local bathymetry and n is the Manning friction coefficient. The source term helps modeling the effects induced on the flow caused by the bathymetry changes in space. Finally, it is also convenient to introduce the free surface water level η := h + b. All the aforementioned variables can be better interpreted by looking at Figure 1.

The idea based on global fluxes consists in incorporating the source term in the flux term and rewriting the conservation law (1) in the following equivalent form:

∂u ∂t + ∂G(u, x) ∂x = 0 such that G(u, x) = q K = q q 2 h + g h 2 2 + R (3)
so that K becomes a global equilibrium variable with

R(x, t) := - x S(u, ξ) dξ = g x h(ξ, t) ∂b(ξ) ∂ξ + n 2 h 7/3 (ξ, t) |q(ξ, t)|q(ξ, t) dξ (4)
System (3) is therefore a hyperbolic system with a global flux, which can be solved with classical techniques for conservation laws without special treatments of the source term. The advantage of the global flux is to have a natural formulation of a quantity R that can be balanced for steady state solutions.

Steady state equilibria

As it is described in [START_REF] Bollermann | A well-balanced reconstruction of wet/dry fronts for the shallow water equations[END_REF][START_REF] Berthon | A fully well-balanced, positive and entropy-satisfying Godunov-type method for the shallow-water equations[END_REF][START_REF] Lukáčová-Medvid'ová | Well-balanced finite volume evolution Galerkin methods for the shallow water equations[END_REF][START_REF] Ricchiuto | An explicit residual based approach for shallow water flows[END_REF][START_REF] Xing | A survey of high order schemes for the shallow water equations[END_REF], the construction and development of effective and accurate numerical methods for the shallow water equations have received much interest in the last decades and it is still ongoing.

In particular, one is interested in schemes that preserve physical quantities or structures from the continuous level. In this paper, we are addressing two types of difficulties: the preservation of moving steady state equilibria and the preservation of the lake at rest solution.

The SWE system (2) is known to admit some steady state solutions whose form depends on the equilibrium between the source terms S and the flux F of the equations. The numerical simulations should be able to capture these behaviors even on coarse grids. Without additional techniques, many methods fail at balancing the source terms and the flux, resulting in small perturbations of the steady state. These perturbations could be then amplified by the method causing a bad approximation of the exact behaviour. This phenomenon is sometimes called numerical storm in such context. To prevent it, one is interested in schemes that are capable of exactly balancing the flux and the source terms to obtain the desired steady-state solution. Numerical schemes enjoying this property are called well-balanced (WB) schemes.

The still water surface is often the first equilibrium taken in consideration. It is given by

u = 0; η(x, t) = h(x, t) + b(x) ≡ η 0 ∈ R, ∀ x ∈ Ω, t ∈ [0, T ]. (5)
It represents a steady-state solution, and is referred to as lake at rest. However, that is only a special case of the moving water equilibrium. The steady state solutions are characterized by the invariants h(x, t)u(x, t) ≡ q 0 and K

(x, t) = q 2 h +g h 2 2 + x x0 g h(ξ, t) ∂b(ξ) ∂ξ + n 2 h 7/3 (ξ, t) |q(ξ, t)|q(ξ, t) dξ ≡ K 0 . (6)
Obviously, the lake at rest ( 5) is a special case of (6) when the velocity reduces to zero. Other balanced states can be found in, inter alia, [START_REF] Ricchiuto | An explicit residual based approach for shallow water flows[END_REF].

It is interesting [START_REF] Michel-Dansac | A well-balanced scheme for the shallow-water equations with topography[END_REF][START_REF] Michel-Dansac | A well-balanced scheme for the shallow-water equations with topography or manning friction[END_REF] to observe that the property (6) in case of smooth solutions and without friction can be rewritten as follows

q ≡ q 0 , (7)
0 = ∂ x q 2 h + g h 2 2 + gh∂ x b = - q 2 0 h 2 ∂ x h + gh∂ x h + gh∂ x b = h∂ x q 2 0 2h 2 + g(h + b) . (8)
Then, the last equation sum up to verify that Υ := q 2 2h 2 + g(h + b) ≡ Υ 0 is constant in space and time. Clearly, this condition does not hold when bathymetry or water height are discontinuous in some points, while ( 6) is always preserved.

Space discretization: Global Flux Finite Volume method

The hyperbolic system considered herein is solved by means of the Method Of Lines (MOL). Hence, in this setting, space and time can be treated independently. This section has the goal of presenting a modified arbitrary high order finite volume framework for balance laws with global fluxes. A second order global flux approach is available in [START_REF] Cheng | A new approach for designing movingwater equilibria preserving schemes for the shallow water equations[END_REF], but it is limited to linear reconstructions. In this section we aim at building an arbitrary high order global flux method based on WENO reconstructions. The computational domain Ω is discretized into N x equispaced control volumes Ω i = [x i-1/2 , x i+1/2 ] of size ∆x centered at x i = i∆x with i = i ℓ , . . . , i r . Considering the system of hyperbolic balance laws described by (1) and (2), for the control volume Ω i we can define the cell average at time t:

Ūi (t) := 1 ∆x x i+1/2 x i-1/2 u(x, t) dx. ( 9
)
The semi-discrete central upwind scheme for the system (3) reads

d Ūi dt + 1 ∆x (H i+1/2 -H i-1/2 ) = 0, (10)
where H i+1/2 is a numerical flux consistent with the global flux G. The global flux differs from the original flux and this makes tricky the development of an upwind scheme based on the solution of Riemann problems by an approximate solver. Thus, we are going to employ an upwind scheme that can be easily applied to problems with global fluxes. In order to have a global flux formulation, all the update terms must depend only on the global flux G itself. In this way, when we reach a numerical steady state, we are sure that G is constant all over the domain. We start from G i cell average values and we reconstruct the global flux at interfaces with a high order WENO procedure. Then, we choose H i+1/2 to be an upwind numerical flux defined only on the global flux:

H i+1/2 = L -1 Λ + L G L i+1/2 + L -1 Λ -L G R i+1/2 . (11)
Here, G L,R i+1/2 are the discontinuous reconstructed point values of the global flux G(U) respectively at the left and right side of the cell interface x i+1/2 . L is the matrix of the left eigenvectors computed from the flux Jacobian of the hyperbolic problem (1) in the Roe averaged state, i.e.,

J(U

* ) = 0 1 -u 2 * + gh * 2u * , with h * = h L +h R 2 , u * = √ h L u L + √ h R u R √ h L + √ h R . (12)
Λ ± correspond to the upwinding weights

Λ + i = 1, if λ i > 0, 0, if λ i < 0, Λ - i = 1, if λ i < 0, 0, if λ i > 0. ( 13
)
It is important to notice that the Jacobian of the flux cannot be directly computed as not all the quantities are available at the interface (we reconstruct only the global flux, not the conserved quantities). Hence, one has to recover the value of h from the global flux itself. To do so in a positive manner, we use the technique proposed in [START_REF] Cheng | A new approach for designing movingwater equilibria preserving schemes for the shallow water equations[END_REF] and reported in Appendix A.

In order to obtain the reconstructed values at the left and right side of interfaces, we will use a high order WENO reconstruction technique that will be explained in Section 3.3, where starting from cell averages, one obtains a nonlinear reconstruction inside a cell. The procedure explained in [START_REF] Cheng | A new approach for designing movingwater equilibria preserving schemes for the shallow water equations[END_REF] is not enough to reach arbitrary high order since it approximates (4) using only the midpoint rule. In particular, we will consider piece-wise continuous reconstructions for all the quantities of interests, i.e., h, q, b, R, G and K. On the other side, S, being the derivative of R, must be defined in a distributional sense. This means that a particular form of S must be chosen when integrating it to obtain R at the interfaces. The details of this procedure will be specified in Section 3.2.

Global flux definitions

Now, to obtain the reconstruction of the global flux at the interfaces, we need, first of all, the values of the global flux at the cell averages, defined as

Ḡi (u, x) = Fi (u) + 0 Ri . ( 14
)
We can compute the cell averaged flux evaluating the flux in some high order quadrature points of the cell, through the WENO reconstruction of the conserved variables in the quadrature points, i.e., Fi (u) = q w q F (ũ(x i,q )).

For the cell average of the global integral source R, we have to perform a reconstruction based on the source terms evaluated, as well, in the quadrature points. Moreover, it will be performed iteratively, from the left boundary of the domain to the right. So that, the cell averages of the integral source can be defined as

Ri ≈ q w q R i,q . (16)
To obtain R in the quadrature points, we have to keep in mind its definition as integral of S (4). So, we can define them using the right interface values of R, i.e.,

R i,q = R R i-1/2 + xi,q x R i-1/2 S(x) dx = R R i-1/2 + θ xi,q x R i-1/2 l θ (x) dx r q θ S(x i,θ ), ( 17
)
where S is a high order reconstruction of S done in quadrature points x i,q for each cell and l θ are Lagrangian polynomials in the quadrature points:

Si (x) = θ l θ (x)S(x i,θ ). ( 18
)
This definition necessitates the values of R at the right interface. In order to define a global R, we need to define it on both sides of the interfaces. In particular, we can define R up to a constant, so we impose that R at the left extrema is equal to 0, i.e.,

R R i ℓ -1/2 = 0. (19)
Then, we proceed with the integration from left to right, obtaining the values of R at the left interfaces from the right interface ones:

R L i+1/2 = R R i-1/2 + x L i+1/2 x R i-1/2 S(u(x), x) dx = R R i-1/2 + ∆x Si , i > i ℓ , (20)
with Si being the cell average of S computed as Si := 1 ∆x

x L i+1/2 x R i-1/2 S(u, ξ) dξ ≈ q w q S(x i,q ). ( 21
)
Finally, we need to give a recursive definition of the right interface values as

R R i+1/2 = R L i+1/2 + [[R i+1/2 ]], (22)
where [[R i+1/2 ]] denotes the jump of R on x i+1/2 . At this level, we still need to define how we compute the source in the quadrature points S(x i,θ ) and the jump of the integral source

[[R i+1/2 ]].
Since S is defined only in a distribution sense, the definition of the jump needs a careful choice. Essentially, S on the interfaces is a Dirac delta function as we need to compute the derivative of the bathymetry which is discontinuous. The definition of such term in order to have a high order scheme is not uniquely determined.

