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Research article

How to finance sustainable tourism: Factors influencing the attitude and willingness to pay green taxes among university students

  • Received: 09 July 2024 Revised: 21 August 2024 Accepted: 08 October 2024 Published: 01 November 2024
  • JEL Codes: H21, O23, L83, Z33

  • Green taxes are an instrument for the development of many destinations where overtourism generates different externalities, helping to alleviate them and create sustainable tourism. Funds raised through green taxes can be used to finance conservation, environmental restoration, and sustainable development initiatives. However, these taxes are often unknown to tourists visiting a city and can often generate mistrust and even discomfort when they are forced to pay them. In terms of the management implications for destinations, green taxes should be seen and conveyed as a means to achieve both economic and environmental sustainability of destinations and not yet another tax to be borne by the tourist. For this reason, the aim of this study is to explore the factors that affect both the positive attitude and the willingness to pay these taxes. Thus, the opinion about green taxes of 120 university students from different countries were collected to use a structural equation model (SEM) to try to provide answers to the different hypotheses put forward. Young people represent a growing part of the tourism market, shaping the trends and practices of the sector, making them central to the future of tourism. The study seeks to deepen theoretical knowledge on this subject and to provide a series of conclusions and recommendations for education regarding green taxes. In addition, our study on green taxes has a direct relationship with the sustainable development goals promulgated by the United Nations, as both seek to promote balanced economic, social, and environmental development.

    Citation: Cristina Ortega-Rodríguez, Julio Vena-Oya, Jesús Barreal, Barbara Józefowicz. How to finance sustainable tourism: Factors influencing the attitude and willingness to pay green taxes among university students[J]. Green Finance, 2024, 6(4): 649-665. doi: 10.3934/GF.2024025

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  • Green taxes are an instrument for the development of many destinations where overtourism generates different externalities, helping to alleviate them and create sustainable tourism. Funds raised through green taxes can be used to finance conservation, environmental restoration, and sustainable development initiatives. However, these taxes are often unknown to tourists visiting a city and can often generate mistrust and even discomfort when they are forced to pay them. In terms of the management implications for destinations, green taxes should be seen and conveyed as a means to achieve both economic and environmental sustainability of destinations and not yet another tax to be borne by the tourist. For this reason, the aim of this study is to explore the factors that affect both the positive attitude and the willingness to pay these taxes. Thus, the opinion about green taxes of 120 university students from different countries were collected to use a structural equation model (SEM) to try to provide answers to the different hypotheses put forward. Young people represent a growing part of the tourism market, shaping the trends and practices of the sector, making them central to the future of tourism. The study seeks to deepen theoretical knowledge on this subject and to provide a series of conclusions and recommendations for education regarding green taxes. In addition, our study on green taxes has a direct relationship with the sustainable development goals promulgated by the United Nations, as both seek to promote balanced economic, social, and environmental development.



    The dynamics of network flows is usually modelled by systems of Partial Differential Equations (briefly PDEs), most of time balance laws. The dynamics is defined on a topological graph with evolution on arcs given by system of PDEs, while additional conditions must be assigned at network nodes, e.g. conservation of mass and momentum. There is a large literature devoted to these problems and we refer to [5] for a extensive survey and for additional references.

    In particular, here we focus on water flows on a oriented network of open canals and the model given by Saint-Venant or shallow water equations. The latter form a non linear system of balance laws composed by a mass and momentum balance laws. In water management problems, these equations are often used as a fundamental tool to describe the dynamics of canals and rivers, see [1] and papers in same volume, and various control techniques were proposed, see [2,3,17,15,19,23,26] and references therein. Moreover, the need of dynamic models in water management is well documented, see [25]. The shallow water system is hyperbolic (except when water mass vanishes) and has two genuinely nonlinear characteristic fields. Moreover, it exhibits two regimes: fluvial or sub-critical, when one eigenvalue is negative and one positive, and torrential or super-critical, when both eigenvalues are positive. This is captured by the so called Froude number, see (1). For a complete description of the physics of the problem one needs to supply the equations with conditions at nodes, which represent junctions. The junction conditions are originally derived by engineers in the modeling of the dynamic of canals and rivers. The first and most natural condition is the conservation of water mass which is expressed as the equality between the sum of fluxes from the incoming canals and that from outgoing ones. One single condition is not sufficient to isolate a unique solution, thus different additional condition were proposed in the literature. Physical reasons motivate different choices of conditions, among which the equality of water levels, of energy levels and conservation of energy. For the assessment of coupling conditions on canals networks and for more details on the existence of solutions in the case of subcritical flows, we refer the reader to [9,18,16,20,14,22,24]. For discussion on supercritical flow regimes, we refer the reader to [18] and references there in.

    Then, to construct solutions one may resort to the concept of Riemann solver at a junction, see [13]. A Riemann solver at a junction is a map assigning solutions to initial data which are constant on each arc. Alternatively one may assign boundary conditions on each arc, but, due to the nonlinearity of equations, one has to make sure that boundary values are attained. This amounts to look for solutions with waves having negative speed on incoming channels and positive on outgoing ones: in other words waves do not enter the junction. A Riemann solver with such characteristics is called consistent, see also [12].

