
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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[2] | Xavier Litrico, Vincent Fromion . Modal decomposition of linearized open channel flow. Networks and Heterogeneous Media, 2009, 4(2): 325-357. doi: 10.3934/nhm.2009.4.325 |
[3] | Graziano Guerra, Michael Herty, Francesca Marcellini . Modeling and analysis of pooled stepped chutes. Networks and Heterogeneous Media, 2011, 6(4): 665-679. doi: 10.3934/nhm.2011.6.665 |
[4] | Didier Georges . Infinite-dimensional nonlinear predictive control design for open-channel hydraulic systems. Networks and Heterogeneous Media, 2009, 4(2): 267-285. doi: 10.3934/nhm.2009.4.267 |
[5] | Rudy R. Negenborn, Peter-Jules van Overloop, Tamás Keviczky, Bart De Schutter . Distributed model predictive control of irrigation canals. Networks and Heterogeneous Media, 2009, 4(2): 359-380. doi: 10.3934/nhm.2009.4.359 |
[6] | Nora Aïssiouene, Marie-Odile Bristeau, Edwige Godlewski, Jacques Sainte-Marie . A combined finite volume - finite element scheme for a dispersive shallow water system. Networks and Heterogeneous Media, 2016, 11(1): 1-27. doi: 10.3934/nhm.2016.11.1 |
[7] | Jacek Banasiak, Proscovia Namayanja . Asymptotic behaviour of flows on reducible networks. Networks and Heterogeneous Media, 2014, 9(2): 197-216. doi: 10.3934/nhm.2014.9.197 |
[8] | Avner Friedman . PDE problems arising in mathematical biology. Networks and Heterogeneous Media, 2012, 7(4): 691-703. doi: 10.3934/nhm.2012.7.691 |
[9] | Ye Sun, Daniel B. Work . Error bounds for Kalman filters on traffic networks. Networks and Heterogeneous Media, 2018, 13(2): 261-295. doi: 10.3934/nhm.2018012 |
[10] | Gildas Besançon, Didier Georges, Zohra Benayache . Towards nonlinear delay-based control for convection-like distributed systems: The example of water flow control in open channel systems. Networks and Heterogeneous Media, 2009, 4(2): 211-221. doi: 10.3934/nhm.2009.4.211 |
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
●
●
●
The Froude-number denominator
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
∂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)=(01−v2+gh2v). | (6) |
The eigenvalues of
λ1=v−√gh,λ2=v+√gh, | (7) |
with corresponding eigenvectors
r1=(1λ1),r2=(1λ2). | (8) |
The shallow water equations are strictly hyperbolic away from
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
(R) Centered Rarefaction Waves. Assume
u(x,t)={u− for x<λi(u−)t,Ri(x/t;u−) for λi(u−)t≤x≤λi(u+)t,u+ for x>λi(u+)t, |
where, for the 1-family
R1(ξ;u−):=(19(v−+2√h−−ξ)2127(v−+2√h−+2ξ)(v−+2√h−−ξ)2) |
for
R2(ξ;u−):=(19(−v−+2√h−−ξ)2127(v−2√h−+2ξ)(−v−+2√h−−ξ)2) |
for
(S) Shocks. Assume that the state
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+−q−h+−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
for h<h0,v=R1(h0,v0;h)=v0−2(√gh−√gh0);for h>h0,v=S1(h0,v0;h)=v0−(h−h0)√gh+h02hh0;for h>h0,v=R2(h0,v0;h)=v0−2(√gh0−√gh);for h<h0,v=S2(h0,v0;h)=v0−(h0−h)√gh+h02hh0. | (10) |
Moreover, we define the inverse curves:
for h>h0,v=R−11(h0,v0;h)=v0+2(√gh0−√gh);for h<h0,v=S−11(h0,v0;h)=v0+(h0−h)√gh+h02hh0. | (11) |
Similarly, we set:
for h<h0,v=R−12(h0,v0;h)=v0+2(√gh−√gh0);for h>h0,v=S−12(h0,v0;h)=v0+(h−h0)√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.