Hence, it will be chosen so that the scheme is consistent and balanced with the flux in the lake at rest equilibrium. In the next section, we describe how to obtain the discretization of the source and of the jump of the integral of the source.

Global Flux formulation for lake at rest preservation

By construction, the scheme presented so far is able to preserve a constant global flux for the family of moving equilibria solutions, i.e., q ≡ q 0 and K ≡ K 0 . However, in the lake at rest simulation, the preservation of the constant water free surface does not come for free, in particular we cannot preserve a priori η = h + b ≡ η 0 . Therefore, a special treatment of the source term is required to assure so. The friction term is neglected here, because it vanishes when the speed is zero as in the lake at rest equilibrium. We start defining a high order reconstruction of h, η and b in a cell Ω i using the WENO reconstruction with the same weights for h, η and b. In particular, the weights will be chosen according to η in order to preserve the lake at rest balance. This means that we obtain a reconstructions for the three variables in the quadrature points x i,q that we denote with hi,q , ηi,q and bi,q , and they are such that hi,q = ηi,q -bi,q .

We start by recasting the source term, given that h

(x) = η(x) -b(x), S(u, x) = gh(x)∂ x b(x) = gη(x)∂ x b(x) -g∂ x b 2 (x) 2 . ( 24
)
Equation ( 24) allows us to give proper shape to [START_REF] Gallardo | On a well-balanced high-order finite volume scheme for shallow water equations with topography and dry areas[END_REF], which now reads

R L i+1/2 = R R i-1/2 - x L i+1/2 x R i-1/2 S(u(x), x) dx (25) = R R i-1/2 + g x L i+1/2 x R i-1/2 η(x)∂ x b(x) dx -g (b L i+1/2 ) 2 2 - (b R i-1/2 ) 2 2 ( 26
)
(if η ≡ η 0 ) = R R i-1/2 + gη 0 b L i+1/2 -b R i-1/2 -g (b L i+1/2 ) 2 2 - (b R i-1/2 ) 2 2 . ( 27
)
As we see, the values of the bathymetry at the interfaces of the cells are necessary to compute [START_REF] Kurganov | Finite-volume schemes for shallow-water equations[END_REF]. It is crucial to maintain the well balanced structure in the quadrature points x i,q , hence, we reconstruct b L i+1/2 and b R i-1/2 using a Lagrangian polynomial interpolating the values at the quadrature points, i.e., bi (x) :=

q ℓ q (x) bi,q , and b L i+1/2 = bi (x i+1/2 ), b R i-1/2 = bi (x i-1/2 ). ( 28
)
Equation ( 27), corresponding to the case η ≡ η 0 , is the value that R L i+1/2 assumes at the interface when the lake at rest is preserved. However, to get there, a proper discretization of the integral

x L i+1/2 x R i-1/2 η(x)∂ x b(x)
should be performed. To do so, let us introduce Q(x) := gη(x)∂ x b(x) and let us reconstruct it as the source term in [START_REF] Díaz | High order exactly well-balanced numerical methods for shallow water systems[END_REF] and we can integrate it using ad hoc quadrature formulae.

x L i+1/2 x R i-1/2 Q(x)dx = x L i+1/2 x R i-1/2 θ ℓ θ (x)dx Q i,θ , (29)
where

Q i,θ = ηi,θ q ℓ ′ q (x i,θ ) bi,q . (30)
In particular, we applied the same approach to treat the derivative of the bathymetry. The fully integrated relation for R L i+1/2 now becomes

R L i+1/2 = R R i-1/2 + x L i+1/2 x R i-1/2 θ ℓ θ (x)dx Q i,θ -g (b L i+1/2 ) 2 2 - (b R i-1/2 ) 2 2 (31) = R R i-1/2 + g θ x L i+1/2 x R i-1/2 ℓ θ (x) dx ηi,θ s ℓ ′ s (x θ ) bi,s -g (b L i+1/2 ) 2 2 - (b R i-1/2 ) 2 2 . ( 32
)
Notice that the value of the bathymetry at the interfaces and the derivative of the bathymetry in the quadrature points is given by the same reconstruction [START_REF] Lukáčová-Medvid'ová | Well-balanced finite volume evolution Galerkin methods for the shallow water equations[END_REF]. So, when η ≡ η 0 , we have that

R L i+1/2 =R R i-1/2 + gη 0 x L i+1/2 x R i-1/2 s bi,s ℓ ′ s (x) dx -g (b L i+1/2 ) 2 2 - (b R i-1/2 ) 2 2 =R R i-1/2 + gη 0 (b L i+1/2 -b R i-1/2 ) -g (b L i+1/2 ) 2 2 - (b R i-1/2 ) 2 2 . ( 33
)
Following the same reasoning R i,q introduced in ( 17) can be recast as

R i,q = R R i-1/2 - xi,q x R i-1/2 S(u(x), x) dx (34) = R R i-1/2 + g xi,q x R i-1/2 η(x)∂ x b(x) dx -g (b i,q ) 2 2 - (b R i-1/2 ) 2 2 ( 35
)
(if η ≡ η 0 ) = R R i-1/2 + gη 0 b i,q -b R i-1/2 -g (b i,q ) 2 2 - (b R i-1/2 ) 2 2 . ( 36
)
Once again, this last identity corresponds to the solution at the lake at rest condition but the integral xi,q

x i-1/2 η(x)∂ x b(x) dx has to be discretized to be consistent also for other solutions. Following what we did in [START_REF] Kurganov | Finite-volume schemes for shallow-water equations[END_REF], [START_REF] Minion | Semi-implicit spectral deferred correction methods for ordinary differential equations[END_REF] becomes

R i,q = R R i-1/2 - xi,q x R i-1/2 S(x) dx (37) = R R i-1/2 + θ xi,q x R i-1/2 ℓ θ (x) dx Qi,θ -g (b i,q ) 2 2 - (b R i-1/2 ) 2 2 (38) = R R i-1/2 + g θ xi,q x R i-1/2 ℓ θ (x) dx ηi,θ s ℓ ′ s (x i,θ ) bi,s -g (b i,q ) 2 2 - (b R i-1/2 ) 2 2 . ( 39
)
So that, when η ≡ η 0 , we have

R i,q = R R i-1/2 + gη 0 xi,q x R i-1/2 s ℓ ′ s (x i,θ ) bi,s dx ηi,θ -g (b i,q ) 2 2 - (b R i-1/2 ) 2 2 (40) = R R i-1/2 + gη 0 bi,q -b R i-1/2 -g (b i,q ) 2 2 - (b R i-1/2 ) 2 2 . ( 41
)
We still have not defined the jump of the integral source term [[R i+1/2 ]]. Equipped with [START_REF] Kurganov | Finite-volume schemes for shallow-water equations[END_REF] and [START_REF] Minion | Semi-implicit spectral deferred correction methods for ordinary differential equations[END_REF], considering η ≡ η 0 and q ≡ 0, we can build it such that the cell averages Ki are constant in all the cells. We have that

K i,q = F i,q + R i,q =R R i-1/2 + g (η 0 -b i,q ) 2 2 + gη 0 b i,q -b R i-1/2 -g (b i,q ) 2 2 - (b R i-1/2 ) 2 2 (42) =R R i-1/2 + g η 2 0 2 -✘ ✘ ✘ ✘ gη 0 b i,q + g b 2 i,q 2 + gη 0 ✚ ✚ b i,q -b R i-1/2 -g ✚ ✚ ✚ ✚ (b i,q ) 2 2 - (b R i-1/2 ) 2 2 (43) =R R i-1/2 + g η 2 0 2 -gη 0 b R i-1/2 + g (b R i-1/2 ) 2 2 (44)
and, clearly, Ki = K i,q for all q. Hence, to prove that the scheme is indeed well-balanced for the lake at rest solution, we set Ki+1 -Ki = 0:

Ki+1 -Ki = R R i+1/2 -R R i-1/2 -gη 0 b R i+1/2 + g (b R i+1/2 ) 2 2 + gη 0 b R i-1/2 -g (b R i-1/2 ) 2 2 ! = 0. ( 45
)
Now we compute the differences of the integral sources using [START_REF] Michel-Dansac | A two-dimensional high-order well-balanced scheme for the shallow water equations with topography and manning friction[END_REF] and we get

R R i+1/2 -R R i-1/2 = R L i+1/2 -R R i-1/2 + [[R i+1/2 ]] (46) = gη 0 b L i+1/2 -b R i-1/2 -g (b L i+1/2 ) 2 2 - (b R i-1/2 ) 2 2 + [[R i+1/2 ]]. (47)
Inserting ( 46) in (45), we get

Ki+1 -Ki =gη 0 b L i+1/2 -✟ ✟ ✟ b R i-1/2 -g (b L i+1/2 ) 2 2 -✟ ✟ ✟ ✟ ✟ (b R i-1/2 ) 2 2 + [[R i+1/2 ]] (48) -gη 0 b R i+1/2 + g (b R i+1/2 ) 2 2 + ✘ ✘ ✘ ✘ ✘ gη 0 b R i-1/2 -✟ ✟ ✟ ✟ ✟ g (b R i-1/2 ) 2 2 ! = 0, (49)
which is indeed equal to zero if

[[R i+1/2 ]] ! = gη 0 b R i+1/2 -b L i+1/2 -g (b R i+1/2 ) 2 2 - (b L i+1/2 ) 2 2 .
In order to have a consistent approximation of the jump of the integral source [[R i+1/2 ]], we define it as

[[R i+1/2 ]] := g η R i+1/2 + η L i+1/2 2 b R i+1/2 -b L i+1/2 -g (b R i+1/2 ) 2 2 - (b L i+1/2 ) 2 2 . ( 50
)
Remark 3.1 (Friction). When adding the friction term to the global flux, we do not need to be so careful. Indeed, the friction term is defined as rational of polynomials of conservative variables that can be reconstructed piecewise continuously on the mesh. Hence, they do not contribute to the jump of R at the interfaces. So, it is enough to define

R i,q := R R i-1/2 + g θ xi,q x R i-1/2 ℓ θ (x) dx ηi,θ s ℓ ′ s (x i,θ ) bi,s + g qi,θ |q i,θ |n 2 h7/3 i,θ -g (b i,q ) 2 2 - (b R i-1/2 ) 2 2 ( 51
)
and

R L i+1/2 := R R i-1/2 +g θ x i+1/2 x R i-1/2 ℓ θ (x) dx ηi,θ s ℓ ′ s (x i,θ ) bi,s + g qi,θ |q i,θ |n 2 h7/3 i,θ -g (b L i+1/2 ) 2 2 - (b R i-1/2 ) 2 2 . (52)
In Algorithm 1 we summarize the steps of the reconstruction of the source integral.