    In this paper we are interested in transitions between different flow regimes, when the transition occurs at a junction of a canals network. We assume to have incoming canals which end at the junction and outgoing canals which start at the junction. Thus we formulate a left-half Riemann problem for incoming canals and a right-half Riemann problem for outgoing canals to define the region of admissible states such that waves do not propagate into the junction. This corresponds to identify the regions where Riemann solvers can take values in order to be consistent. Such regions are enclosed by the Lax curves (and inverted Lax curves) and the regime change curves. To help the geometric intuition, we developed pictures showing such curves and the regions they enclose.

    The definitions described above and given in Section 4, are the necessary basis for an analysis on a complex network. Due to the complexity of the problem, we consider as case study the specific case of two identical canals interconnected at a junction (simple junction). We start focusing on conservation of water through the junction and equal height as coupling conditions. It is typically expected the downstream flow to be more regular, thus we consider three cases: fluvial to fluvial, torrential to fluvial and torrential to torrential. In the fluvial to fluvial case there exists a unique solution. However such solution may be different than the solution to the same Riemann problem inside a canal (without the junction) and may exhibit the appearance of a torrential regime. The torrential to fluvial case is more delicate to examine. Three different cases may happen: the solution propagates the fluvial regime upstream, the solution propagates the torrential regime downstream or no solution exits. Finally, in the torrential to torrential case, if the solution exists then it is torrential.

    To illustrate the achieved results we perform simulations using a Runge-Kutta Discontinuous Galerkin scheme [6]. The RKDG method is an efficient, effective and compact numerical approach for simulations of water flow in open canals. Specifically, it is a high-order scheme and compact in the sense that the solution on one computational cell depends only on direct neighboring cells via numerical fluxes, thus allowing for easy handling the numerical boundary condition at junctions. In the first example we show a simulation where an upstream torrential regime is formed starting from special fluvial to fluvial conditions. The second example shows how a torrential regime may propagate downstream.

    We conclude by discussing the possible solutions if the water height condition is replaced by the equal energy condition.

    The paper is organized as follows: in Section 2, we present the model starting from the one-dimensional shallow water equations. In Section 3, we give useful notations and preliminary results that allow to determine the admissible states for the half-Riemann problems discussed in the following Section 4. In Section 5 we study possible solutions at a simple junction for different flow regimes and different junction conditions. Finally, in Section 6 we illustrate the results of the previous section with a couple of numerical tests.

    The most common and interesting method of classifying open-channel flows is by dimensionless Froude number, which for a rectangular or very wide channel is given by the formula:

    Fr=|v|gh, (1)

    where v is the average velocity and h is the water depth. The three flow regimes are:

    Fr<1 subcritical flow or fluvial regime;

    Fr=1 critical flow;

    Fr>1 supercritical flow or torrential regime.

    The Froude-number denominator (gh)1/2 is the speed of an infinitesimal shallow-water surface wave. As in gas dynamics, a channel flow can accelerate from subcritical to critical to supercritical flow and then return to subcritical flow through a shock called a hydraulic jump, see [11] and references there in.

    We are interested in the transition between different flow regimes when it occurs at a junction of a canals network. On each canal the dynamics of water flow is described by the following system of one-dimensional shallow water equations

    (hhv)t+(hvhv2+12gh2)x=0. (2)

    The quantity q=hv is often called discharge in shallow water theory, since it measures the rate of water past a point. We write the system as:

    tu+xf(u)=0, (3)

    where

    u=(hhv),f(u)=(hvhv2+12gh2). (4)

    For smooth solutions, these equations can be rewritten in quasi-linear form

    tu+f(u)xu=0, (5)

    where the Jacobian matrix f(u) is given by

    f(u)=(01v2+gh2v). (6)

    The eigenvalues of f(u) are

    λ1=vgh,λ2=v+gh, (7)

    with corresponding eigenvectors

    r1=(1λ1),r2=(1λ2). (8)

    The shallow water equations are strictly hyperbolic away from h=0 and both 1-th and 2-th fields are genuinely nonlinear (λj(u)rj(u)0, j=1,2). Note that λ1 and λ2 can be of either sign, depending on the magnitude of v relative to gh, so depending on the Froude number.

    Solutions to systems of conservation laws are usually constructed via Glimm scheme of wave-front tracking [10,21]. The latter is based on the solution to Rieman problems:

    {tu+xf(u)=0,u(x,0)={ul if x<0,ur if x>0. (9)

    Here u(x,0)=(h(x,0),q(x,0)) and ul=(hl,ql) and ur=(hr,qr). The solution always consists of two waves, each of which is a shock or rarefaction, thus we first describe these waves.

    (R) Centered Rarefaction Waves. Assume u+ lies on the positive i-rarefaction curve through u, then we get

    u(x,t)={u for x<λi(u)t,Ri(x/t;u) for λi(u)txλi(u+)t,u+ for x>λi(u+)t,

    where, for the 1-family

    R1(ξ;u):=(19(v+2hξ)2127(v+2h+2ξ)(v+2hξ)2)

    for ξ[v+h,v+2h), and for the second family

    R2(ξ;u):=(19(v+2hξ)2127(v2h+2ξ)(v+2hξ)2)

    for ξ[λ2(u),).

    (S) Shocks. Assume that the state u+ is connected to the right of u by an i-shock, then calling λ=λi(u+,u) the Rankine-Hugoniot speed of the shock, the function

    u(x,t)={u if x<λtu+ if x>λt

    provides a piecewise constant solution to the Riemann problem. For strictly hyperbolic systems, where the eigenvalues are distinct, we have that

    λi(u+)<λi(u,u+)<λi(u),λi(u,u+)=q+qh+h.

