A Generalized Finite Difference-Volume Hybrid Method Applied to Shallow-Water Equations

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==A Generalized Finite Difference-Volume Hybrid Method Applied to Shallow-Water Equations==

'''G.&nbsp;Tinoco-Guerrero<sup>a,b</sup>, F.&nbsp;J.&nbsp;Domínguez-Mota<sup>a</sup>, J.&nbsp;G.&nbsp;Tinoco-Ruiz<sup>a</sup>, J.&nbsp;S.&nbsp;Lucas-Martínez<sup>a</sup>, N.&nbsp;S.&nbsp;Tinoco-Guerrero<sup>a</sup>, M.&nbsp;Leppäranta<sup>b</sup>,I.&nbsp;Mammarella<sup>b</sup>'''
-->

==Abstract==

Due to the importance of the shallow-water equations in models of real-life phenomena, in recent years the study and model of problems that involve them have been the object of interest of many people. By reason of this, it is imperative to have efficient numerical methods to obtain an approximation of the solutions of the shallow-water equations.

Several authors have worked in approximations using the well-known finite volume and finite element methods, nevertheless, even when these methods compute good approximations to real-life behavior, the computational cost is usually high, which could be a limitation to the application of these methods.

This paper presents an explicit Generalized Finite Difference-Volume Hybrid approximation to the solution of the shallow-water equations, solved on irregular regions meshed with logically rectangular grids; the numerical results show the accuracy obtained with a low-cost implementation. The proposed scheme is a hybridization of a generalized finite difference scheme with the finite volume method.

'''keywords''' hybrid method, finite difference, finite volume, shallow-water equations, irregular regions

==1 Introduction==

In nature, there exist many types of flow that can be characterized as shallow-water flows. The main characteristic of these kinds of flows is that the vertical scales are much smaller than the horizontal ones. This happens in many regions all over the world, such as lakes, some rivers and, in some special cases, in parts of the oceans.

Since the flow in these cases is almost horizontal, a great number of simplifications can be done in the physical and mathematical formulation taking into account that the value of the pressure is essentially the hydrostatic one. It is important to remark that, even with these simplifications, the formulations are not two-dimensional, since the bottom friction must be taken into account in the boundary layers. In the literature, these <math display="inline">3D</math> effects are often not essential for the models, and it is only necessary to consider the two-dimensional depth-averaged form <span id='citeF-1'></span>[[#cite-1|[1]]].

Due to the simplicity of the finite difference methods, some important advances have been done in the approximation to the solution of these equations; Po-Wei and Chia-Ming present in <span id='citeF-2'></span>[[#cite-2|[2]]] a generalized finite difference method that can produce very good results, with the limitation that the average flow direction has to be known ''a priori'' in order to use the generalized finite difference method.

Young discusses a meshless method in <span id='citeF-3'></span><span id='citeF-4'></span>[[#cite-3|[3,4]]] that can produce very good results in very irregular domains by using a local radial-basis-functions differential-quadrature approach; the results presented in his papers have very good quality and can be applied to real-life scenarios. One more time, even when this method produces very good results, even for inflow problems, the computational cost can be very high.

One of the most common problems that are presented in many works is the treatment of discontinuities and singularities; it is well known that one way to overcome these problems is using a ''conservative form'' of the equations. Nevertheless, Ulrik Skre Fjordhol <span id='citeF-5'></span>[[#cite-5|[5]]], Bruno Gabutti <span id='citeF-6'></span>[[#cite-6|[6]]], and Carlos Parés <span id='citeF-7'></span>[[#cite-7|[7]]] have proposed accurate numerical schemes to approximate the solution of non-conservative Hyperbolic Equations, discussing upwind like and splitting schemes for solving numerically this version of the equations. Following the previous ideas, in this paper we present numerical schemes for the '''conservative form''' and '''non-conservative form''' of the shallow-water equations; the main aim of this work is to obtain an explicit method that can be applied to irregular domains.

In order to do that, let us first consider the problem of obtaining an approximation to the solution of the '''conservative form''' of the shallow-water equations

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>    \frac{\partial h}{\partial t} + \frac{\partial }{\partial x}(au) + \frac{\partial }{\partial y}(av) = 0,   </math>
|}
|}

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>    \frac{\partial }{\partial t}(au) + \frac{\partial }{\partial x}(au^2) + \frac{\partial }{\partial y}(auv) - fav + ga\frac{\partial h}{\partial x} + c_fu\sqrt{u^2+v^2} = 0,   </math>
|}
|}

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>    \frac{\partial }{\partial t}(av) + \frac{\partial }{\partial x}(auv) + \frac{\partial }{\partial y}(av^2) + fau + ga\frac{\partial h}{\partial y} + c_fv\sqrt{u^2+v^2} = 0,   </math>
|}
|}

in a simply connected planar domain <math display="inline">\Omega </math> defined by a polygonal boundary, where <math display="inline">h</math> is the water level, <math display="inline">u</math> and <math display="inline">v</math> are the velocity fields in the <math display="inline">x</math> and <math display="inline">y</math> directions respectively, <math display="inline">a</math> is the water-body depth, <math display="inline">f</math> is the Coriolis force and <math display="inline">c_f</math> represents the external forces (see figure ([[#img-1|1]])).