Algorithm 1 Source integral reconstruction R i l -1/2 := 0 for i = i l , . . . , i r do Reconstruct the variables h, η and b in the quadrature point of the cell, obtaining hi,θ , ηi,θ and bi,θ with the same weights Reconstruct q in the quadrature points obtaining qi,θ Define R i,q as in (51

) Define R L i+1/2 as in (52) Define [[R i+1/2 ]] as in (50) Define R R i+1/2 := R L i+1/2 + [[R i+1/2 ]] end for

Weighted Essentially Non-Oscillatory (WENO) method

Scalar reconstruction

The WENO method is a high order accurate reconstruction of the solution at some prescribed points. In particular, we are interested in finding the value of the solution at cell interfaces x i+1/2 given the cell averages of the solutions in the neighboring cells. We will denote the high order accurate reconstructed variables as u L i+1/2 and u R i+1/2 at the left and the right side of the interface, respectively. The left value is obtained by reconstructing the variable in Ω i , while the right value is recovered performing the same reconstruction in Ω i+1 . In order to obtain a high order p reconstruction, with p odd, we consider p cells that constitute a stencil:

{Ω lx , l x = i -r + 1, . . . , i + r -1}, (53)
where 2r -1 = p. As an example, WENO reconstruction with order 5 (WENO5) will use a stencil of length p = 5 with r = 3 low order reconstructions. The involved cells are from i -2 to i + 2.

In practice WENO considers a high order reconstruction polynomials p HO that fulfills all the constraints 1 ∆x

x i-j+1/2 x i-j-1/2 p HO (x) dx = u i-j , j = -r + 1, . . . , r -1, (54)
and r low order (r -1) polynomials p m (x) m = 0, . . . , r -1 that fulfill 1 ∆x

x i-r+j+m+1/2 x i-r+j+m-1/2 p m (x)dx = u i-j+m , j = 1, . . . , r. (55)
The WENO reconstruction aims at combining the low order polynomials in order to obtain the high order reconstruction in case all the low order polynomials are non-oscillatory, while it will prefer the least oscillatory polynomial in case some of these polynomials show oscillations.

In order to achieve the high order accuracy (p), for every quadrature point or flux point ξ in which we want to reconstruct the polynomial, we look for some optimal weights d m such that m d m p m (ξ) = p HO (ξ). However, these optimal weights are not enough to control oscillations and Gibbs phenomena, since "Linear numerical schemes for solving PDEs, having the property of not generating new extrema (monotone scheme), can be at most first-order accurate" [START_REF] Godunov | A difference scheme for numerical solution of discontinuous solution of hydrodynamic equations[END_REF]. Hence, we weigh these optimal weights by

α m = d m (β m + ǫ) 2 , ω m = α m r-1 k=0 α k , m = 0, . . . , r -1, ( 56
)
where ǫ is needed to avoid division by 0 (10 -30 in the simulations) and β m are the smoothness indicator defined by

β m = r-1 k=1 x i+1/2 x i-1/2 d k dx k p m (x) 2 ∆x 2k-1 dx , m = 0, . . . , r -1. (57)
In this way, we obtain the WENO weights ω m (ξ) that gives the reconstruction in a specific point ξ:

ũ(ξ) = r-1 m=0 ω m (ξ)p m (ξ). ( 58
)
The values of the optimal weights d m and the formulae for computing β m can be found in [START_REF] Jiang | Efficient implementation of weighted ENO schemes[END_REF][START_REF] Balsara | Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy[END_REF] up to r = 6.

In the following, we will test the reconstruction with orders p = 3 and p = 5.

Time discretization

After the description of the space discretization, we will introduce an arbitrarily high order time integration method. We use the MOL splitting of spatial and time discretization, so we can describe the time discretization for the semidiscretized ODE resulting after the spatial discretization. We will use the Deferred Correction (DeC) method [START_REF] Daniel | Iterated deferred corrections for initial value problems[END_REF][START_REF] Dutt | Spectral Deferred Correction Methods for Ordinary Differential Equations[END_REF][START_REF] Minion | Semi-implicit spectral deferred correction methods for ordinary differential equations[END_REF] as described in [START_REF] Abgrall | High order schemes for hyperbolic problems using globally continuous approximation and avoiding mass matrices[END_REF][START_REF] Abgrall | High order asymptotic preserving deferred correction implicit-explicit schemes for kinetic models[END_REF][START_REF] Torlo | Hyperbolic problems: high order methods and model order reduction[END_REF][START_REF] Veiga | DeC and ADER: Similarities, Differences and a Unified Framework[END_REF]. The DeC is a family of one step methods with arbitrarily high order of accuracy.

Deferred Correction method

The original DeC formulation was introduced in [START_REF] Daniel | Iterated deferred corrections for initial value problems[END_REF], then developed and studied in its different forms in [START_REF] Dutt | Spectral Deferred Correction Methods for Ordinary Differential Equations[END_REF][START_REF] Minion | Semi-implicit spectral deferred correction methods for ordinary differential equations[END_REF][START_REF] Christlieb | Integral deferred correction methods constructed with high order Runge-Kutta integrators[END_REF][START_REF] Liu | Strong stability preserving property of the deferred correction time discretization[END_REF], while a slightly different form was presented in [START_REF] Abgrall | High order schemes for hyperbolic problems using globally continuous approximation and avoiding mass matrices[END_REF] with applications to finite element methods. In [START_REF] Abgrall | High order schemes for hyperbolic problems using globally continuous approximation and avoiding mass matrices[END_REF], the DeC is presented as an iterative procedure that involves two operators. The iteration process mimic the Picard-Lindelöf proof at the discrete level with a fixed-point iterative method. At each iteration of this process one order of accuracy is gained and after a finite amount of iterations the aimed accuracy is reached. Let us consider the initial value problem

y ′ (t) = f (y(t)), y(t 0 ) = y 0 , (59)
where y : R → R S and f : R S → R S , where f can be the semidiscretized operator presented above.

The first operator we introduce is L 1 , a low order approximation of the integral version of the ODE (59) and the solution of L 1 (y) = 0 is easy to obtain, e.g. explicit Euler method. The second operator, L 2 , is a high order discretization of the ODE and the solution of L 2 (y) = 0 might be difficult to obtain, as an implicit RK method. The DeC procedure combines the two operators exploiting the accuracy of L 2 and the simplicity of L 1 obtaining an explicit (or easy-to-solve) high order method. Consider a time step [t n , t n+1 ] and let us subdivide it into M sub-time steps {[t n,m-1 , t n,m ]} M m=1 where the boundary points coincide with the extrema of the time step, i.e., t n = t n,0 and t n,M = t n+1 . We consider a discretization of the variables in the sub-time nodes t n,m denoted as y n,m as explained in Figure 2. We define the L 2 operator as L 2 (y n,0 , . . . , y n,M ) :=

t n = t n,0 y n = y n,0 t n,1 y n,1 t n,m y n,m t n,M = t n+1 y n,M
       y n,M -y n,0 -∆t M r=0 θ M r f (y n,r ), . . . y n,1 -y n,0 -∆t M r=0 θ 1 r f (y n,r ), ≈        y n,M -y n,0 - t n,M
t n,0 f (y(s))ds, . . .

y n,1 -y n,0 - t n,1 t n,0 f (y(s))ds, ( 60
)
where θ m r is the integral of the r-th Lagrangian basis function defined on the sub-time nodes over the interval [t n,0 , t n,m ] [23, 1] normalized by the factor ∆t. This operator is a high order discretization of the integral form of the ODE in each sub-time step. Depending on the chosen sub-time nodes, the order of accuracy varies, for example, with equispaced nodes we can obtain a scheme with order M + 1, while with Gauss-Lobatto nodes we obtain a 2M -th accurate scheme, i.e., the Lobatto IIIA schemes [START_REF] Hairer | Solving ordinary differential equations[END_REF]. The system of equations L 2 = 0 is a (possibly) nonlinear system of equations which might be very complicated to solve as f is, in our case, the semidiscretization of a nonlinear PDE. That is why we introduce L 1 a simplification of the L 2 operator. The L 1 operator in this context consists in a explicit Euler discretization in each sub-time step, written as

L 1 (y n,0 , . . . , y n,M ) :=        y n,M -y n,0 -β M ∆tf (y n,0 ), . . . y n,1 -y n,0 -β 1 ∆tf (y n,0 ), ( 61
)
where β m = t n,m -t n,0 ∆t . The solution of L 1 = 0 is straightforward. To simplify the notation of the DeC iterative process, we define a vector of all the variables y at each sub-time node (we omit the superscript n as we will always refer to the time step [t n , t n+1 ]): y := (y n,0 , . . . , y n,M ) ∈ R M×S , such that (62)

L 1 (y) := L 1 (y n,0 , . . . , y n,M ) and L 2 (y) := L 2 (y n,0 , . . . , y n,M ).