    To determine a solution for problems on a network, we need to analyze in detail the shape of shocks and rarefaction curves and, more generally, of Lax curves (which are formed by joining shocks and rarefaction ones, see [4]). We start fixing notations and illustrating the shapes of curves.

    For a given point (h0,v0), we use the following notations for shocks and rarefaction curves:

    for h<h0,v=R1(h0,v0;h)=v02(ghgh0);for h>h0,v=S1(h0,v0;h)=v0(hh0)gh+h02hh0;for h>h0,v=R2(h0,v0;h)=v02(gh0gh);for h<h0,v=S2(h0,v0;h)=v0(h0h)gh+h02hh0. (10)

    Moreover, we define the inverse curves:

    for h>h0,v=R11(h0,v0;h)=v0+2(gh0gh);for h<h0,v=S11(h0,v0;h)=v0+(h0h)gh+h02hh0. (11)

    Similarly, we set:

    for h<h0,v=R12(h0,v0;h)=v0+2(ghgh0);for h>h0,v=S12(h0,v0;h)=v0+(hh0)gh+h02hh0. (12)

    We will also consider the regime transition curves: the 1-critical curve is given by

    v=C+(h)=gh (13)

    and the 2-critical curve by

    v=C(h)=gh. (14)

    In Figure 1 we illustrate the shape of these curves.

    Figure 1.  Shocks, rarefaction and critical curves 10-14 on the plane (h,v) (up) and on the plane (h,q) (down).

    To construct a solution to a Riemann problem (ul,ur), we define the Lax curves. Given a left state ul=(hl,ql) the Lax curve is given by:

    ϕl(h):=R1(hl,vl;h)S1(hl,vl;h). (15)

    For the right state ur=(hr,qr), we define the inverse Lax curve:

    ϕr(h):=R12(hr,vr;h)S12(hr,vr;h). (16)

    Remark 1. The Riemann problem for shallow water equations 2 with left state ul and right state ur has a unique solution if and only if the Lax curves ϕl(h) and ϕr(h) have a unique intersection. In that case, the intersection will be called the middle state um. As shown in Figure 2, the function v=ϕl(h) is strictly decreasing, unbounded and starting at the point vl+2ghl and, v=ϕr(h) is strictly increasing, unbounded, with minimum vr2ghr. Thus, the Riemann problem for shallow water has a unique solution in the region where

    vl+2ghlvr2ghr. (17)
    Figure 2.  Graph of ϕl and ϕr defined in 15 and 16 respectively.

    When working with (h,q) variables we use the following notations for Lax curves and regime transition curves:

    ˜ϕl(h)=hϕl(h),˜ϕr(h)=hϕr(h)and˜C+(h)=hC+(h),˜C(h)=hC(h).

    Moreover, for a given value (hi,vi) (or (hi,qi)) we set

    Fi=vighi,or˜Fi=qihighi. (18)

    In this subsection we study in detail the properties of the function q=˜ϕl(h). For a given left state ul=(hl,ql),

    ˜ϕl(h)={h(vl+2ghl2gh),0<hhl,h(vlg2hl(hhl)h+hlh),h>hl

    with

    limh0+˜ϕl(h)=0andlimh+˜ϕl(h)=.

    By computing its first and second derivatives,

    ˜ϕl(h)={vl+2ghl3gh,0<hhl,vlg2hl(4h2+hlhh2l2h(h+hl)),h>hl

    and

    ˜ϕl(h)={32gh,0<hhl,g2hl(8h3+12hlh+3h2lh+h3l4h(h+hl)h(h+hl)),h>hl,

    we can conclude that ˜ϕlC2(]0,+[). Specifically, in ]0,+[ ˜ϕl<0, ˜ϕl(h) is strictly decreasing and ˜ϕl is a strictly concave function. As

    ˜ϕl(0)=vl+2ghl and limh+˜ϕl(h)=

    we investigate two different cases:

    Case 1. If vl2ghl, ˜ϕl is a strictly negative decreasing concave function and specifically ˜ϕl(h)<˜C(h) for h>0. Therefore, in this case ˜ϕl never intersects the critical curves ˜C+ and ˜C.

    Case 2. If vl>2ghl, the function ˜ϕl admits a maximum point hmaxl, so it is increasing in (0,hmaxl) an decreasing in (hmaxl,+). In this case the function ˜ϕl intersects the two critical curves ˜C+ and ˜C. The intersection points vary with the choice of the left state ul, see Figure 3. To compute this points, we distinguish the following subcases: ul is such that 2Fl1 or such that Fl>1.

    Figure 3.  Graph of q=˜ϕl(h) for different values of left state ul and its intersections with critical curves q=˜C+(h) and q=˜C(h). The left state ul have been chosen such that: Fl>1 (dotted green line), |Fl|<1 (blue dashed line) and 2Fl<1 (red dotted line).

    Case 2.1. For 2<Fl1, it is the rarefaction portion of ˜ϕl to intersect the 1-critical curve ˜C+ at u+l,R=(h+l,R,˜C+(h+l,R)) with

    h+l,R=19g(vl+2ghl)2, (19)

    and we have that the maximum point hmaxl is such that ˜ϕl(hmaxl)˜C+(hmaxl), i.e. hmaxlh+l,R. In the special case Fl=1 we get hmaxlhl. Moreover,

    ˜ϕl(h)=0h=14g(vl+2ghl)2.