<div id='img-1'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Review_838453312870-fig01.png|450px|'''Definition of bottom and free surface.''']]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 1:''' '''Definition of bottom and free surface.'''
|}

In these equations, the change of variables

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>    q = h,  r = au,  s=av,   </math>
|}
|}

leads to the expressions

<span id="eq-1"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\frac{\partial q}{\partial t} + \frac{\partial r}{\partial x} + \frac{\partial s}{\partial y} = 0,       </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (1)
|}

<span id="eq-2"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\frac{\partial r}{\partial t} + \frac{\partial }{\partial x}\left(\frac{r^2}{a}\right)+ \frac{\partial }{\partial y}\left(\frac{rs}{a}\right)- fs + ga\frac{\partial q}{\partial x} + c_fr\frac{\sqrt{r^2+s^2}}{a^2} = 0,       </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (2)
|}

<span id="eq-3"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\frac{\partial s}{\partial t} + \frac{\partial }{\partial x}\left(\frac{rs}{a}\right)+ \frac{\partial }{\partial y}\left(\frac{s^2}{a}\right)+ fr + ga\frac{\partial q}{\partial y} + c_fs\frac{\sqrt{r^2+s^2}}{a^2} = 0,</math>
|-
| style="text-align: center;" | 
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (3)
|}

here, the unknowns are the conservative quantities <math display="inline">q</math>, <math display="inline">r</math> and <math display="inline">s</math>, that represent the mass and momentum of the physical problem.

After differentiation and some algebra, equations ([[#eq-1|1]]) - ([[#eq-3|3]]) can be rewritten as

<span id="eq-4"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\frac{\partial q}{\partial t} + \frac{\partial r}{\partial x} + \frac{\partial s}{\partial y} = 0,       </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (4)
|}

<span id="eq-5"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\frac{\partial r}{\partial t} + \frac{2r}{a}\frac{\partial r}{\partial x} + ga\frac{\partial q}{\partial x} + \frac{r}{a}\frac{\partial s}{\partial y} + \frac{s}{a}\frac{\partial r}{\partial y} - fs + c_f \frac{r}{a^2}\sqrt{r^2+s^2} = 0,       </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (5)
|}

<span id="eq-6"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\frac{\partial s}{\partial t} + \frac{r}{a}\frac{\partial s}{\partial x} + \frac{s}{a}\frac{\partial r}{\partial x} + \frac{2s}{a}\frac{\partial s}{\partial y} + ga\frac{\partial q}{\partial y} + fr + c_f \frac{s}{a^2}\sqrt{r^2+s^2} = 0.</math>
|-
| style="text-align: center;" | 
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (6)
|}

This is the '''non-conservative form''' or '''differential form''' of the shallow-water equations.

==2 Proposed hybrid schemes==

The proposed schemes arise from the integral form of equations ([[#eq-2|2]]) and ([[#eq-3|3]]), ([[#eq-5|5]]) and ([[#eq-6|6]]), and a finite difference approximation of equations ([[#eq-1|1]]) and ([[#eq-4|4]]).

===2.1 Hybrid schemes on rectangular regions===

On rectangular regions, the space region can be discretized by taking a grid formed by uniform cells <math display="inline">C_{(i, j)}</math> defined as

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>C_{(i, j)}  =  \left[x_{\left(i-\frac{1}{2}\right)},x_{\left(i+\frac{1}{2}\right)}\right]\times \left[y_{\left(i-\frac{1}{2}\right)},y_{\left(i+\frac{1}{2}\right)}\right],    </math>
|}
|}

as shown on figure ([[#img-2|2]]).

<div id='img-2'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Review_838453312870-fig02.png|570px|'''Rectangular meshed region Ω.''']]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 2:''' '''Rectangular meshed region <math>\Omega </math>.'''
|}

====2.1.1 Hybrid scheme for the conservative form====

With this discretization of the region, the proposed scheme approximates the value of <math display="inline">q</math> at the center of each cell by using a classical finite difference approximation. For this case, the partial derivatives can be approximated at a central point <math display="inline">p_{(i, j)}</math> as

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>      \frac{\partial q}{\partial t} \approx \frac{q_{(i, j)}^{k+1} - q_{(i, j)}^{k}}{\Delta t},     </math>
|}
|}

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>      \frac{\partial r}{\partial x} \approx  \frac{r^{k}_{(i+\frac{1}{2}, j)}-r^{k}_{(i-\frac{1}{2}, j)}}{\Delta x},     </math>
|}
|}

and

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>      \frac{\partial s}{\partial y} \approx \frac{s^{k}_{(i, j+\frac{1}{2})} - s^{k}_{(i, j-\frac{1}{2})}}{\Delta y}.     </math>
|}
|}

where the subscripts and superscripts represent the spatial position on the grid and the time level, respectively.

Taking into account these approximations, the first equation can be approximated as

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>      \frac{q_{(i, j)}^{k+1} - q_{(i, j)}^{k}}{\Delta t} + \frac{r^{k}_{(i+\frac{1}{2}, j)}-r^{k}_{(i-\frac{1}{2}, j)}}{\Delta x} + \frac{s^{k}_{(i, j+\frac{1}{2})} - s^{k}_{(i, j-\frac{1}{2})}}{\Delta y} = 0.     </math>
|}
|}

From here, solving for <math display="inline">q^{k+1}_{(i, j)}</math>, the approximation

<span id="eq-7"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>q^{k+1}_{(i, j)} = q^{k}_{(i, j)} - \Delta t\left[\frac{r^{k}_{(i+\frac{1}{2}, j)}-r^{k}_{(i-\frac{1}{2}, j)}}{\Delta x} + \frac{s^{k}_{(i, j+\frac{1}{2})} - s^{k}_{(i, j-\frac{1}{2})}}{\Delta y}\right],           </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (7)
|}

can be obtained.

Then, the values of <math display="inline">r</math> and <math display="inline">s</math> are approximated with a ''hybrid Finite Difference-Volume scheme'' on the edges of each cell. In order to do so the integral form of equations ([[#eq-2|2]]) and ([[#eq-3|3]]) must be taken into account.