(63)

Finally we can introduce the iterations of the DeC algorithm. The aim is to approximate the solution of the L 2 operator y * , i.e., L 2 (y * ) = 0, with successive iterations, where, at each iteration, we gain one order of accuracy. Let us denote the iterations with the superscript (k) for k = 0, . . . , K, so that each variable y (k) ∈ R (M+1)×S is composed by the entries y n,m,(k) ∈ R S with m = 0, . . . , M . The DeC algorithm reads DeC Algorithm y n,0,(k) := y(t n ), k = 0, . . . , K, y n,m,(0) := y(t n ), m = 1, . . . , M,

L 1 (y (k) ) = L 1 (y (k-1) ) -L 2 (y (k-1) ) with k = 1, . . . , K, ( 64
)
where K is the number of iterations that we want to compute. In particular, after K iterations the order of accuracy of the method will be the minimum between K and the accuracy of L 2 . As an example, to obtain order 5, we need K = 5 and for equispaced sub-time nodes M = 4 (for Gauss-Lobatto M = 3 suffices). A more explicit formulation of iteration step of (64) is the following

y n,m,(k) -y n,0 -∆t M r=0 θ m r f (y n,r,(k-1) ) = 0. ( 65
)
More details, proofs and properties of the DeC can be found in [START_REF] Veiga | DeC and ADER: Similarities, Differences and a Unified Framework[END_REF][START_REF] Abgrall | Relaxation deferred correction methods and their applications to residual distribution schemes[END_REF][START_REF] Torlo | Hyperbolic problems: high order methods and model order reduction[END_REF] and references therein.

Numerical Simulations

The arbitrary high order well-balanced WENO finite volume scheme based on Flux Globalization has been tested and validated on several test cases to assess convergence properties and performances. In Section 5.1 we study the lake at rest steady state case to verify the well-balancedness of the scheme. In Section 5.2 we add a perturbation to the previous tests to compare the quality of well-balanced schemes and not wellbalanced ones. In Section 5.3 we focus on more general steady state with nonzero global fluxes and we test the method on subcritical, supercritical and transcritical flows. Finally, in Section 5.4 we add also the friction to the previous tests.

For smooth tests we will perform a convergence analysis to assess the order of accuracy of our methods. In particular, we will validate the algorithm with WENO3 and WENO5 reconstructions.

Lake at rest

Firstly, we consider the lake at rest solution characterized by the initial data

h(x, 0) = 1 -b(x), q(x, 0) ≡ 0 (66)
over the computational domain [0, 25] with subcritical inlet/outlet at the two boundaries. The bathymetry employed contains a sinusoidal bump damped at the boundaries (see Figure 3 for better visualization of solution and bathymetry) and it reads b(x) = 0.05 sin (x -12.5) exp 1

-(x -12.5) 2 . ( 67
)
Let us remark that the classical bathymetry provided in [START_REF] Delestre | Swashes: a compilation of shallow water analytic solutions for hydraulic and environmental studies[END_REF] for many tests is only piecewise polynomial, but globally only C 0 . Hence, it is not suited to test the accuracy of very high order methods. On the other side, the bathymetry (67) is C ∞ and it has values smaller than machine precision at the boundaries. The gravitational constant is considered to be g = 1 and the simulation is run until the final time T = 1 with N e = {25, 50, 100, 150, 200, 400, 800} uniform cells using both the non-well-balanced (non-WB) and well-balanced (WB) version of the algorithm to assess the convergence and well-balancing properties. The convergence tests performed with WENO3 and WENO5 are listed in Table 1. It can be noticed that the error decay for the non-WB simulations matches the order of the reconstruction for both WENO3 and WENO5, while, for the WB cases, the scheme is able to preserve the exact solution up to machine precision.

Small perturbation of the lake at rest solution

For this test case, we analyze the perturbation of the lake at rest solution characterized by

h(x, 0) = 1 -b(x) + 10 -5 , if 9 < x < 10 0, otherwise , q(x, 0) ≡ 0 (68)
over the computational domain [0, 25] with subcritical inlet/outlet at the two boundaries. The bathymetry is a rescaling of (67) and it is defined as b(x) = 0.5 sin (x -12.5) exp 1 -(x -12.5) 2 . (

A slightly different bathymetry, with respect to Section 5.1, has been chosen in order to introduce more noise in the non-WB simulation and appreciate more the method capabilities. The simulation was run using a mesh with 150 cells. Figure 4 shows the evolution over time of the perturbation added over the lake at rest solution computed with the GF-WENO5 WB scheme. To better present the results, the plots show the relative variable hh eq where h eq represents the lake at rest solution without perturbation provided in Equation (66). We then compare the results in Figure 4 with those computed using the classical WENO5 approach, pointed out in Figure 5. It should be noticed that the classical approach fails at correctly reproducing the perturbation. Indeed, it generates a discretization error of several orders of magnitude higher than the perturbation itself, eventually spoiling the final result. Contrary to that, the GF-WENO5 WB scheme correctly reproduce the perturbation which splits into two waves traveling at opposite directions and interacting with the bathymetry. The obtained result is fairly accurate, also considering the coarse mesh used for this study.

Steady states without friction (n = 0)

Here, we test the method for some moving equilibria steady state problems. We run the tests up to convergence towards steady state in different situations. In the subcritical and supercritical tests the bathymetry is smooth and equal to (67), as we want to assess the high order accuracy of the schemes. For the transcritical tests we use a modification of the bathymetry used in [START_REF] Delestre | Swashes: a compilation of shallow water analytic solutions for hydraulic and environmental studies[END_REF], to study a very similar discontinuous problem. Depending on the initial and boundary conditions set at the borders of the domain, the flow may be supercritical, subcritical or transcritical. The meshes used are with N e ∈ {25, 50, 75, 100, 125, 150, 250, 500}. We consider the following three sets of initial and boundary conditions: • Supercritical flow

h(x, 0) = 2 -b(x), q(x, 0) ≡ 0, h(0, t) = 2, q(0, t) = 24, (70)
• Subcritical flow

h(x, 0) = 2 -b(x), q(x, 0) ≡ 0, q(0, t) = 4.42, h(25, t) = 2, (71)
• Transcritical flow b(x) =    0.2 exp 1 - 1 1-( |x-10| 5 ) 2 , if |x -10| < 5, 0,
else, h(x, 0) = 0.33b(x), q(x, 0) ≡ 0, q(0, t) = 0.18, h(25, t) = 0.33.

(72)

The gravitational constant is set to g = 9.812 for all these tests. For these three cases, we compare the results obtained using a classical WENO finite volume scheme and the new approach based on flux globalization. As already mentioned, for supercritical and subcritical cases, the bathymetry (67) allows us to perform convergence tests for very high order methods, when also the flow is smooth. For the transcritical case with shock, only a qualitative analysis of the test is performed. Hence, when supercritical and subcritical flows are of interest, we can study the convergence properties of the new scheme by finding the exact solution given by the non-linear equations taken from [START_REF] Delestre | Swashes: a compilation of shallow water analytic solutions for hydraulic and environmental studies[END_REF]. Both the WB and non-WB versions of the scheme have been run to compare the influence of the formulation on the ability of preserving the balanced steady state solution. Finally, we also run the same test cases with the classical WENO3 and WENO5 schemes.

Convergence curves for supercritical and subcritical flows are depicted in Figures 6 and7, respectively. All curves, both for WENO3 and WENO5, show the correct third and fifth order accuracy. However, it should be noticed that the GF formulation allows a much better prediction of the solution characterized by a discretization error that is by far lower compared to the classical method. Introducing then the WB formulation allows even better convergence trends with slightly lower errors with the advantage of preserving also the lake-at-rest solution. In some cases, e.g. supercritical flow GF-WENO5 WB and GF-WENO3 non-WB, we observe a superconvergence of the methods, which might be caused by the extra constraints that the flux globalization and WB techniques impose. The behavior is still under investigation.

After the convergence analysis, we focus on a more qualitative analysis of the results computed with the WENO5 and GF-WENO5 methods. To do so we introduce a set of four representative variables that we are going to use to compare the classical WENO5 with the new GF-WENO5 WB. The three main variables that we used to assess the new method are η, q and K (K is only defined for GF formulation). In particular, it should be noticed that, by construction, the methods based on flux globalization preserve moving equilibria, i.e., q x ≡ 0, K x ≡ 0. This means that the GF-WENO5 will approximate these quantities up to the order of the residual of the time derivative at the end of the simulation. In our case, we are able to preserve the constant q and K up to ∼ 10 -9 . The last variable we decided to study is the aforementioned Υ, which corresponds to the smooth formulation of the more general global flux introduced herein. Although our approach is developed to preserve other equilibria, this variable allows to get more insights about the capability of the new algorithm since, for smooth flows, the analytical Υ should be constant at equilibrium. Solutions for supercritical and subcritical flows are shown in Figures 8 and9. For both cases, it is clear that q and K are well-preserved, and Υ is much better predicted with respect to the one computed through the classical WENO5 method. The test case that stands out more among the three situation considered here is the transcritical flow with shock in Figure 10. The WENO5 method introduces spurious oscillation where the shock occurs that are then propagated in the rest of the computational domain making the overall solution spoiled. By using the new GF-WENO5 approach, oscillations are not present and the correct solution is recovered before and after the shock. It can be noticed that, when discontinuous flows are of interest, Υ is not globally constant but features a jump where the shock occurs, see Figure 10.

Steady states with friction (n = 0.05)

In this section we focus on the supercritical and subcritical flows studied in Section 5.3 including the friction term in the source term. As for the previous cases, also when friction with constant Manning coefficient n is present, we can obtain moving equilibria. Again the quantity that are preserved at equilibrium are q and K. That is why it is interesting to perform simulations similar to the previous ones comparing standard methods with GF ones. We consider the subcritical case defined in (70) and the supercritical case defined in (71) with the same bathymetry, on which we add the friction term with Manning coefficient n = 0.05. In the supercritical case, the friction term implies a slow down of the physical speed from left to right and a consequent increasing of η from left to right. In the subcritical case, conversely, we expect h to decrease from left to right and the speed to increase. The variable Υ is not conserved and, for these tests, there is not another constant variable that can be easily computed analytically, hence, we do not plot it. We display in Figures 11 and12 both the solutions computed with the new GF-WENO5 WB scheme and the classical WENO5. It should be noticed that both schemes obtain valid and consistent results with the expected solution. The difference between the schemes is remarkable and the global flux variables clearly highlights it. The WENO5 case, without the global flux, is characterized by strong spurious oscillations around the area where the effect of the bathymetry is stronger. On the other side, the GF-WENO5 results are very precise and are able to preserve the global flux variables up to ∼ 10 -9 .