    Notice that for 2<Fl1 the rarefaction portion of ˜ϕl intersects ˜C at ul,R=(hl,R,˜C(hl,R)) with

    hl,R=1g(vl+2ghl)2; (20)

    while for 1<Fl1, the shock portion of ˜ϕl intersects ˜C at ul,S=(hl,S,˜C(hl,S)) with hl,S given by the following condition:

    vl(hl,Shl)ghl,S+hl2hlhl,S+ghl,S=0; (21)

    Case 2.2. For Fl>1, the maximum value of ˜ϕl is reached by the shock portion and so hmaxl>hl. The shock portion intercepts the critical curve ˜C+ at u+l,S=(h+l,S,˜C+(h+l,S)) with

    h+l,Ssuch thatvl(h+l,Shl)gh+l,S+hl2hlh+l,Sgh+l,S=0; (22)

    and the curve ˜C at ul,S defined in 21.

    Notice that for hhl the equation

    ql=˜ϕl(h)=hS1(hl,vl;h)

    has two solutions:

    h=hlandh=hl=hl2(1+1+8F2l). (23)

    Moreover, for Fl>1, we have hl>h+l, where the height h+l=(q2l/g)13 is given by the intersection between ˜C+ and the horizontal line q=ql. That is:

    (q2lg)13<hl2(1+1+8F2l),forFl>1.

    Indeed, using the relation q2l=gh3lF2l, if Fl>1 then 2F43lF23l1>0. So, for (hl,ql) supercritical, the point (hl,ql) is subcritical, i.e. |Fl|<1.

    Here we study the properties of the function ˜ϕr(h), given by:

    ˜ϕr(h)={h(vr2ghr+2gh),0<hhr,h(vr+g2hr(hhr)h+hrh),h>hr.

    By straightforward computations we get its derivatives:

    ˜ϕr(h)={vr2ghr+3gh,0<hhr,vr+g2hr(4h2+hrhh2r2h(h+hr)),h>hr.
    ˜ϕr(h)={32gh,0<hhr,g2hr(8h3+12hrh+3h2rh+h3r4h(h+hr)h(h+hr)),h>hr.

    Then, ˜ϕr(h)C2(]0,+[) and in ]0,+[ ˜ϕl>0, ˜ϕl(h) is strictly increasing and ˜ϕl is a strictly convex function. As

    ˜ϕr(0)=vr2ghr and limh+˜ϕr(h)=+

    we investigate two different cases:

    Case 1. If vr2ghr, ˜ϕr is a strictly positive increasing convex function and specifically ˜ϕr(h)>˜C+(h), for h>0. Therefore, ˜ϕr never intersects the critical curves ˜C+ and ˜C.

    Case 2. If vr<2ghl, the function ˜ϕr admits a minimum point hminr. In this case the function ˜ϕr intersects the two critical curves ˜C+ and ˜C. As done before we distinguish two subcases:

    Case 2.1. For 1<Fr2, the rarefaction portion of ˜ϕr intersects the 2-critical curve ˜C at ur,R=(hr,R,˜C(hr,R)), with

    hr,R=19g(vr+2ghr)2 (24)

    and we have that the minimum point hminr is such that ˜ϕr(hminr)˜C(hminr), i.e. hminrhr,R. In the special case Fr=1 we get hminrhr. Moreover,

    ˜ϕr(h)=0h=14g(vr+2ghr)2.

    For 1<Fr2 the rarefaction portion of ˜ϕr intersects ˜C+ at u+r,R=(h+r,R,˜C+(h+r,R)), with

    h+r,R=1g(vr+2ghr)2, (25)

    while for 1<Fr1, the shock portion of ˜ϕr intersects ˜C+ at u+r,S=(h+r,S,˜C+(h+r,S)), with h+r,S such that

    vr+(h+r,Shr)gh+r,S+hr2hrh+r,S+gh+r,S=0. (26)

    Case 2.2. For Fr<1, the minimum value of ˜ϕr is reached by the shock portion and so hminr>hr. The shock portion intercepts the critical curves ˜C at ur,S=(hr,S,˜C(hr,S)), with

    hr,S such that vr+(hrh+r,S)ghr,S+hr2hrhr,S+ghr,S=0

    and curve ˜C+ at u+r,S given by 26.

    Notice that the equation

    qr=˜ϕr(h)=hS12(hr,vr;h),hhr,

    has two solutions:

    h=hrandh=hr=hr2(1+1+8F2r). (27)

    Moreover, for Fr<1 we have hr>hr=(q2r/g)13, where the height hr is given by the intersection between ˜C and the horizontal line q=qr. So, for (hr,qr) supercritical, the point (hr,qr) is subcritical.

    [The case of an incoming canal] We fix a left state and we look for the right states attainable by waves of non-positive speed.

    Fix ul=(hl,ql), we look for the set N(ul) of points ˆu=(ˆh,ˆq) such that the solution to the Riemann problem

    {tu+xf(u)=0,u(x,0)={ul if x<0ˆu if x>0 (28)

    contains only waves with non-positive speed. We distinguish three cases:

    Case A: the left state ul is such that |˜Fl|<1;

    Case B: the left state ul is such that ˜Fl>1;

    Case C: the left state ul is such that ˜Fl<1.

    For this case we refer to Figure 4.We identify the set NA(ul) as the union of three regions IA1, IA2 and IA3 defined in the plane (h,q). The first region is identified by all points that belong to the curve ˜ϕl(h) such that ˜C(h)˜ϕl(h)˜C+(h), i.e.