The integral form of equation ([[#eq-2|2]]) evaluated over an arbitrary cell <math display="inline">C_{(i, j)}</math> is

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>      \int _{\mathcal{C}_{(i, j)}}\left[\frac{\partial r}{\partial t} + \frac{\partial }{\partial x}\left(\frac{r^2}{a}\right)+ \frac{\partial }{\partial y}\left(\frac{rs}{a}\right)- fs + ga\frac{\partial q}{\partial x} + c_fr\frac{\sqrt{r^2+s^2}}{a^2}\right]dA = 0.     </math>
|}
|}

In this expression the partial derivatives can be replaced by their finite difference approximations, calculated on the edges of the cell,

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>      \frac{\partial r}{\partial t} \approx \frac{r_{(i+\frac{1}{2}, j)}^{k+1} - r_{(i+\frac{1}{2}, j)}^{k}}{\Delta t},     </math>
|}
|}

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>      \frac{\partial }{\partial x}\left(\frac{r^2}{a}\right)\approx  \frac{\left(r^{k}_{(i+\frac{3}{2}, j)}\right)^2-\left(r^{k}_{(i-\frac{1}{2}, j)}\right)^2}{2a\Delta x},     </math>
|}
|}

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>      \frac{\partial }{\partial y}\left(\frac{rs}{a}\right)\approx \frac{r^{k}_{(i+\frac{1}{2},j+1)}s^{k}_{(i+\frac{1}{2},j+1)} - r^{k}_{(i+\frac{1}{2},j-1)}s^{k}_{(i+\frac{1}{2},j-1)}}{2a\Delta y},     </math>
|}
|}

and

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>      \frac{\partial q}{\partial x} \approx \frac{q^{k}_{\left(i+1, j\right)} - q^{k}_{\left(i, j\right)}}{\Delta x},     </math>
|}
|}

to obtain

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>      \int _{\mathcal{C}_{(i, j)}}\left[\frac{r^{k+1}_{(i+\frac{1}{2}, j)} - r^{k}_{(i+\frac{1}{2}, j)}}{\Delta t} + \frac{\left(r^{k}_{(i+\frac{3}{2}, j)}\right)^2-\left(r^{k}_{(i-\frac{1}{2}, j)}\right)^2}{2a\Delta x} + \frac{r^{k}_{(i+\frac{1}{2},j+1)}s^{k}_{(i+\frac{1}{2},j+1)} - r^{k}_{(i+\frac{1}{2},j-1)}s^{k}_{(i+\frac{1}{2},j-1)}}{2a\Delta y} - \right.     </math>
|}
|}

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>      \left.fs_{i+\frac{1}{2},j}^{k} + ga\frac{q^{k}_{\left(i+1, j\right)} - q^{k}_{\left(i, j\right)}}{\Delta x}  + \frac{c_f r^{k}_{(i+\frac{1}{2}, j)}}{a^2}\sqrt{\left(r^{k}_{(i+\frac{1}{2}, j)}\right)^2+\left(s^{k}_{(i+\frac{1}{2}, j)}\right)^2}\right]dA = 0.     </math>
|}
|}

Since this integral must vanish for every cell <math display="inline">C_{(i, j)}</math>, the expression

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>      \frac{r^{k+1}_{(i+\frac{1}{2}, j)} - r^{k}_{(i+\frac{1}{2}, j)}}{\Delta t} + \frac{\left(r^{k}_{(i+\frac{3}{2}, j)}\right)^2-\left(r^{k}_{(i-\frac{1}{2}, j)}\right)^2}{2a\Delta x} + \frac{r^{k}_{(i+\frac{1}{2},j+1)}s^{k}_{(i+\frac{1}{2},j+1)} - r^{k}_{(i+\frac{1}{2},j-1)}s^{k}_{(i+\frac{1}{2},j-1)}}{2a\Delta y}     </math>
|}
|}

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>      fs_{i+\frac{1}{2},j}^{k} + ga\frac{q^{k}_{\left(i+1, j\right)} - q^{k}_{\left(i, j\right)}}{\Delta x}  + \frac{c_f r^{k}_{(i+\frac{1}{2}, j)}}{a^2}\sqrt{\left(r^{k}_{(i+\frac{1}{2}, j)}\right)^2+\left(s^{k}_{(i+\frac{1}{2}, j)}\right)^2} = 0.     </math>
|}
|}

can be obtained. Now here, solving for <math display="inline">r^{k+1}_{(i+\frac{1}{2}, j)}</math>,

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>      r^{k+1}_{(i+\frac{1}{2}, j)} = r^{k}_{(i+\frac{1}{2}, j)} - \Delta t \left[\frac{\left(r^{k}_{(i+\frac{3}{2}, j)}\right)^2-\left(r^{k}_{(i-\frac{1}{2}, j)}\right)^2}{2a\Delta x} + \frac{r^{k}_{(i+\frac{1}{2},j+1)}s^{k}_{(i+\frac{1}{2},j+1)} - r^{k}_{(i+\frac{1}{2},j-1)}s^{k}_{(i+\frac{1}{2},j-1)}}{2a\Delta y} -\right.     </math>
|}
|}

<span id="eq-8"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\left.fs_{i+\frac{1}{2},j}^{k} + ga\frac{q^{k}_{\left(i+1, j\right)} - q^{k}_{\left(i, j\right)}}{\Delta x}  + \frac{c_f r^{k}_{(i+\frac{1}{2}, j)}}{a^2}\sqrt{\left(r^{k}_{(i+\frac{1}{2}, j)}\right)^2+\left(s^{k}_{(i+\frac{1}{2}, j)}\right)^2}\right]           </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (8)
|}

Following a similar logic, the integral form of equation ([[#eq-3|3]])

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>      \int _{\mathcal{C}_{(i, j)}}\left[\frac{\partial s}{\partial t} + \frac{\partial }{\partial x}\left(\frac{rs}{a}\right)+ \frac{\partial }{\partial y}\left(\frac{s^2}{a}\right)+ fr + ga\frac{\partial q}{\partial y} + c_fs\frac{\sqrt{r^2+s^2}}{a^2}\right]dA = 0,     </math>
|}
|}

can be treated to obtain an approximation to <math display="inline">s^{k+1}_{(i+\frac{1}{2}, j)}</math>,