Conclusion

In this paper, we presented a novel arbitrary high-order well-balanced finite-volume method based on flux globalization for the shallow water equations with source terms. The high order accuracy is obtained through a WENO reconstruction performed on the variables of interest, i.e. free surface level and global fluxes. The flux globalization allows to exactly preserve the constant fluxes in moving water equilibria. The new scheme has been also designed to preserve the "lake-at-rest" solution. This was possible by introducing a particular quadrature procedure for the source flux and considering a jump of the global fluxes at each interface. Several tests have been performed to assess the properties of the scheme. The preservation of the constant flux has been verified on academic moving water test cases (subcritical, transcritical and supercritical), while the well-balancedness with respect to the lake at rest equilibrium has been tested on the classical "lake-at-rest" and "perturbed lake-at-rest" cases. Once the space discretization is well implemented, the introduction of additional source terms is straightforward, e.g. Manning friction, as demonstrated in the simulation section.

With the presented scheme, we were able to outperform classical methods in many situations, obtaining much more accurate solutions and useful properties also at the discrete level. Moreover, the high order accuracy of scheme allows to obtain precise solutions also when the equilibria are not reached.

There are several extensions to this work that could be carried out. The authors are already studying a DGSEM-like formulation to simplify the global flux procedure, more complicated balance laws where other moving equilibria are object of interest, multi-dimensional problems on which the global flux must be carefully defined, and path-conservative strategies.

If (74) is satisfied, then we have to deal with two possibilities. First, if q L i+1/2 = 0, we obtain the unique positive solution

h L i+1/2 = 2 K L i+1/2 -R L i+1/2
g , while if q L i+1/2 = 0, we solve Eq. ( 73) for h L i+1/2 and obtain the following three solutions:

h L i+1/2 = 2 √ P cos 1 3 [Θ + 2πk] , k = 0, 1, 2, (76)
where

P := 2 K L i+1/2 -R L i+1/2
3g and Θ := arccos

  - q L i+1/2 2 g P 3/2    . ( 77
)
It can be shown that one of these roots is negative, whilst the other two roots, corresponding to the subcritical and supercritical cases, are positive. We choose the one closer to the corresponding value of h L i+1/2 given in (75).

Figure 1 :

 1 Figure 1: Shallow Water Equations: definition of the variables.

Figure 2 :

 2 Figure 2: Time interval divided into sub-time steps

Figure 3 :

 3 Figure 3: Lake at rest solution: η (red) and b (blue).

5 Figure 4 :

 54 Figure 4: Small perturbation of the lake at rest solution computed with the GF-WENO5 WB scheme: hheq (red) and rescaled b (blue).

5 Figure 5 :

 55 Figure 5: Small perturbation of the lake at rest solution computed with the WENO5 scheme: hheq (red) and rescaled b (blue).

Figure 6 :

 6 Figure 6: Supercritical flow: convergence tests with WENO3 and WENO5.

Figure 7 :

 7 Figure 7: Subcritical flow: convergence tests with WENO3 and WENO5.

Figure 9 :

 9 Figure 9: Subcritical flow: relevant variables computed with GF-WENO5 (red continuous line) and WENO5 (black dashed line) schemes.

Figure 10 :

 10 Figure 10: Transcritical flow: relevant variables computed with GF-WENO5 (red continuous line), WENO5 (black dashed line) schemes and b (blue dotted line).

Figure 11 :

 11 Figure 11: Supercritical flow with friction: relevant variables computed with GF-WENO5 (red continuous line) and WENO5 (black dashed line) schemes.

Figure 12 :

 12 Figure 12: Subcritical flow with friction: relevant variables computed with GF-WENO5 (red continuous line) and WENO5 (black dashed line) schemes.

Table 1 :

 1 Lake at rest: errors and estimated order of accuracy (EOA) with WB and non-WB schemes and WENO3 and WENO5 reconstructions.

			Non-WB			WB
		h		q		h		q
	Ne	L2 error	EOA	L2 error	EOA	L2 error	EOA	L2 error	EOA
			GF-WENO3			GF-WENO3
	25	1.0384E-4	-	4.7943E-5	-	9.8858E-14	-	1.2228E-15	-
	50	1.5496E-5	2.67	9.2488E-6	2.31	9.8667E-14	-	1.4249E-15	-
	100	1.2117E-6	3.62	3.6777E-7	4.59	9.8276E-14	-	1.6041E-15	-
	150	2.6776E-7	3.69	1.5898E-7	2.05	1.9644E-13	-	3.3908E-15	-
	200	9.6323E-8	3.53	7.6469E-8	2.53	1.9619E-13	-	3.6713E-15	-
	400	8.2671E-9	3.53	6.0441E-9	3.65	2.9360E-13	-	6.1689E-15	-
	800	6.8811E-10	3.58	4.7122E-10	3.67	5.8655E-13	-	1.3035E-14	-
			GF-WENO5			GF-WENO5
	25	5.1800E-5	-	6.1657E-5	-	9.8947E-14	-	1.3247E-15	-
	50	4.4066E-6	3.45	1.5244E-6	5.18	9.8661E-14	-	1.4060E-15	-
	100	6.7998E-7	2.66	3.5908E-7	2.06	9.8289E-14	-	1.5992E-15	-
	150	1.5437E-7	3.63	8.8535E-8	3.42	1.9639E-13	-	3.4157E-15	-
	200	4.1973E-8	4.50	2.3725E-8	4.55	1.9611E-13	-	3.7034E-15	-
	400	1.3952E-9	4.89	7.5991E-10	4.95	2.9357E-13	-	6.2007E-15	-
	800	4.3120E-11	5.01	2.2633E-11	5.06	5.8648E-13	-	1.3039E-14	-

Acknowledgements D. T. has been funded by a postdoctoral fellowship in the CARDAMOM team in INRIA-Bordeaux Sud-Ouest and by a postdoctoral fellowship in SISSA. M. Ciallella is funded by an Inria PhD fellowship in the CARDAMOM team in INRIA-Bordeaux.

A Positive reconstruction of water height at interfaces

From the cell average a high order WENO reconstruction is performed on the fluxes to have q L,R i+1/2 and K L,R i+1/2 . Equipped with q L,R i+1/2 , K L,R i+1/2 , R L,R i+1/2 , the point values h L,R i+1/2 can be obtained by solving the nonlinear equation coming from the definition of the global variable K in (3):

Let us solve the depressed cubic equation ( 73) for h L i+1/2 (the solution for h R i+1/2 is obtained with the same procedure). First of all, it can be noticed that (73) does not have any positive solution unless the determinant is greater than zero, meaning that

If (74) is not satisfied we reconstruct η L i+1/2 and then compute h L i+1/2 given that


Mathieu Beaudouin

Keywords: Tangut, agreement, syntax, dependency, clause chaining, topic/comment, modality

. Second, cases of non agreement are analyzed. Apart from the case of nonlocal contexts, we see that the absence of agreement occurs in nonfinite forms resulting from a dependency pattern, such as clause chain ing, topic/comment context, and semantically dependent modality.

Introduction

The rules of agreement

As known since [START_REF] Kepping | Subject and object agreement in the Tangut verb[END_REF], the Tangut verb encodes a sizeable quantity of in formation at both sides of the root. Preverbs (traditionally called "prefixes") to the left of the root provide knowledge regarding orientation and TAME. 2 To the right of the root, one notably finds suffixes carrying agreement with the arguments of the verb. These display forms responding to different types of indexation systems.

Intransitive verbs

With intransitive verbs, the scheme is quite simple: the verb, having just one unique argument (the subject), agrees with that unique argument. The suffix will be -ŋa² 𗧓 for the first person singular, -nja² 𘉞 for the second person singular, and -nji² 𗐱 for the plural of both the first and second person (examples 1a to 1c). 3 (1) a. 𘘮 𘙊 𘙊 𗄼 𗧓 •jɨ² QUOT 'They [used to] say: "We are the descendants of Tai Bo".' (Leilin 04.33.A.3, cited in Shi et al. 1993: 275) The verb with a thirdperson subject is to be treated apart, as it never shows agreement, as seen in example [START_REF]西夏語の研究 -西夏語の再構成と西夏文字の解読 Seikago no kenkyū -Seikago no saikōsei to seika moji no kaidoku [Tangut Studies -Decipherment of the Tangut Script and Reconstruction of the Tangut Language[END_REF]. Finally, one must note that the occurrence of reported speech does not impede agreement, as shown by examples (1c) and (3).4

3 Before Kepping's seminal work, the main (and inexact) interpretation regarding two of these suffixes was that they were related to modality. Nevsky (1960, posth.: 146-147) saw in 𗐱 -nji² and 𘉞 -nja² suffixes indicating assertion, in the manner of Japanese べし beshi, ヨ yo or ヤ ya, an interpretation Nishida (1964Nishida ( -1966) ) accepted with some reservations. Sofronov (1968: 217-218) stayed inside this modal explanational frame, and described suffixes encoding an "appellative mood" ("aпеллятивное наклонение"). He also conjectured that the choice between 𗐱 -nji² and 𘉞 -nja² was related to a hierarchy between the speaker and the addressee. The first person suffix 𗧓 -ŋa² was not mentioned at all until Kepping, probably because of the graphic indifferentiation with the homograph pronoun 𗧓 ŋa². •jɨ² QUOT 'The servants said: "You are a benevolent emperor".' (Leilin 03.10B.1, cited in Shi et al. 1993: 259)

Transitive verbs

With transitive verbs, the pattern is far more complex. In local scenarios (between first and second person), agreement always occurs with the patient, and alignment is of an ergativeabsolutive nature (𗟻𗧓 pʰji¹ŋa² 'You send me', 𗟻𘉞 pʰji¹ nja² 'I send you'). However, in mixed contexts (i.e., in the interaction between first/second and third person), the agreement occurs with the first or second person, no matter its syntactic role (i.e., agent or patient), the only difference being in the theme of the root: A (normal) for 3 > 1/2 configurations, B for 1/2 > 3 ones. In example (4) (mixed context, 2 > 3), the agreement is with the agent, conversely to example (5) (mixed context, 3 > 1), where the agreement is with the patient.