    IA1={(ˆh,ˆq):h+l,Rˆhhl,S,ˆq=˜ϕl(ˆh)}, (29)
    Figure 4.  Left-half Riemann problem, Section 4.1. Region NA(ul)=IA1IA2IA3 defined by 29-31. Following our notation ˜S2(ul,S;h)=hS2(hl,S,C(hl,S);h).

    where the points h+l,R and hl,S are given in 19 and 21 respectively. The second region is defined as follows

    IA2={(ˆh,ˆq):0<ˆhhl,S, ˆqˆhS2(hl,S,C(hl,S);ˆh)}{(ˆh,ˆq):ˆh>hl,S, ˆq˜C(ˆh)}. (30)

    The last region IA3 is defined by the set of all possible right states ˆu that can be connected by a 2-shock with non-positive speed to an intermediate state um lying on ˜ϕl(h) curve such that |˜Fm|1 and

    λ(um,ˆu)=qmˆqhmˆh0.

    To define this region, we have to look for values q=hS2(hm,vm;h) as h<hm such that qqm. That is,

    qmq=(hmh)(vm+g2hmh(h+hm))0,h<hm.

    This inequality is verified for 1Fm<0 and for all hhm with hm given by

    hm=hm2(1+1+8F2m).

    We obtain (see Figure 4),

    IA3={(ˆh,ˆq): for all (hm,qm) which vary on ˜ϕl such that 1˜Fm<0,  0<ˆhhm, ˆq=ˆhS2(hm,vm;ˆh)}. (31)

    For this case we refer to Figure 5. It is always possible to connect the left value ul to a value ul by a 1-shock with zero speed (vertical shock). Specifically, we set hl>hl such that ql=ql, i.e.

    hl=12(1+1+8F2l)hl
    Figure 5.  Left-half Riemann problem, Section 4.1. Region NB(ul)=I,A1I,A2I,A3 given in 32.

    as previously computed in 23. Moreover, as previously observed at the end of subsection 3.0.1 the value (hl,ql) falls in the subcritical region, then

    NB(ul)=NA(ul){ˆu=(ˆh,ˆq):h+l,Sˆhhl, ˆq=˜ϕl(ˆh)}, (32)

    where NA(ul) falls under the previous Case A.

    For this case we refer to Figure 6. We have that: if Fl2 then

    NC(ul)={(ˆh,ˆq):ˆh>0, ˆq<˜C(ˆh)};
    Figure 6.  Left-half Riemann problem, Section 4.1. Region NC(ul) bounded by q=˜S2(ul,R;h) and q=˜C(h) as defined in 33.

    otherwise if 2<Fl<1 the intersection point ul,R between hR1(hl,vl;h) and ˜C(h) given in 20 is different from zero and defines the admissible region

    NC(ul)={(ˆh,ˆq):0<ˆhhl,R, q<ˆhS2(hl,R,vl,R;ˆh)}{(ˆh,ˆq):ˆh>hl,R, ˆq<˜C(ˆh)}. (33)

    [The case of an outgoing canal] We fix a right state and we look for the left states attainable by waves of non-negative speed. For sake of space the figures illustrating these cases will be postponed to the Appendix.

    Fix ur=(hr,qr), we look for the set P(ur) of points ˜u=(˜h,˜q) such that the solution to the Riemann problem

    {tu+xf(u)=0,u(x,0)={˜u if x<0ur if x>0 (34)

    contains only waves with non-negative speed. As in the previous case we identify three cases:

    Case A: the right value ur is such that |˜Fr|<1.

    Case B: the right value ur is such that ˜Fr>1

    Case C: the right value ur is such that ˜Fr<1.

    For this case we refer to Figure 12 in the Appendix. We identify the set PA(ur) as the union of three regions OA1, OA2 and OA3 defined in the plane (h,q). The first region is defined by all points that belong to the curve ˜ϕr(h) such that ˜C(h)˜ϕr(h)˜C+(h), i.e.

    OA1={(˜h,˜q):hr,R˜hh+r,S, ˜q=˜ϕr(˜h)}, (35)
    Figure 12.  Right-half Riemann problem, Section 4.2. Region PA(ur)=OA1OA2OA3 defined by 35-37 where ur is such that |˜Fr|<1.

    where the points hr,R and h+r,S are given in 24 and 26 respectively.

    The second region is such that

    OA2={(˜h,˜q):0<˜hh+r,S, ˜q˜hS11(h+r,S,v+r,S;˜h)}{˜hh+r,S, ˜q˜C+(˜h)}. (36)

    The third region is defined by the set of all possible left states ˜u that can be connected by a 1-shock with non-negative speed to a middle state um lying on ˜ϕr(h) curve such that |˜Fm|1 and

    λ(um,˜u)=qm˜qhm˜h0.