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>      s^{k+1}_{(i+\frac{1}{2}, j)} = s^{k}_{(i+\frac{1}{2}, j)} - \Delta t \left[\frac{r^{k}_{(i+\frac{3}{2},j)}s^{k}_{(i+\frac{3}{2},j)} - r^{k}_{(i-\frac{1}{2},j)}s^{k}_{(i-\frac{1}{2},j)}}{2a\Delta x} + \frac{\left(s^{k}_{(i+\frac{1}{2}, j+1)}\right)^2-\left(s^{k}_{(i+\frac{1}{2}, j-1)}\right)^2}{2a\Delta y} -\right.     </math>
|}
|}

<span id="eq-9"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\left.f r^{k}_{(i+\frac{1}{2},j)} + ga\frac{q^{k}_{\left(i,j+1\right)} - q^{k}_{\left(i,j\right)}}{\Delta y}  + \frac{c_f s^{k}_{(i+\frac{1}{2}, j)}}{a^2}\sqrt{\left(r^{k}_{(i+\frac{1}{2}, j)}\right)^2+\left(s^{k}_{(i+\frac{1}{2}, j)}\right)^2}\right].           </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (9)
|}

Now, equations ([[#eq-7|7]]), ([[#eq-8|8]]) and ([[#eq-9|9]]) define a ''hybrid Finite Difference-Volume scheme'' for the '''conservative form''' of the shallow-water Equations, in rectangular regions.

====2.1.2 Hybrid scheme for the non-conservative form====

In the case of the non-conservative form of shallow-water equations, a similar approach, as the one taken for the conservative form, can be chosen. In this occasion we could to take into account a finite difference scheme for equation ([[#eq-4|4]]),

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>      \frac{\partial q}{\partial t} + \frac{\partial r}{\partial x} + \frac{\partial s}{\partial y} = 0,     </math>
|}
|}

and the integral form of equations ([[#eq-5|5]]) and ([[#eq-6|6]]),

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>      \frac{\partial r}{\partial t} + \frac{2r}{a}\frac{\partial r}{\partial x} + ga\frac{\partial q}{\partial x} + \frac{r}{a}\frac{\partial s}{\partial y} + \frac{s}{a}\frac{\partial r}{\partial y} - fs + c_f \frac{r}{a^2}\sqrt{r^2+s^2} = 0,     </math>
|}
|}

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\begin{array}{l}      \frac{\partial s}{\partial t} + \frac{r}{a}\frac{\partial s}{\partial x} + \frac{s}{a}\frac{\partial r}{\partial x} + \frac{2s}{a}\frac{\partial s}{\partial y} + ga\frac{\partial q}{\partial y} + fr + c_f \frac{s}{a^2}\sqrt{r^2+s^2} = 0.\\     \end{array}</math>
|}
|}

Doing this, a scheme defined by

<span id="eq-10"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>q^{k+1}_{(i, j)} = q^{k}_{(i, j)} - \Delta t\left[\frac{r^{k}_{(i+\frac{1}{2}, j)}-r^{k}_{(i-\frac{1}{2}, j)}}{\Delta x} + \frac{s^{k}_{(i, j+\frac{1}{2})} - s^{k}_{(i, j-\frac{1}{2})}}{\Delta y}\right],           </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (10)
|}

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>      r^{k+1}_{(i+\frac{1}{2}, j)} =  r^{k}_{(i+\frac{1}{2}, j)} - \Delta t\left[\frac{2r_{(i+\frac{1}{2}, j)}^{k}}{a}\frac{r^{k}_{(i+\frac{3}{2}, j)} - r^{k}_{(i-\frac{1}{2}, j)}}{\Delta x} + ga\frac{q^{k}_{\left(i+1, j\right)} - q^{k}_{\left(i, j\right)}}{\Delta x} + \right.     </math>
|}
|}

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>      \left.\frac{r_{(i+\frac{1}{2}, j)}^{k}}{a}\frac{s^{k}_{(i+\frac{1}{2}, j+1)} - s^{k}_{(i+\frac{1}{2}, j-1)}}{\Delta y} + \frac{s_{(i+\frac{1}{2}, j)}^{k}}{a}\frac{r^{k}_{(i+\frac{1}{2}, j+1)} - r^{k}_{(i+\frac{1}{2}, j-1)}}{\Delta y} - \right.     </math>
|}
|}

<span id="eq-11"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\left.fs^{k}_{(i+\frac{1}{2}, j)} + c_f\frac{r^{k}_{(i+\frac{1}{2}, j)}}{a^2}\sqrt{(r^{k}_{(i+\frac{1}{2}, j)})^2+(s^{k}_{(i+\frac{1}{2}, j)})^2}\right],           </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (11)
|}

and

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>      s^{k+1}_{(i+\frac{1}{2}, j)} =  s^{k}_{(i+\frac{1}{2}, j)} - \Delta t\left[\frac{r_{(i+\frac{1}{2}, j)}^{k}}{a}\frac{s^{k}_{(i+\frac{3}{2}, j)} - s^{k}_{(i-\frac{1}{2}, j)}}{\Delta x} + \frac{s_{(i+\frac{1}{2}, j)}^{k}}{a}\frac{r^{k}_{(i+\frac{3}{2}, j)} - r^{k}_{(i-\frac{1}{2}, j)}}{\Delta x} + \right.     </math>
|}
|}

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>      \left.\frac{2s_{(i+\frac{1}{2}, j)}^{k}}{a}\frac{s^{k}_{(i+\frac{1}{2}, j+1)} - s^{k}_{(i+\frac{1}{2}, j-1)}}{2\Delta y} + ga\frac{q^{k}_{\left(i, j+1\right)} - q^{k}_{\left(i, j\right)}}{\Delta y} + \right.     </math>
|}
|}

<span id="eq-12"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\left.fr^{k}_{(i+\frac{1}{2}, j)} + c_f\frac{s^{k}_{(i+\frac{1}{2}, j)}}{a^2}\sqrt{(r^{k}_{(i+\frac{1}{2}, j)})^2+(s^{k}_{(i+\frac{1}{2}, j)})^2}\right].           </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (12)
|}

can be obtained. This scheme can be applied to the '''non-conservative form''' of the shallow-water equations, in rectangular regions.