(4) It is also to be noted, as already observed by [START_REF] Kepping | Тангутский язык: Морфология [The Tangut Language: Morphology[END_REF], that the possessor of a possessed argument often triggers agreement in the verb. In example (7), the verb agrees with the hand's owner, i.e., the woman, and not the hand itself, as such a coreference would not cause agreement. The Tangut verb's complete paradigm is summarized in figure 1 (mixed contexts where Stem B occurs are colored in orange). 'The woman told the king: "There is someone among the persons in the hall who grabbed my hand."' (Leilin 07.22.A.3, cited in Shi et al. 1993: 302)

The dual suffixes

Tangut also has two suffixes indexing dual arguments: the first person 𘙌 -kjɨ¹ (ex ample 8a), and the second person 𘂆 -tsjɨ¹ (example 8b). [START_REF] Nishida | 西夏語文法新探 Xīxiàyǔ wénfǎ xīntàn[END_REF] was the first to reveal the existence of the first person dual suffix and since then, Arakawa (2018) and Zhang (Ms) found new examples, together with occurrences of a sec ond person suffix for the latter. Note that in the two examples given below, the coreference is enforced by pronouns. The place occupied by 𘂆 tsjɨ¹ in example (8b), between the stem and the inferential 𗭪 sji², is also entirely consistent with the Tangut template order, as shown in [START_REF] Jacques | The structure of the Tangut verb[END_REF]. As the quantity of exam ples available for these two dual suffixes is still too scarce (a sufficient amount of contexts of direct speech with a dual argument should be constituted first), they are not included in the analysis for the moment.

Is agreement optional or mandatory?

The two examples given below can illustrate the question of whether or not agree ment is optional or mandatory. As one can see, in (9a), all verb forms are indexed with a personal agreement suffix, whereas in (9b), none are marked.

Facing such diversity, the positions held by scholars have been subject to evo lution. Awareness of Tangut verbal agreement is as stated above relatively recent, as the discovery of the phenomenon dates to [START_REF] Kepping | Subject and object agreement in the Tangut verb[END_REF], and the explanation of stem alternation to [START_REF] Gong | 〈西夏語動詞的人稱呼應與音韻轉換〉 Xīxiàyǔ dòngcí de rénchēng hūyìng yǔ yīnyùn zhuǎnhuàn [Rhyme alternation and personal agreement in the Tangut verb[END_REF]. Since then, some scholars have expressed the view that agreement was optional (Ahrens 1990, LaPolla 1992b 6 ). [START_REF] Jacques | Tangut, Gyalrongic, Kiranti and the nature of person indexation in Sino-Tibetan/Trans-Himalayan[END_REF], on the other hand, has proposed that indexation should be seen as mandatory with transitive verbs, nonfinite exceptions being found in "converbal chains". ( 9 Shi et al. 1993: 279) The view I support here is that agreement is primarily mandatory not only in transitive verbs, but also in intransitive ones, and that the verbal forms not fol lowing the pattern above are nonfinite. The key proof here is that if the cases of nonagreement can be explained by the presence of invariant contexts which, typologically speaking, also often accompany nonfinite forms, agreement should be seen as mandatory.

The primary material of the present analysis consists of two xyloɡraphs, the 𗴮𘊳 Djịj¹ bo¹ ('Forest of Categories', also known as 類林 'Lèilín', edition of [START_REF] Shi | 類林研究 Lèilín yánjiù [Study of Leilin[END_REF]) and the 𗰗𗍫𗂧 Ɣạ² njɨɨ¹ lhjịj ('Twelve Kingdoms', edition of

Clues to the mandatoriness of agreement

In dependent clauses, the emergence of a nonfinite form is something that can happen in a language as familiar as English, in a sentence like 'Who is the boy writing on the desk?'. In the relative clause, the action nominal 'writing' is used in place of the structure with the relative pronoun 'who' in 'Who is the boy who is writing on the desk?'.

In Tangut, a verb inside a syntactically dependent clause (as a relative clause or a completive clause) displays agreement. In example (10) which cites a story from the 左轉 Zuǒ Zhuàn (Lord Xuan, 15), the verb inside the headinternal rela tive clause 𗍳𗴷𗴺𗂬𗶹𗠔𘉞 (between brackets) agrees with the secondperson agent of the verb 𗂬𗶹𗠔 'let marry' (that person also being the addressee of the whole sentence). I explain the absence of agreement of the copula of the first foreground clause in section 3.

(10) These two examples cannot prove the mandatory nature of Tangut verbal agree ment, but are still substantial clues. I am now going to analyze and categorize the cases where the Tangut verb does not agree with its argument(s).

𘒺 𘓐 𗔬 𗐪 𗗙 𘒣 𘘣 𗧓

Clause chaining dependency

The label "clause chaining" refers in the field of syntax analysis to the succes sion of foreground clauses, i.e., not subordinate to one another [START_REF] Dooley | Exploring clause chaining[END_REF] inside the same sentence, with an operator dependency (van Valin 2005), typolog ically prenuclear in SOV languages (i.e., with a controlling position on the right). Jacques (2016) used the term "converbal chain" to depict that phenomenon; how ever, I prefer the label "clause chaining," as converbs show a similar pattern, but not only-as I think to be the case in Tangut-limited to coordination, the defining parameter of converbal chains being the existence of a dedicated operator called the "converb" (Haspelmath & König 1995).

One could note that to validate the existence of clause chaining, the first thing to do is to prove coordination. Haspelmath 2007 proposed a set of conditions we will use visàvis example (12).

• no intercalation rule: no clause can be contained inside another; • temporal iconicity rule: the linear order must be in accordance with the chronology of the facts;

• no cataphora rule: the first coordinate clause cannot contain a reference that appears only in the second clause;

• no focus rule: no coordinate can be focused;

• no extraction rule: it must be impossible to extract an interrogative from the second coordinate clause to put it in the first position.

As one can see in example (12) which translates a story from the Book of the Later Han (Volume 27), all the conditions listed ahead apply to the phenomenon: first of all, all the clauses (separated by a coordinator in the translation) are inde pendent and cannot be contained by any of the other clauses; then the temporal iconicity is respected between each clause (the first clause referring to the past cause of the presence of the messenger, the second to his present state, and the last to a desire necessarily concerning a possible future); then none of the arguments of the first clause can be regarded as a cataphora of an argument first appearing in the second clause; furthermore, the focus on one clause would be strange, as neither of the other clauses could follow such a focus; eventually, the response to the question 'What are you holding?' definitely cannot be 'The situ sent me'. (12) Shi et al. 1993: 309) Thus, we are facing coordination, in a pattern similar to many languages which mark the last verb of a chain of nonfinite forms (as one of the uses of the て 'te' form in Japanese, or the "absolutive" case seen in Indian languages). The origi nality of Tangut lies only in the fact that the operator dependency concerns person, the agreement phenomenon being at the same time able to define finiteness.

In example (13), in which the pattern is similar to example (10) in the last sec tion (nonagreement in the first foreground clause), clause chaining explains once more the absence of indexation of 𗪆 sjwɨ¹. The verbs included in the comple tive clause are to be seen as generic forms not related to the parameter of person, maybe nominalized by 𘆄 njɨ². 08.12.B.7, cited in Shi et al. 1993: 309) Eventually, if one recalls the Tangut verb's usual agreement with a possessed argument (see example 7), clause chaining can shed new light on the semantic coreferences we have to establish. In example ( 14), 𗂟 should agree with the second person, and absence of indexation should then be seen as a consequence of clause chaining, and not of the fact that the agent (the king) is a third person. 'The king will no doubt cut your belly and pull the pearl out of it.' (Leilin 04.02A.2, cited in Shi et al. 1993: 267)

Conditional dependency

The conjunction 𗓱 tjij¹ usually precedes a nonfinite form. In a case such as in example ( 15), though, it is not easy to know for sure if it is a nonfinite form coreferent with the agent, or if it is a generic form similar to that of the following verb 𗒍𗍷 dzjwɨ²dzjɨj² (a hypothesis I favor here). Nevertheless, there are other examples, as in ( 16), where the knowledge of the context allows the establishment of a coreference between the subject and the verb, and where the absence of an agreement therefore has to be explained. (15) 'He sent someone at Tian Heng's place and made him say: "If you come, I will make you prince, but if you don't come I will certainly send someone to have you executed."' (Leilin 09.07.B.4, cited in Shi et al. 1993: 314) There is usually no agreement after the conjunction 𗓱 tjij¹, whose conditional structure can therefore be seen at the origin of a dependency which is itself at the origin of the absence of agreement. However, the indexation of the main clause's verb is a prerequisite for such a form to occur. In example (17), the verb follow ing the conjunction displays agreement, as the main clause's verb does not allow for the establishment of a coreference with the addressee/agent of the verb 𘝪𗈜 dźjow¹ka² 'move away'. (17)

Topic/comment dependency

As said above, indexation is probably primarily mandatory both in transitive and intransitive verbs. Table (1) illustrates this statement by listing all the forms of the copula ŋwu² in direct or semidirect speech in the Leilin.