    To define this region we have to look for values q=hS11(hm,vm;h) for h<hm such that qmq. Then, following the same reasoning done for the left-half Riemann problem we get

    OA3={(˜h,˜q): for all (hm,qm) which vary on ˜ϕr such that 0<˜Fm1  0<˜hhm, ˜q=˜hS11(hm,vm;˜h)}. (37)

    For this case we refer to Figure 13 in the Appendix. If F2 then

    PB(ur)={(˜h,˜q): ˜h0, ˜q˜C+(˜h)};
    Figure 13.  Right-half Riemann problem, Section 4.2. Region PB(ur) bounded by q=˜S2(ul,R;h) and q=˜C+(h) as defined in 38 where ur is such that ˜Fr>1.

    otherwise, if 1F<2 the intersection point u+r,R between R12(ur;h) and C+(h), given in 24 is different from zero and defines the admissible region

    PB(ur)={(˜h,˜q): 0<˜hh+r,R, ˜q˜hS11(ur,R;˜h)}{(˜h,˜q),˜h>hr,R, ˜q˜C+(˜h)}. (38)

    For this case we refer to Figure 14 in the Appendix. It is always possible to connect the right value ur to a value ur by a 2-shock with zero speed (vertical shock). Specifically, we set hr>hr such that qr=qr, i.e.

    hr=12(1+1+8F2r)hr
    Figure 14.  Right-half Riemann problem, Section 4.2. Region PC(ur)=O,A1O,A2O,A3 given in 39 where ur is such that ˜Fr<1..

    as done in 27. Moreover, as previously observed at the end of subsection 3.0.2, the point (hr,qr) is subcritical and then

    PC(ur)=PA(ur){(h,q):hr,Shhr, q=˜ϕr(h)}, (39)

    where PA(ur) falls under the previous Case A.

    We consider here as case study a fictitious network formed by two canals intersecting at one single point, which artificially represents the junction. The junction is straight and separates two equal canals, one is the continuation of the other. This simple scenario appears to be like considering a problem for one straight canal, but by adding a fictitious junction we mimic a network and we provide the first analysis necessary for addressing more complicated networks for which the solution strongly depends on the given assumptions at the junction.

    We name the canals such that 1 is the incoming canal and 2 is the outgoing ones. We indicate by ub=(hb,vbhb)=(hb,qb) the traces at the junction. The flow is given by the one-dimensional shallow-water equations 2 in each canal coupled with special conditions at the junction. Our aim is to define and solve Riemann problems at the junction.

    A Riemann Problem at a junction is a Cauchy Problem with initial data which are constant on each canal incident at the junction. So, assuming constant initial conditions u01, u02 on canal 1 and 2 respectively, the Riemann solution consists of intermediate states ub1 and ub2 such that ub1N(u01) and ub2P(u02) and verifying given coupling conditions. Here, we are interested in evaluating possible solutions for different types of initial data belonging to different flow regimes. We start assuming, as junction conditions, the conservation of mass

    qb1=qb2 (40)

    and equal heights

    hb1=hb2. (41)

    In the following we study the boundary solution ub at the junction in the following interesting cases: AA, the water flow is fluvial in both incoming and outgoing canals around the junction; BA, the water flow is torrential with positive velocity in the incoming canal while it is fluvial in the outgoing one; BB, the water flow is torrential with positive velocity in both incoming and outgoing canals.

    Case AA (Fluvial Fluvial). Here we assume to have a left state ul and a right state ur such that |Fl|<1 and |Fr|<1. The solution at the junction consists of two waves separated by intermediate states ub1 and ub2 such that

    {ub1NA(ul),ub2PA(ur),qb1=qb2=qb,hb1=hb2, (42)

    with hb1,hb2>0.

    Proposition 1. Under the subcritical condition on ul and ur, the system 42 admits a unique solution.

    Proof. We distinguish two cases:

    Case 1. The two curves ˜ϕl and ˜ϕr intersect inside the subcritical region (Figure 7). In this case the solution is trivially the intersection point.

    Figure 7.  Case Fluvial Fluvial, system 42. In this case curves ˜ϕl and ˜ϕr intersect inside the subcritical region. The solution is the intersection point ub.

    Case 2. The two curves ˜ϕl and ˜ϕr do not intersect inside the subcritical region. If hr<hl (specifically h+r,S<h+l,R, see Figure 8) then the only point that verifies the junction conditions 42 is the critical point u+l,R given in 19, thus

    ub1=ub2=u+l,R. (43)
    Figure 8.  Case Fluvial Fluvial, system 42: curves ˜ϕl and ˜ϕr have empty intersection inside the subcritical region and hr<hl. The solution is the critical point u+l,R.

    If hr>hl (specifically hr,R>hl,S, see Figure 9) the only point that verifies the junction conditions 42 is the critical point ur,R given in 24, then

    ub1=ub2=ur,R. (44)
    Figure 9.  Case Fluvial Fluvial, system 42: curves ˜ϕl and ˜ϕr have empty intersection inside the subcritical region and hl<hr. The solution is the critical point ur,R.

    Remark 2. Notice that the proposed procedure may give a solution which is different from the classical solution of the Riemann problem on a single channel, given by the intersection point of ϕl and ϕr curves, see Remark 1.

    Case BA (Torrential Fluvial). Here we assume to have a left state ul and a right state ur with Fl>1 and |Fr|<1. The solution at the junction consists of two waves separated by intermediate states ub1 and ub2 such that

    {ub1NB(ul),ub2PA(ur),qb1=qb2=qb,hb1=hb2, (45)

    with hb1,hb2>0. Next Proposition provides the results about solutions, while the illustrating figures are postponed to the Appendix.

    Proposition 2. System 45 admits a solution if the two regions NB and PA intersect in the subcritical set {(h,q):h>0, ˜C(h)q˜C+(h)} or if ulPA(ur).