===2.2 Hybrid schemes on non-rectangular regions===

The hybrid schemes for non-rectangular regions are obtained analogously as the ones for rectangular regions, with the difference that, in this case, instead of replacing the spatial partial derivatives for their finite difference approximation, they are replaced for the generalized finite difference approximations. To learn further about generalized finite difference method see <span id='citeF-8'></span><span id='citeF-9'></span><span id='citeF-10'></span><span id='citeF-11'></span><span id='citeF-12'></span><span id='citeF-13'></span>[[#cite-8|[8,9,10,11,12,13]]].

====2.2.1 Hybrid scheme for the conservative form====

For the case of the integral form of ([[#eq-2|2]]) and ([[#eq-3|3]]),

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>      \int _{\mathcal{C}_{(i, j)}}\left[\frac{\partial r}{\partial t} + \frac{\partial }{\partial x}\left(\frac{r^2}{a}\right)+ \frac{\partial }{\partial y}\left(\frac{rs}{a}\right)- fs + ga\frac{\partial q}{\partial x} + c_fr\frac{\sqrt{r^2+s^2}}{a^2}\right]dA = 0,     </math>
|}
|}

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>      \int _{\mathcal{C}_{(i, j)}}\left[\frac{\partial s}{\partial t} + \frac{\partial }{\partial x}\left(\frac{rs}{a}\right)+ \frac{\partial }{\partial y}\left(\frac{s^2}{a}\right)+ fr + ga\frac{\partial q}{\partial y} + c_fs\frac{\sqrt{r^2+s^2}}{a^2}\right]dA = 0     </math>
|}
|}

the spatial partial derivatives can be replaced by their generalized finite difference approximation, while the temporal partial derivatives will be replaced with their finite difference approximation.

In order to address the definition of generalized finite difference, it is convenient to consider, for each case, the approximation to the first order operator

<span id="eq-13"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>L_{qrs} = Aq_{x} + Bq_{y} + Cq + Dr_{x} + Er_{y} + Fr + Gs_{x} + Hs_{y}+ Is,           </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (13)
|}

where <math display="inline">A</math>, <math display="inline">B</math>, <math display="inline">C</math>, <math display="inline">D</math>, <math display="inline">E</math>, <math display="inline">F</math>, <math display="inline">G</math>, <math display="inline">H</math> and <math display="inline">I</math> are given functions; the operator at some point <math display="inline">p_0 = (x_0, y_0)</math> can be approximated using values of <math display="inline">\phi </math> in some neighbor points

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>      p_i = (x_i, y_i)i = 1, 2, \dots , m.     </math>
|}
|}

Thus, a finite difference scheme at <math display="inline">p_0</math> is a linear combination

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>      L_0 = \Gamma _0\phi (p_0) + \Gamma _1\phi (p_1) + \dots + \Gamma _m\phi (p_m),     </math>
|}
|}

where <math display="inline">\Gamma _0, \dots , \Gamma _m \in I\!\!R</math> are suitable weights.

Since operator ([[#eq-13|13]]) is partially separable, it can be rewritten as

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>      L_{qrs} = L_{q} + L_{r} + L_{s}     </math>
|}
|}

where

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>      L_{q} = Aq_{x} + Bq_{y} + Cq,     </math>
|}
|}

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>      L_{r} = Dr_{x} + Er_{y} + Fr,     </math>
|}
|}

and

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>      L_{s} = Gs_{x} + Hs_{y} + Is.     </math>
|}
|}

Each operator can be approximated separately, ''i.e.''

<span id="eq-14"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>L_{0q} = \Gamma _{10} q(p_0) + \Gamma _{11} q(p_1) + ... + \Gamma _{1m} q(p_m) = \sum _{i=0}^{m}\Gamma _{1i} q(p_i),           </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (14)
|}

<span id="eq-15"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>L_{0r} = \Gamma _{20} r(p_0) + \Gamma _{21} r(p_1) + ... + \Gamma _{2m} r(p_m) = \sum _{i=0}^{m}\Gamma _{2i}r(p_i),           </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (15)
|}

and

<span id="eq-16"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>L_{0s} = \Gamma _{30} s(p_0) + \Gamma _{31} s(p_1) + ... + \Gamma _{3m} s(p_m) = \sum _{i=0}^{m}\Gamma _{3i}s(p_i).</math>
|-
| style="text-align: center;" | 
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (16)
|}

A finite difference scheme <math display="inline">L_0</math> is consistent if the local truncation <math display="inline">\tau (p_0)</math> error satisfies

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>      \tau (p_0)=\left[L\phi \right]_{p_0} - L_0\rightarrow 0     </math>
|}
|}

as <math display="inline">p_1,\dots ,p_m \rightarrow p_0</math> <span id='citeF-14'></span><span id='citeF-15'></span>[[#cite-14|[14,15]]].

In this case, this means that

<span id="eq-17"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>Aq_{x} + Bq_{y} + Cq - \sum _{i=0}^{m}\Gamma _{1i} q(p_i) \to 0,           </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (17)
|}

<span id="eq-18"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>Dr_{x} + Er_{y} + Fr - \sum _{i=0}^{m}\Gamma _{2i} r(p_i) \to 0,           </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (18)
|}

and

<span id="eq-19"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>Gs_{x} + Hs_{y} + Is - \sum _{i=0}^{m}\Gamma _{3i} s(p_i) \to 0,           </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (19)
|}

Expanding ([[#eq-17|17]]), ([[#eq-18|18]]) and ([[#eq-19|19]]) in Taylor series and regrouping terms, they can be written as