I already analyzed the last form listed in the table as a consequence of clause chaining. However, the agreement of the copula 𘟂 ŋwu² seems to behave accord ing to parameters also related to the topic/comment (thematic/rhematic) structure of the clause. Example ( 18) is illustrative of what one can see as a topical depen dency: in Table (1), each time the copula doesn't agree with its subject, the subject is followed by the topical clitic 𗫂 =tja¹.

One can observe that a topic/comment structure can be coupled in two cases with clause chaining, as in example ( 19), where the last verb of the main clause should be understood as generic ('it is not possible'). In that case, it is still dif ficult to determine which one of the two processes is responsible for the lack of agreement. However, the explanation of nonindexation as a consequence of the appearance of the topical 𗫂 remains consistent if we look at the other cases. 07.16.B.4, cited in Shi et al. 1993: 301) There are still two cases that do not display agreement when the topical clitic does not appear. I treat the case of the interroɡative sentence (07.16.B.4) in the next section. Regarding the sentence 07.22.A.1 reproduced in example (20),9 if we con sider the possibility that the last phrase is really reported speech, I hypothesize that the nonfinite copula is due to a dependency similar to that of the topic/comment structure, but at an extraclausal level (or thetical, see [START_REF] Kaltenböck | Outside the clause: form and function of extra-clausal constituents Studies in Language Companion Series[END_REF] Shi et al. 1993: 302) This form of dependency of a verbal form on a preceding sentence exists, for example, in East Greenlandic Inuit, where it has been defined by Tersis (2010: 588) "as a form of situational dependency at the paragraph level." I reproduce her example in (21); one can see that the attributive, ordinarily used as a nounmodifier thirdperson participial (e.g., tikkak niitiq 'man eating', lit. man eatATTRIB.SG) also occurs at a discursive level in a dependent clause. In Tangut, as for clause chaining, the dependency only concerns the parameter of person and could be linked to the context of enunciation (example 9b of the introduction). 'The children are used to going to Tasiilaq, there they stay at the school.'

This isolated example can only lead to conjectures, and other occurrences will be needed to enforce this hypothesis. Still, the distribution pointed out for the copula 𘟂 ŋwu² is already able to show the role of the topic/comment structureat least at the intra-clausal level -in the occurrence of non-finite forms in Tangut.

Modal dependencies

Some non-indexed forms of the Tangut language seem potentially due to modality. For example, in imperative contexts, both indexation and non-indexation (example 22) seem possible. Jacques (2017) pointed out that the imperative in rGyalrong in the second person forms (singular 2 or 2 > 3) does not allow for the presence of an agreement affix. This distribution could lead to seeing nonindexation as a primer state in Tangut, and the indexed forms as resulting from analogy with other forms (in a process still at work at the epoch of Tangut texts).

However, the descriptions given by Lai (2017: 463) and Honkasalo (2019: 621) of the imperative in Khroskyabs (Wobzi) for the former and Horpa (Geshiza) for the latter show a strict [directional -verb -suffix of agreement] pattern. Knowing that these languages are closer to Tangut (Lai et al. 2020, Beaudouin accepted), such a distribution tends to point towards the loss of indexation as a feature acquired by Tangut. 'Give him back his rank and raise his emoluments higher than those he had before!' (12K, 132.26.02, cited in Solonin 1995: 40) In the last example (23), one can see that an interrogation with exclamative overtones can cause the same phenomenon. Note that the interrogation contains in that particular case a specific illocutionary force linked to the speaker's surprise, an element perhaps related to the imperative by the media of exclamation. Further investigation covering the impact of modality on the agreement in Tangut should certainly bring more discoveries. (23)

Conclusion

This overview of the limited contexts where agreement does not appear reveals the importance of dependency in our understanding of Tangut grammar. Indeed, dependency is the objective explicative tool that describes the computation of a coreference from the speaker's subjective point of view. This coreference is anaphoric in the case of topic/comment (perhaps thetic) structures or modal statements as imperative or interrogation, and cataphoric in the case of clause chaining and conditional dependency. Apart from these situations, all the verbs used in direct speech are indexed, and agreement should then be seen as primarily mandatory, non-agreement being limited to the specific contexts listed in the present study.

2 '

 2 come slowly.'(Leilin 07.21.A.5, cited in Shi et al. 1993: 302) b. 𗍳 𗟜 𘕿 𗞞 𗶷 𘉞 nji² Go quickly!'(Leilin 07.21.A.5, cited in Shi et al. 1993: 302) c. 𗧦 𗣊 𗕛 𘌜 𗗙 𘈷 𘃾 𘟂 𗐱

'

  He sat three times on the vehicle and three times got down from it.' (12K, 132.21.02, cited in Solonin 1995: 39)

  𘙌 𗅮 𗶧 𘇂 𘓐 𗖌 [𗧓] 𗗙 𗁅 𘙌 𗜈 𘃡 𗧓 PFV:IN-holding.LV:do[ᴀ]-1SG

( 14 )

 14 𘟙 𘙌 𗅆 [𘀍] 𗗙 𘀟 𗂟 𗊏 𘖗 𘃡 𘉞

  homehaveATTRIB3PL

( 22 )

 22 𗫸 𘉀 𘛃 𗞞 𘓯 𗣬ø 𗟶 𗡅 𗪘 𗸒 𗈪 𗁦 𗣬ø IMP:UP-raise=do[ʙ]ø

  5(8) a. 𗪯 𗦉 𗗙 𗤄 𘒣 𗧓 𗍏 𘒏 𗦮 𗑠 𘓂 𘙌 𗵽 𘙌 𘃞 𘘣

	gji²bjij²=•jij¹		ljɨ ̣¹kjɨ¹
	śjwo²kjɨ¹=ljɨ¹ be.beautiful1DU=EXCLAM	Junping=COMIT QUOT •jɨ²	which.one

wife=ANTIERG •jɨʳ¹dạ² ask ŋa² 1SG sju² Xu gja¹pʰjij¹=rjiʳ² '...he asked his wife: "Who from Xu Junping or me is more beauti ful?"' (12K, 132.51.06, cited in Solonin 1995: 45) b. 𗍳 𘆄 𗵐 𗵐 𗷝 𘏷 𘂆 𗭪 nji²njɨ² 2.HON-PL nej²nej² safely mjij²djii¹tsjɨ¹sji² NEG.PFV-divide-2DU-IFR 'It seems you two safely did not divide.' (Jinglü yixiang, Juan 15, cited in Zhang Ms)

  ) a. 𗍳 𗟜 𘕿 𗞞 𗶷 𘉞 𘘮 𘙊 𘙊 𗄼 𗧓

	nji² 2.HON	dzjɨr¹ɣa² fastLOC	dja²ɕjɨ¹nja² IMP-go 1 -2SG	mjo² 1.HUM	lwẹ²lwẹ²
	ɕji¹ before bji² servant nji²=rjir² 2.HON=COMIT •wji¹ø do[ᴀ]ø nji²=do² 2.HON=LOC lhjịj² country •o² alcohol dʑjwɨ¹ neighbour tjwij¹ cup khjɨ² tsə̣ ¹	sjij¹ today khu¹ø nji²=•jij¹ 2.HON=GEN offerø nji² zjọ² 2.HON life
	ten.thousand

slowly lja¹ŋa² come1SG 'Go quickly. Me, I'll come slowly.' (Leilin 07.21.A.5, cited in Shi et al. 1993: 302) b. 𗪘 𗍳 𗑠 𗂧 𘎆 𗫈 𗍳 𗗙 𗂸 𘃡 𗍳 𘋩 𗊖 𗞙 𗼒 𗍳 𘗽 𗕑 𘌞 𗄼 autumn lja¹ come 'Yesterday I was a neighbor country to you, and today I am your ser vant. I raise my cup to you and wish you a ten thousand autumns life.' (Leilin 05.17B.1, cited in

  [𗫂] { 𗍳 𗴷 𗴺 𗂬 𗶹 𗠔 𘉞 } 𗗙 𗔀 𘟂 𘘦 𗉝 𘐄 𗳒 𗍳 𗗙 𘘖 𘆏 𗆐 𗧓 Similarly, in the following sentence, perhaps written in semidirect speech, the verb inside the completive 𗼻𘎵𘐉𗐱 agrees with the semantic agent 𗂸. As one can see, the form used in the English translation is the infinitive, i.e., a nonfinite form.(11) 𘟙 𗂸 𗗙 { 𗼻 𘎵 𘐉 𗐱 } 𘘣 𘉞

	njij²	bji²=•jij¹	{	ljɨ ̣²	zjịj²	•wjạ²nji²	}	•jɨ²nja²
	king	servant=ANTIERG	{	soil	taxes	remit12PL	}	say2SG
	'King, tell the servants { to remit the taxes }.' (12K 132.46.06, left un
	translated in Solonin 1995)
	nar²dzjwo² old.man •jar² marry ɕji²phjo²nja² •we² Wei khjow²=•jij¹ Ke=ANTIERG go 2 CAUS[ʙ]-2SG }=•jij¹ }=GEN dạ²•jɨ² say •wja¹ father ŋa²=[tja¹] 1SG=[TOP] ŋwu²ø=tɕhjwo¹ { { nji² 2.HON [ɕjɨ² la²mja¹ aunt COPø=because [grass lhwa⁰]=ŋwu² tie]=INS nji²=•jij¹ ljị² ljɨɨ¹ ljịj²ŋa² 2.HON=ANTIERG favour give.back come[ʙ]1SG

'The old man said to Wei Ke: "I am the father of your aunt, 7 the one you made wed once more; due to that reason I returned your favour to you by fastening the grass [under the feet of the enemy].'