    Proof. We distinguish two cases:

    Case 1. The two curves hS1(hl,vl;h) and ˜ϕr(h) intersect inside the subcritical region (Figure 15 of the Appendix). If ulPA(ur) the solution is trivially the intersection point. On the contrary, if ulPA(ur) we have two possible solutions: the intersection point inside the subcritical region or the starting supercritical value ul. So, we may obtain both fluvial or torrential regime. Coherently with the solution that would be obtained in a single channel we choose the fluvial regime assigning as solution the intersection point inside the subcritical region.

    Figure 15.  Case Torrential Fluvial, system 45. In this case the curve hS1(h,v;h) and ˜ϕr(h) intersect inside the subcritical region. The solution is the intersection point ub..

    Case 2. The two curves hS1(hl,vl;h) and ˜ϕr(h) do not intersect inside the subcritical region. We distinguish the two subcases shown in Figure 16 and in Figure 17 of the Appendix.

    Figure 16.  Case Torrential Fluvial, system 45. In this case the curve hS1(h,v;h) and ˜ϕr(h) have empty intersection inside the subcritical region. This configuration is an example in which system 45 does not admit a solution..
    Figure 17.  Case Torrential Fluvial, system 45. In this case the curve hS1(h,v;h) and ˜ϕr(h) have empty intersection inside the subcritical region. The solution is the point ur,R.

    Case 2.1. Referring to Figure 16, the point ul may fall or not in the PB(ur) region. If ulPA(ur) the only possible solution is ul self and ub1=ub2=ul; if on the contrary ulPA(ur), system 45 does not admit a solution.

    Case 2.2. Referring Figure 17, if ulPA(ur) the only possible solution is the critical point ur,R defined in 24, then ub1=ub2=ur,R; If ulPA(ur), both ul self and the critical point ur,R are admissible solutions. Coherently with the solution obtained in the previous subcase, we select again ul as solution and we assign ub1=ub2=ul. So, the torrential regime propagates on the outgoing canal.

    Case BB (Torrential Torrential). As shown in Figure 18 of the Appendix, the two admissible regions NB and PB may have empty intersection. So, in this case there exists a solution at the junction which verifies the conditions of conservation of the mass and of equal heights, if and only if ulPB(ur), i.e. ub1=ub2=ul.

    Figure 18.  Case Torrential Torrential. In this case the two admissible regions NB and PB have empty intersection.

    Assuming different conditions at the junction give rise to new possible solutions. In canals network problems, it is usual to couple the conservation of the mass with the conservation of energy at the junctions. The specific energy E is a useful parameter in channel flow and it is defined as

    E=h+v22g. (46)

    For a given flow rate, there are usually two states possible for the same specific energy. Studying E as a function of h for constant q, there is a minimum value of E at a certain value of h called the critical depth,

    h=hc=(q2g)13. (47)

    Critical depth hc corresponds to some critical channel velocity vc defined by Fc=1. For E<Emin solution does not exists, and thus such a flow is impossible physically. For E>Emin solutions are possible: large depth with |F|<1 subcritical flow, and small depth with |F|>1 supercritical flow.

    In our case, assuming equal energy at the junction gives

    v212+gh1=v222+gh2. (48)

    Moreover, assuming qb1=qb2=qb to be constant (and known) we get from 48

    gh1F212+gh1=gh31F212h22+gh2,

    where F1=v1/gh1 and where we used the following relations

    v21=gh1F21 and v22=v21h21h22=gh31F21h22.

    Then, we have two possible solution for the heights values at the junction:

    hb1=hb2  (equal heigths) (49)

    or

    hb2hb1=F214(1+1+8F21). (50)

    So, for hb1hb2 we get new possible solutions at the junction with (hb1,qb) subcritical and (hb2,qb) supercritical or vice-versa. Specifically, case TorrentialFluvial and case TorrentialTorrential may admit solution even if their admissible regions have empty intersection in the subcritical set.

    Remark 3. In the case of a simple junction, the natural assumption (consistent with the dynamic of shallow-water equations) should be to assume the conservation of the momentum. With our notation, the relation 49 or 48 sholud be replaced by the following:

    q21h1+12gh21=q22h2+12gh21. (51)

    By the same reasoning used before in the case of the conservation of energy, from 51 we get

    (h2h1)3(2F21+1)(h2h1)+2F21=0.

    Then, we have again two possible relations for the heights values at the junction:

    hb1=hb2  (equal heigths) (52)

    or

    hb2hb1=12(1+1+8F21). (53)

    So again, for hb1hb2 we get new possible solutions at the junction. Case Torrential Fluvial and case TorrentialTorrential may have solution and specifically we have that in Figure 16, points (hl,ql) and (hl,ql) verify 53 (see 23) and if ulPA(ur) they are candidate to be the new solution. Even in the case described in Figure 18, if ulPB(ur) the points (hl,ql) and (hl,ql) are the solution at the junction.

    Let us conclude observing that for appropriate values of (hl,ql), for Torrential Fluvial we would get the same solution considering our simple network as a simple canal, i.e. a stationary shock called hydraulic jump characterized indeed by the conservation of the momentum in the transition from a supercritical to subcritical flow [11].

    In this Section we illustrate the results of Section 5 by means of numerical simulations. We first give a sketch of the adopted numerical procedure and then we focus on two numerical tests which illustrate the regime transitions from fluvial to torrential and viceversa. The latter depend on well chosen initial conditions for Riemann problems at the junction.