<span id="eq-20"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>L_{q}(p_0) - L_0  \approx  \left[C(p_0) - \sum _{i=0}^{m}\Gamma _{1i}\right]q(p_0) + \left[A(p_0)-\sum _{i=1}^{m}\Gamma _{1i}\Delta x_i\right]q_x(p_0) + </math>
|-
| style="text-align: center;" | <math>              \left[B(p_0) - \sum _{i=1}^{m}\Gamma _{1i}\Delta y_i\right]q_y(p_0),      </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (20)
|}

<span id="eq-21"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>L_{q}(p_0) - L_0  \approx  \left[F(p_0) - \sum _{i=0}^{m}\Gamma _{2i}\right]r(p_0) +  \left[D(p_0) - \sum _{i=1}^{m}\Gamma _{2i}\Delta x_i\right]r_x(p_0) + </math>
|-
| style="text-align: center;" | <math>              \left[E(p_0) - \sum _{i=1}^{m}\Gamma _{2i}\Delta y_i\right]r_y(p_0).      </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (21)
|}

and

<span id="eq-22"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>L_{q}(p_0) - L_0  \approx  \left[I(p_0) - \sum _{i=0}^{m}\Gamma _{3i}\right]s(p_0) + \left[G(p_0) - \sum _{i=1}^{m}\Gamma _{3i}\Delta x_i\right]s_x(p_0) + </math>
|-
| style="text-align: center;" | <math>              \left[H(p_0) - \sum _{i=1}^{m}\Gamma _{2i}\Delta y_i\right]s_y(p_0).      </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (22)
|}

where <math display="inline">\Delta x_i = x_i - x_0</math>, <math display="inline">\Delta y_i = y_i - y_0</math> for <math display="inline">i = 0,1,\dots ,m</math>.

The consistency condition yields the undetermined systems

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>C(p_0) - \sum _{i=0}^{m}\Gamma _{1i}    =  0,</math>
|-
| style="text-align: center;" | <math>       A(p_0)-\sum _{i=1}^{m}\Gamma _{1i}\Delta x_i  =  0,</math>
|-
| style="text-align: center;" | <math>       B(p_0) - \sum _{i=1}^{m}\Gamma _{1i}\Delta y_i  =  0,</math>
|-
| style="text-align: center;" | 
|}
|}

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>F(p_0) - \sum _{i=0}^{m}\Gamma _{2i}    =  0,</math>
|-
| style="text-align: center;" | <math>       D(p_0) - \sum _{i=1}^{m}\Gamma _{2i}\Delta x_i  =  0,</math>
|-
| style="text-align: center;" | <math>       E(p_0) - \sum _{i=1}^{m}\Gamma _{2i}\Delta y_i  =  0,</math>
|-
| style="text-align: center;" | 
|}
|}

and

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>I(p_0) - \sum _{i=0}^{m}\Gamma _{3i}     =  0,</math>
|-
| style="text-align: center;" | <math>      G(p_0) - \sum _{i=1}^{m}\Gamma _{3i}\Delta x_i   =  0,</math>
|-
| style="text-align: center;" | <math>      H(p_0) - \sum _{i=1}^{m}\Gamma _{2i}\Delta y_i   =  0.     </math>
|}
|}

To find the <math display="inline">\Gamma </math> values that satisfy these conditions requiress to solve the systems

<span id="eq-23"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\left(       \begin{array}{cccc}1 & 1      & ... & 1\\        0 & \Delta x_{1}    & ... & \Delta x_{m}\\        0 & \Delta y_{1}    & ... & \Delta y_{m}\\       \end{array}      \right)      \left(       \begin{array}{c}\Gamma _{10}\\        \Gamma _{11}\\        \Gamma _{12}\\        .\\        .\\        .\\        \Gamma _{1m}\\       \end{array}      \right)      =      \left(       \begin{array}{c}C(p_0)\\        A(p_0)\\        B(p_0)\\       \end{array}      \right),           </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (23)
|}

<span id="eq-24"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\left(       \begin{array}{cccc}1 & 1      & ... & 1\\        0 & \Delta x_{1}    & ... & \Delta x_{m}\\        0 & \Delta y_{1}    & ... & \Delta y_{m}\\       \end{array}      \right)      \left(       \begin{array}{c}\Gamma _{20}\\        \Gamma _{21}\\        \Gamma _{22}\\        .\\        .\\        .\\        \Gamma _{2m}\\       \end{array}      \right)      =      \left(       \begin{array}{c}F(p_0)\\        D(p_0)\\        E(p_0)\\       \end{array}      \right),           </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (24)
|}

and

<span id="eq-25"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\left(       \begin{array}{cccc}1 & 1      & ... & 1\\        0 & \Delta x_{1}    & ... & \Delta x_{m}\\        0 & \Delta y_{1}    & ... & \Delta y_{m}\\       \end{array}      \right)      \left(       \begin{array}{c}\Gamma _{30}\\        \Gamma _{31}\\        \Gamma _{32}\\        .\\        .\\        .\\        \Gamma _{3m}\\       \end{array}      \right)      =      \left(       \begin{array}{c}I(p_0)\\        G(p_0)\\        H(p_0)\\       \end{array}      \right),           </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (25)
|}

it has to be noted that each system has <math display="inline">3</math> equations and <math display="inline">m+1</math> unknowns so, in general, in a suitable grid, there is a non trivial kernel. To select a solution, we use a subset of the normal equations of the corresponding least-squares problem; then, considering the last <math display="inline">2</math> equations in each system

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>      \left(       \begin{array}{ccc}        \Delta x_{1} & ... & \Delta x_{m}\\        \Delta y_{1} & ... & \Delta y_{m}\\       \end{array}      \right)      \left(       \begin{array}{c}        \Gamma _{11}\\        \Gamma _{12}\\        .\\        .\\        .\\        \Gamma _{1m}\\       \end{array}      \right)      =      \left(       \begin{array}{c}        A(p_0)\\        B(p_0)\\       \end{array}      \right),     </math>
|}
|}