To construct a solution to a Riemann problem
ϕl(h):=R1(hl,vl;h)∪S1(hl,vl;h). | (15) |
For the right state
ϕr(h):=R−12(hr,vr;h)∪S−12(hr,vr;h). | (16) |
Remark 1. The Riemann problem for shallow water equations 2 with left state
vl+2√ghl≥vr−2√ghr. | (17) |
When working with
˜ϕl(h)=hϕl(h),˜ϕr(h)=hϕr(h)and˜C+(h)=hC+(h),˜C−(h)=hC−(h). |
Moreover, for a given value
Fi=vi√ghi,or˜Fi=qihi√ghi. | (18) |
In this subsection we study in detail the properties of the function
˜ϕl(h)={h(vl+2√ghl−2√gh),0<h≤hl,h(vl−√g2hl(h−hl)√h+hlh),h>hl |
with
limh→0+˜ϕl(h)=0andlimh→+∞˜ϕl(h)=−∞. |
By computing its first and second derivatives,
˜ϕ′l(h)={vl+2√ghl−3√gh,0<h≤hl,vl−√g2hl(4h2+hlh−h2l2√h(h+hl)),h>hl |
and
˜ϕ′′l(h)={−32√gh,0<h≤hl,−√g2hl(8h3+12hlh+3h2lh+h3l4h(h+hl)√h(h+hl)),h>hl, |
we can conclude that
˜ϕ′l(0)=vl+2√ghl and limh→+∞˜ϕl(h)=−∞ |
we investigate two different cases:
Case 1. If
Case 2. If
Case 2.1. For
h+l,R=19g(vl+2√ghl)2, | (19) |
and we have that the maximum point
˜ϕl(h)=0⇔h=14g(vl+2√ghl)2. |
Notice that for
h−l,R=1g(vl+2√ghl)2; | (20) |
while for
vl−(h−l,S−hl)√gh−l,S+hl2hlh−l,S+√gh−l,S=0; | (21) |
Case 2.2. For
h+l,Ssuch thatvl−(h+l,S−hl)√gh+l,S+hl2hlh+l,S−√gh+l,S=0; | (22) |
and the curve
Notice that for
ql=˜ϕl(h)=hS1(hl,vl;h) |
has two solutions:
h=hlandh=h∗l=hl2(−1+√1+8F2l). | (23) |
Moreover, for
(q2lg)13<hl2(−1+√1+8F2l),forFl>1. |
Indeed, using the relation
Here we study the properties of the function
˜ϕr(h)={h(vr−2√ghr+2√gh),0<h≤hr,h(vr+√g2hr(h−hr)√h+hrh),h>hr. |
By straightforward computations we get its derivatives:
˜ϕ′r(h)={vr−2√ghr+3√gh,0<h≤hr,vr+√g2hr(4h2+hrh−h2r2√h(h+hr)),h>hr. |
˜ϕ′′r(h)={32√gh,0<h≤hr,√g2hr(8h3+12hrh+3h2rh+h3r4h(h+hr)√h(h+hr)),h>hr. |
Then,
˜ϕ′r(0)=vr−2√ghr and limh→+∞˜ϕr(h)=+∞ |
we investigate two different cases:
Case 1. If
Case 2. If
Case 2.1. For
h−r,R=19g(−vr+2√ghr)2 | (24) |
and we have that the minimum point
˜ϕr(h)=0⇔h=14g(−vr+2√ghr)2. |
For
h+r,R=1g(−vr+2√ghr)2, | (25) |
while for
vr+(h+r,S−hr)√gh+r,S+hr2hrh+r,S+√gh+r,S=0. | (26) |
Case 2.2. For
h−r,S such that vr+(hr−h+r,S)√gh−r,S+hr2hrh−r,S+√gh−r,S=0 |
and curve
Notice that the equation
qr=˜ϕr(h)=hS−12(hr,vr;h),h≥hr, |
has two solutions:
h=hrandh=h∗r=hr2(−1+√1+8F2r). | (27) |
Moreover, for
[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
{∂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
● Case B: the left state
● Case C: the left state
For this case we refer to Figure 4.We identify the set
IA1={(ˆh,ˆq):h+l,R≤ˆh≤h−l,S,ˆq=˜ϕl(ˆh)}, | (29) |
where the points
IA2={(ˆh,ˆq):0<ˆh≤h−l,S, ˆq≤ˆhS2(h−l,S,C−(h−l,S);ˆh)}⋃{(ˆh,ˆq):ˆh>h−l,S, ˆq≤˜C−(ˆh)}. | (30) |
The last region
λ(um,ˆu)=qm−ˆqhm−ˆh≤0. |
To define this region, we have to look for values
qm−q=(hm−h)(vm+√g2hm√h(h+hm))≤0,h<hm. |
This inequality is verified for
h∗m=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<ˆh≤h∗m, ˆq=ˆhS2(hm,vm;ˆh)}. | (31) |
For this case we refer to Figure 5. It is always possible to connect the left value
h∗l=12(−1+√1+8F2l)hl |
as previously computed in 23. Moreover, as previously observed at the end of subsection 3.0.1 the value
NB(ul)=NA(ul)∖{ˆu=(ˆh,ˆq):h+l,S≤ˆh≤h∗l, ˆq=˜ϕl(ˆh)}, | (32) |