(Leilin 07.18.A.3, cited in Shi et al. 1993: 301) 8

  𗧓 𗫂 𗬐 𘊴 𘙇 𗷆ø 𗥌 𘝞 𗖌 𗜈ø 𗥑 𗕌 𗑠 𗄛 𗧠 𗧓

	ŋa²=tja¹ 1SG=TOP ber¹ kiẹj²ŋa² sə¹thu¹ situ	rjɨr²phjii¹ø PFV:DIR-send[ᴀ]ø	njij¹•jwɨr² letter	gjɨ² one	zow²ø holdø	xu¹ʑjĩ¹=rjir² wife=COMIT
	see	desire1SG
	'The situ sent me, I'm holding a letter and I want to see my wife.' (Leilin
	08.11.A6, cited in

  (13) { 𗒱 𗷁 𘈷 𗦉 𗤋 [𘆄] } 𗅋 𗪆ø 𗧀 𘓺 𗯴 𗦳 𗤋 𗗙 𗪆 𗧓

	{ { [ŋwər¹khju¹ ɕjwi¹ age dzaa¹ decrease [world dzju² gji²bjij² wife sovereign mjij¹]=•jij¹ mjij¹ NEG.COP NEG.COP]=ANTIERG [njɨ²] etc. } } mji¹sjwɨ¹ø NEG-preoccupyø sjwɨ¹ŋa² preoccupy1SG lew¹ only
	'I don't care about such things as getting old or not having a spouse; I
	just worry about the fact that there isn't a sovereign in that world.' (Leilin

  [𗓱] 𗅋 𘃡 𗌭 𘙌 𗅆 𗬩 𗖵 𗒍 𗍷

	[tjij¹] [if]	mji¹•wji¹ø NEG-do[ᴀ]ø	ku¹ then	kjɨ¹djɨj² necessarily	kie¹=bju¹ rules=according.to	dzjwɨ²dzjɨj² judge
	'If it is not done, then it will be judged in accordance to the rules.' (Leilin
	07.03A.5, cited in Shi et al. 1993: 297)
	(16)
	thjɨj¹	xiwəj¹=do²	dzjwo²	zeew²	bju²	ɕjɨ¹phji¹	[tjij¹]
	Tian	Heng=TERM	person	send	announce	go 1 CAUS[ᴀ]	[if]

𘄱 𘌴 𘋩 𘓐 𘒫 𘗒 𗶷 𗟻

[𗓱]

𗄼ø 𗌭 𘙌 𗅆 𘟙 𗣫 𘉁 𘛺 𗟻 𘉞

[𗅋]

𗄼ø 𗌭 𘙌 𗅆 𘓐 𘒫 𗳭 𗟻 𘉞

  𗓱 𘄴 𗂧 𗑠 𘝪 𗈜 𘉞 𗌭 𗾥 𗼻 𘍞 𗋂 𘏚 𗰓 𗁁 𘘣

	tjij¹	tsʰji¹	lʰjịj=rjir²	dźjow¹ka²nja²	ku¹	sja²	ljɨ ̣²•iọ¹	śjwo¹tjị²
	[if] ljọ² INTRG Qi	State=COMIT •wjij² EXV.on •jɨ² QUOT	separate2	then	Xie	land	useNMLZ

'If you move away from Qi's State, then what will be the use of Xie's land?

' (12K, 132.33.04, cited in Solonin 1995: 42)

Table 1 :

 1 Agreement of 𘟂 ŋwu² COP in the Leilin (18) 𘍑 𗍾 𗗙 𗞞 𗤄 𘍑 𗍾 𘒣 𗧓 [𗫂] 𗷺 𗾔 𘚐 𗝠 𘕰 𗯴 𗠫 𘓐 𘟂 𘘣 𗠫 𘓐 𘒣 𗧓 [𗫂] 𘄴 𗂧 𘓐 𘟂 𗤳 𘍑 𗦻 𗍾 𘝞 𗫦 𗶹 𘕕 𗤒 𗨻 𗫈 𗆮 𗧠 𗡅 𘒑 𗞞 𗋃 𗖵 𗰜 𗂧 𗦍 𘃡 𘅇 𗩱

	Form	Finiteness Affixes	Context	Ref.
	2SG 2SG 2SG 2SG 1SG 1SG 1.PL 1SG 2SG 1SG 1.PL	F F F F NF NF F NF NF NF F	𘉞 nja² main 𘉞 nja² main 𘉞 nja² main 𘉞 nja² completive / comment / comment 𗐱 nji² main / comment / interrog. / comment / clause ch. 07.16.B.4 03.10B.1 03.10B.2 03.10B.6 03.11A.2 03.23A.4 03.35B.7 04.33A.4 07.17.A.6 07.16.B.4 𗐱 nji² main 07.21.B.7
	1.PL 1SG 1SG 1SG	NF F F NF	/ 𗧓 ŋa² 𗧓 ŋa² /	comment main (pron.) main (pron.) comment / clause ch. 07.18.A.3 07.22.A.1 08.11.A.7 08.13.B.4
	ljɨj¹ Ling sow¹ mulberry •wjij²=•jij¹ Zhe=ANTIERG sji¹phu²=khju¹ dja²•jɨr¹ PFV-ask tree=SUBE dʑjwiw² ljɨj¹ Ling starving dzjwo² •wjij² Zhe person	dạ² say ŋwu²ø ŋa²=[tja¹] 1SG=[TOP] COPø •jɨ² QUOT	mja¹ former	be² day
	'He asked Lingzhe; the latter answered: "I'm the starving man who was
	under the mulberry tree last time."' (Leilin 07.17.A.6, cited in Shi et al.
	1993: 301)
	(19)
	dʑjwiw² starving ljɨj¹ mjiij² dzjwo² person •wjij² dạ² say •jwɨr² ŋa²=[tja¹] 1SG=[TOP] ɣiew¹ tshji¹ Qi ɕji²ø lhjịj² country sọ¹ kjiw¹ dzjwo² person •we² ŋwu²ø COPø sjij¹	mə² clan.name lhjwo¹
	Ling, kiẹj²ø desireø name dzji¹dʑjwij² Zhe, literature food dja²sji¹=bju¹ study PFV-to.end=because go 2 ø three mər² year lhjịj² do origin country today X•wji¹ return X.do[ᴀ] tji¹njwi² (ø) PROHIB-can (ø)

  ). (20) 𗏁 𘓐 𘒣 𗷺 𗾔 𘆝 𘝚 𗇋 𘟂 𗐱 𘘣 [𗉔 𗤀 𘓐 𘟂 𘃞] 𘘣

	ŋwə¹ five ɕjij¹ dynasty dzjwo² person dzjwo² dạ² say person ŋwu²ø mja¹ be² rjijr¹ kjwɨɨr¹=mjijr² other day horse steal=NMLZ COPø ljɨ¹] ASSERT •jɨ² QUOT	ŋwu²nji² COP-12PL	•jɨ² QUOT	[tɕjiw¹ Zhou

First possible translation: 'We are the five people who stole the horse last time; we are from Zhou!' (Leilin 07.22.A.1) Alternative translation: "'We are the five people who stole the horse last time." They were from Zhou.'

(Leilin 07.22.A.1, cited in

  𗍳 𘕣 𗍊 𘓐 𘟂ø

	nji² 2.HON	•wa²=sju² INTRG=sort	dzjwo² people	ŋwu²ø COPø
	'What sort of person are you??' (Leilin 07.16B.3, cited in Shi et al. 1993: 301)

The transcription used in the present paper is the reconstruction of[START_REF] Gong | 西夏語文研究論文集 Xīxià yǔwén lùnwénjí [Collected papers on the Tangut language[END_REF], as given in Li (2008)'s TangutChinese dictionary.

The template follows an order shown by[START_REF] Jacques | The structure of the Tangut verb[END_REF] and includes ancient directional pre fixes, interrogation prefixes, negation prefixes, modal prefixes, and monosyllabic incorporated nouns.

The nature of reported speech in Tangut is probably semidirect (Type II of reported speech in Aikhenvald 2007), or hybrid[START_REF] Tournadre | Manuel de tibétain standard[END_REF], like in Horpa languages[START_REF] Jacques | Semi-direct speech in rGyalrongic languages: Rtau vs Japhug[END_REF], even if I translate it as direct speech in my examples. In example (3), a more

One can still note that these dual forms tend to argue in favor of the Tangut indexation system as a primary state, and not as a feature acquired through the grammaticalization of former pro nouns. Indeed, these suffixes, conversely to other agreement particles, do not have pronominal counterparts (apart from a bisyllabic interrogative pronoun 𘓂𘙌 ljɨ ̣¹kjɨ¹ where the second part 𘙌 kjɨ¹ can reasonably be assessed to be related to the first person dual; however in that case the pronoun itself could also be the result of a suffixation process).

[START_REF] Solonin | 𗰗𗍫𗂧 / Двенадцать царств [The Twelve Kingdoms[END_REF]. These two documents offer the advantage of being relatively close approximations of oral speech, with the style of the translation being quite distinct from the original Chinese text.6 This optionality was for LaPolla an argument to support his view of an 'isolating' protoSino Tibetan, of view today not shared by a majority of scholars in the field of SinoTibetan historical linguistics.

I translate here the main meaning of 𗴷𗴺, 'aunt' in Tangut, although 妾 in Chinese shows that 𗴷𗴺 refers here to a concubine of ego's deceased father.

I use the label "antiergative" (LaPolla 1992a) in place of "oblique" essentially because of the ability of the clitic 𗗙 =•jij¹ to mark the semantic object of a prototypic transitive verb, a distribution not incompatible with the canonical conception of oblique case (cf., differential object marking), but which adds complexity when the label "antiergative" confers 𗗙 •jij¹ a label coherent both syntactically and semantically.

It would have been easier not to consider this example, as it does not enter in the topic/comment structure at a clausal level, and as[START_REF] Shi | 類林研究 Lèilín yánjiù [Study of Leilin[END_REF] analyze the phrase 𗉔 𗤀 𘓐 𘟂 𘃞 tɕjiw¹ ɕjij¹ dzjwo² ŋwu² as a narrative component. However, the verb 𘘣 •jɨ² indicates reported speech quite consistently, while the assertive is often found in (semi)directspeech. I thus consider here the possibility of speech produced by the thieves themselves. More examples will be necessary to be completely sure of the exact meaning of this sentence.

rue de Lille, 75007 PARIS mathieu.beaudouin@inalco.fr

Abbreviations