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>          \left(           \begin{array}{cccc}            \Delta x_{1} & ... & \Delta x_{m}\\            \Delta y_{1} & ... & \Delta y_{m}\\           \end{array}          \right)          \left(           \begin{array}{c}            \Gamma _{21}\\            \Gamma _{22}\\            .\\            .\\            .\\            \Gamma _{2m}\\           \end{array}          \right)          =          \left(           \begin{array}{c}            D(p_0)\\            E(p_0)\\           \end{array}          \right),         </math>
|}
|}

and 

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>          \left(           \begin{array}{cccc}            \Delta x_{1} & ... & \Delta x_{m}\\            \Delta y_{1} & ... & \Delta y_{m}\\           \end{array}          \right)          \left(           \begin{array}{c}            \Gamma _{31}\\            \Gamma _{32}\\            .\\            .\\            .\\            \Gamma _{3m}\\           \end{array}          \right)          =          \left(           \begin{array}{c}            G(p_0)\\            H(p_0)\\           \end{array}          \right),         </math>
|}
|}

which can be solved through a reduced Cholesky factorization, the values of <math display="inline">\Gamma _{11},\dots ,\Gamma _{1m}, \Gamma _{21},\dots ,\Gamma _{2m}, \Gamma _{31},\dots ,\Gamma _{3m}</math> can be obtained.

After this, in order to obtain the values of <math display="inline">\Gamma _{10}</math>, <math display="inline">\Gamma _{20}</math> y <math display="inline">\Gamma _{30}</math>, the first equations in ([[#eq-23|23]]), ([[#eq-24|24]]) and ([[#eq-25|25]]), can be used

<span id="eq-26"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\Gamma _{10} = C(p_0) - \Gamma _{11} ... - \Gamma _{1m},           </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (26)
|}

<span id="eq-27"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\Gamma _{20} = F(p_0) - \Gamma _{21} ... - \Gamma _{2m},           </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (27)
|}

<span id="eq-28"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>\Gamma _{30} = G(p_0) - \Gamma _{31} ... - \Gamma _{3m}.           </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (28)
|}

The resulting <math display="inline">\Gamma </math> coefficients define the scheme ([[#eq-14|14]] - [[#eq-16|16]]).

This scheme can be used to approximate the operators

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>L_{q}   =  Aq_{x} + Bq_{y} + Cq,</math>
|-
| style="text-align: center;" | <math>      L_{r}   =  Dq_{x} + Eq_{y} + Fq,</math>
|-
| style="text-align: center;" | <math>      L_{s}   =  Gq_{x} + Hq_{y} + Iq.     </math>
|}
|}

in order to get, for each equation separately, the approximations

<span id="eq-29"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>r^{k+1}_{0} = r_{0}^{k} + \sum _{i=0}^{m}\left[\Gamma _{1i} q_{i}^{k} + \Gamma _{2i} r_{i}^{k} + \Gamma _{3i}s_{i}^{k} - fs^{k}_{i} + \frac{c_f r^{k}_{i}}{a^2}\sqrt{(r^{k}_{i})^2+(s^{k}_{i})^2}\right],           </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (29)
|}

<span id="eq-30"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>s^{k+1}_{0} = s_{0}^{k} + \sum _{i=0}^{m}\left[\Gamma _{1i} q_{i}^{k} + \Gamma _{2i} r_{i}^{k} + \Gamma _{3i}s_{i}^{k} + fr^{k}_{i} + \frac{c_f s^{k}_{i}}{a^2}\sqrt{(r^{k}_{i})^2+(s^{k}_{i})^2}\right],           </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (30)
|}

Now, for equation ([[#eq-1|1]])

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>      \frac{\partial q}{\partial t} + \frac{\partial r}{\partial x} + \frac{\partial s}{\partial y} = 0,     </math>
|}
|}

a similar path must be taken in order to get a generalized finite difference approximation

<span id="eq-31"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>q^{k+1}_{0} = q^{k}_{0} + \sum _{i=0}^m \left[\Gamma _{2i} r_{i}^{k} + \Gamma _{3i} s_i^{k}\right],           </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (31)
|}

Equations ([[#eq-29|29]]), ([[#eq-30|30]]) and ([[#eq-31|31]]) define a ''hybrid Generalized Finite Difference-Volume scheme'', for the '''conservative form''', for non-rectangular regions.

====2.2.2 Hybrid scheme for the non-conservative form====

In an analogous way, for the non-conservative form of the shallow-water equations, if the integral form of equations ([[#eq-5|5]]) and ([[#eq-6|6]]) is taken into account

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>      \int _{\mathcal{C}_{(i, j)}}\left[\frac{\partial r}{\partial t} + \frac{2r}{a}\frac{\partial r}{\partial x} + ga\frac{\partial q}{\partial x} + \frac{r}{a}\frac{\partial s}{\partial y} + \frac{s}{a}\frac{\partial r}{\partial y} - fs + c_f \frac{r}{a^2}\sqrt{r^2+s^2}\right]dA = 0,     </math>
|}
|}

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>      \int _{\mathcal{C}_{(i, j)}}\left[\frac{\partial s}{\partial t} + \frac{r}{a}\frac{\partial s}{\partial x} + \frac{s}{a}\frac{\partial r}{\partial x} + \frac{2s}{a}\frac{\partial s}{\partial y} + ga\frac{\partial q}{\partial y} + fr + c_f \frac{s}{a^2}\sqrt{r^2+s^2}\right]dA = 0,     </math>
|}
|}

and a generalized finite difference scheme is applied to equation ([[#eq-4|4]]),

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>      \frac{\partial q}{\partial t} + \frac{\partial r}{\partial x} + \frac{\partial s}{\partial y} = 0,     </math>
|}
|}

the a scheme defined by

<span id="eq-32"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>r^{k+1}_{0} = r_{0}^{k} + \sum _{i=0}^{m}\left[\Gamma _{1i} q_{i}^{k} + \Gamma _{2i} r_{i}^{k} + \Gamma _{3i}s_{i}^{k} - fs^{k}_{i} + \frac{c_f r^{k}_{i}}{a^2}\sqrt{(r^{k}_{i})^2+(s^{k}_{i})^2}\right],           </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (32)
|}