where
For this case we refer to Figure 6. We have that: if
NC(ul)={(ˆh,ˆq):ˆh>0, ˆq<˜C−(ˆh)}; |
otherwise if
NC(ul)={(ˆh,ˆq):0<ˆh≤h−l,R, q<ˆhS2(h−l,R,v−l,R;ˆh)}⋃{(ˆh,ˆq):ˆh>h−l,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
{∂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
● Case B: the right value
● Case C: the right value
For this case we refer to Figure 12 in the Appendix. We identify the set
OA1={(˜h,˜q):h−r,R≤˜h≤h+r,S, ˜q=˜ϕr(˜h)}, | (35) |
where the points
The second region is such that
OA2={(˜h,˜q):0<˜h≤h+r,S, ˜q≥˜hS−11(h+r,S,v+r,S;˜h)}⋃{˜h≥h+r,S, ˜q≥˜C+(˜h)}. | (36) |
The third region is defined by the set of all possible left states
λ(um,˜u)=qm−˜qhm−˜h≥0. |
To define this region we have to look for values
OA3={(˜h,˜q): for all (hm,qm) which vary on ˜ϕr such that 0<˜Fm≤1 0<˜h≤h∗m, ˜q=˜hS−11(hm,vm;˜h)}. | (37) |
For this case we refer to Figure 13 in the Appendix. If
PB(ur)={(˜h,˜q): ˜h≥0, ˜q≥˜C+(˜h)}; |
otherwise, if
PB(ur)={(˜h,˜q): 0<˜h≤h+r,R, ˜q≥˜hS−11(u−r,R;˜h)}⋃{(˜h,˜q),˜h>h−r,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
h∗r=12(−1+√1+8F2r)hr |
as done in 27. Moreover, as previously observed at the end of subsection 3.0.2, the point
PC(ur)=PA(u∗r)∖{(h,q):h−r,S≤h≤h∗r, q=˜ϕr(h)}, | (39) |
where
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
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
qb1=qb2 | (40) |
and equal heights
hb1=hb2. | (41) |
In the following we study the boundary solution
Case A
{ub1∈NA(ul),ub2∈PA(ur),qb1=qb2=qb,hb1=hb2, | (42) |
with
Proposition 1. Under the subcritical condition on
Proof. We distinguish two cases:
Case 1. The two curves
Case 2. The two curves
ub1=ub2=u+l,R. | (43) |
If
ub1=ub2=u−r,R. | (44) |
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
Case B
{ub1∈NB(ul),ub2∈PA(ur),qb1=qb2=qb,hb1=hb2, | (45) |
with
Proposition 2. System 45 admits a solution if the two regions
Proof. We distinguish two cases:
Case 1. The two curves
Case 2. The two curves
Case 2.1. Referring to Figure 16, the point
Case 2.2. Referring Figure 17, if
Case B
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=h+v22g. | (46) |
For a given flow rate, there are usually two states possible for the same specific energy. Studying
h=hc=(q2g)13. | (47) |
Critical depth
In our case, assuming equal energy at the junction gives
v212+gh1=v222+gh2. | (48) |
Moreover, assuming
gh1F212+gh1=gh31F212h22+gh2, |
where
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
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
Let us conclude observing that for appropriate values of
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
∫Cmw(x)∂tUdx=∫Cmf(U)∂xw(x)dx−(ˆfm+12w−m+12−ˆfm−12w+m−12). | (55) |
Terms
u1(x,t)=k∑l=0ˆu1,lm(t)ψlm(x), | (56) |
where
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.
ul≈Ul=limx→x−M+12U1(x,⋅) |
with
ur≈Ur=limx→x+,212U2(x,⋅), |
with
ˆfM+12≐f(ub1)for the canal 1 ,ˆf12≐f(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:
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
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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