    We consider again a network formed by two canals intersecting at one single point, which represents the junction. Following [6], we use a high order Runge-Kutta Discontinuous Galerkin scheme to numerically solve system 3 on both canals 1 and 2:

    tu1+xf(u1)=0,for x<0,tu2+xf(u2)=0,for x<0. (54)

    The 1D domain of each canal is discretized into cells Cm=[xm12,xm+12], with xm=(xm12+xm+12)/2, m=1,,M, being M the total number of computational cells. The DG method for 3 is formulated by multiplying the equation system by some test functions w, integrating over each computational cell, and performing integration by parts. Specifically, we seek the approximation U=(u1,u2) with uiWΔx={w:w|CmPk(Cm),m=1,,M}, i=1,2, where Pk(Cm) is the space of polynomials of degree at most k on cell Cm, such that  wWΔx

    Cmw(x)tUdx=Cmf(U)xw(x)dx(ˆfm+12wm+12ˆfm12w+m12). (55)

    Terms w±m+12 denote left and right limits of the function values and the numerical fluxes ˆfm±12=f(Um±12,U+m±12) are approximate Riemann solvers. In our simulations, we use the Lax-Friedrich flux. For implementation, each component of the approximate solution U, e.g. u1, on mesh Cm is expressed as

    u1(x,t)=kl=0ˆu1,lm(t)ψlm(x), (56)

    where {ψlm(x)}kl=0 is the set of basis functions of Pk(Cm). Specifically, we choose the Legendre polynomials as local orthogonal basis of Pk(Cm) and take the test function w(x) in eq. 55 exactly as the set of basis functions ψlm(x), l=0k, assuming the polynomial degree k=2. The equation system 55 can then be evolved in time via the method of lines approach by a TVD RK method. More implementation details for RKDG methods can be found in the original paper [7] and the review article [8].

    Once the numerical procedure on both canals has been settled, the two systems in 55 have to be coupled with boundary conditions. At the junction the boundary values is settled as follows: at each time step and at each RK stage via the method-of-line approach, we set as left state in 42 (or 45) the approximate solution from canal 1 at the left limit of the junction, i.e.

    ulUl=limxxM+12U1(x,)

    with [xM12,xM+12] being the right-most cell in the 1D discretization of the incoming canal 1, and as right state in 42 (or 45) the approximate solution from canal 2 at the right limit of the junction, i.e.

    urUr=limxx+,212U2(x,),

    with [x12,x32] being the left-most cell in the 1D discretization of the outgoing canal 2. With this two values, we compute the intermediate states ub1 and ub2 by solving 42 (or 45) and, preserving the mass we directly assign the numerical fluxes at the junction as

    ˆfM+12f(ub1)for the canal 1 ,ˆf12f(ub2)for the canal 2 .

    Finally, in our simulations we assume Neumann boundary conditions at the free extremity of the channels.

    Applying this numerical procedure, in Figure 10 and 11 we give two examples which illustrate the solution that is obtained in the regime transitions from fluvial to torrential and viceversa. In Figure 10, we assume to have a starting configuration given by the following subcritical constant states: ul=(0.25,0.025) on canal 1 (left) and ur=(2.5,0.25) on canal 2 (right). We are in the situation showed in Figure 9 and the boundary value at the junction is defined by the critical value urR(1.088,3.55). The solution is a backward water movement along canal 1 with a torrential regime. In Figure 11, we assume as initial state on canal 1 the supercritical constant value ul=(0.2,3) and on canal 2 the subcritical constant value ur=(1.8,4). We are in the configuration given in Figure 16 and the boundary value at the junction is defined by the starting point ul. The solution is a forward movement with positive velocity, so the torrential regime propagates on canal 2.

    Figure 10.  Numerical test case for the configuration given in Fig. 9.
    Figure 11.  Numerical test case for the configuration given in Fig. 16.

    This paper deals with open canal networks. The interest stems out of applications such as irrigation channels water management. We base our investigations on the well-known Saint-Venant or shallow water equations. Two regimes exist for this hyperbolic system of balance laws: the fluvial, corresponding to eigenvalues with different sign, and the torrential, corresponding to both positive eigenvalues. Most authors focused the attention on designing and analysing network dynamics for the fluvial regime, while here we extend the theory to include regime transitions. After analyzing the Lax curves for incoming and outgoing canals, we provide admissibility conditions for Riemann solvers, describing solutions for constant initial data on each canal. Such analysis allows to define uniquely dynamics according to a set of conditions at junctions, such as conservation of mass, equal water height or equal energy. More precisely, the simple case of one incoming and outgoing canal is treated showing that, already in this simple example, regimes transitions appear naturally at junctions. Our analysis is then visualized by numerical simulations based on Runge-Kutta Discontinuous Galerkin methods.

    M. Briani is a member of the INdAM Research group GNCS.

    Here we collect additional figures illustrating attainable regions for half-riemann problems and solutions for a simple channel. Figures 12-14 refer to the right-half Riemann problem described in Section 4.2. They show the regions of admissible states such that waves on the outgoing canals do not propagate into the junction, given a right state ur such that |˜Fr|<1, ˜Fr>1 and ˜Fr<1 respectively.

    Figures 15-18 refer to Section 5 in which we study the possible solutions at a simple junction for different flow regimes, assuming the conservation of mass and equal heights at the junction. Specifically, Figures 15-17 illustrate the possible configurations and their associated solution that may occur during the transition from torrential to fluvial regime. The last Figure 18 shows instead the only possible configuration that admits a solution for the torrential flow regime.



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