<span id="eq-33"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>s^{k+1}_{0} = s_{0}^{k} + \sum _{i=0}^{m}\left[\Gamma _{1i} q_{i}^{k} + \Gamma _{2i} r_{i}^{k} + \Gamma _{3i}s_{i}^{k} + fs^{k}_{i} + \frac{c_f s^{k}_{i}}{a^2}\sqrt{(r^{k}_{i})^2+(s^{k}_{i})^2}\right],           </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (33)
|}

<span id="eq-34"></span>
{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>q^{k+1}_{0} = q^{k}_{0} + \sum _{i=0}^m \left[\Gamma _{2i} r_{i}^{k} + \Gamma _{3i} s_i^{k}\right],           </math>
|}
| style="width: 5px;text-align: right;white-space: nowrap;" | (34)
|}

can be found.

Now equations ([[#eq-32|32]]), ([[#eq-33|33]]) and ([[#eq-34|34]]) define a ''hybrid Generalized Finite Difference-Volume scheme'', for the '''non-conservative form''', for non-rectangular regions.

Now, an adequate selection of <math display="inline">m</math>, the number of points, used by the scheme has to be done in order to represent different characteristics accurately. In this work, the selection of <math display="inline">m</math> has been done as follows: <math display="inline">m = 4</math> for <math display="inline">q</math>, and <math display="inline">m = 6</math> for <math display="inline">r</math> and <math display="inline">s</math>. The chosen stencils are shown in figure ([[#img-3|3]]).     <div id='img-3'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Review_838453312870-fig03.png|600px|'''Different stencils used by the scheme.''']]
|[[Image:Review_838453312870-fig05.png|292px|]]
|- style="text-align: center; font-size: 75%;"
| (3) '''Different stencils used by the scheme.'''
|-
| colspan="2"|[[Image:Review_838453312870-fig04.png|420px|]]
|}

==3 Numerical tests==

For the numerical tests, three different regions were selected: The unit square for the classical finite difference-finite volume hybrid scheme (denoted as QUAD), a widely used geometry denoted as DOME <span id='citeF-16'></span>[[#cite-16|[16]]], that is a concave region limited by the lines <math display="inline">x=0</math>, <math display="inline">x=1</math>, <math display="inline">y = 0</math> and <math display="inline">y = \frac{3}{4}+\frac{1}{4}\sin{\left(\pi \left(\frac{1}{2}-2x\right)\right)}</math> and an approximation to Zirahuen's Lake in Mexico (an endorheic basin), denoted as ZIRA. The corresponding normalized meshes with <math display="inline">40\times{40}</math> cells can be seen in figures ([[#img-4|4]]-[[#img-6|6]]).

<div id='img-4'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Review_838453312870-fig06.png|360px|Regular mesh for the unitary square.]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 4:''' Regular mesh for the unitary square.
|}

<div id='img-5'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Review_838453312870-fig07.png|360px|Mesh for DOME region.]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 5:''' Mesh for DOME region.
|}

<div id='img-6'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Review_838453312870-fig08.png|360px|Mesh for Zirahuen's Lake region.]]
|- style="text-align: center; font-size: 75%;"
| colspan="1" | '''Figure 6:''' Mesh for Zirahuen's Lake region.
|}

For each region 2 different tests were done, the first one using the proposed scheme for the '''conservative form''' of the Equations, and the second using the scheme for the '''non-conservative form'''.

The initial condition, for both tests, for <math display="inline">q</math> was chosen to be a droplet with different center, according to the region, as follows:

* QUAD: center  in <math display="inline">(.5,.3)</math>.
* DOME: center in <math display="inline">(.2,.4)</math>
* ZIRA: center in <math display="inline">(.35,.4)</math>.

The initial conditions for <math display="inline">r</math> and <math display="inline">s</math> where fixed as <math display="inline">0</math> for all the regions. Also, reflective boundary conditions were chosen for <math display="inline">r</math> and <math display="inline">s</math>.

To produce stable calculations <span id='citeF-17'></span>[[#cite-17|[17]]], the time discretization was chosen by considering

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>     \Delta t = c\frac{\min (\Delta x)\min (\Delta y)}{\min (\Delta x) + \min (\Delta y)}    </math>
|}
|}

where <math display="inline">\min \Delta x</math> and <math display="inline">\min \Delta y</math> are the minimum values of <math display="inline">\Delta x</math> and <math display="inline">\Delta y</math> over the region and <math display="inline">c = 0.5\frac{t}{d}</math>. With this, the time interval <math display="inline">(0s,5s)</math> was divided into <math display="inline">50,000</math> steps.

The total amounts of mass and momentum, over all the domain, are given at a time step <math display="inline">k</math> as

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>     G^k_q = \int _\Omega \left(q^k\right)dx dy,    </math>
|}
|}

and

{| class="formulaSCP" style="width: 100%; text-align: left;" 
|-
| 
{| style="text-align: left; margin:auto;width: 100%;" 
|-
| style="text-align: center;" | <math>     G^k_v = \int _\Omega \left(\sqrt{\left(r^k\right)^2+\left(s^k\right)^2}\right)dx dy.    </math>
|}
|}

In our case, they were approximated by means of a numerical quadrature.

The set of figures ([[#img-7|7]]) and ([[#img-8|8]]) show the results for the test using the '''conservative form''' of the equations in QUAD region, while the set of figures ([[#img-9|9]]) and ([[#img-10|10]]) show the results '''non-conservative form''' one for the same region. Following the same idea, the results for the region DOME are shown in the sets of figures ([[#img-11|11]]) to ([[#img-14|14]]), the same for ZIRA region in the sets ([[#img-15|15]]) to ([[#img-18|18]]). These sets of figures are shown as follows: the figures on the left show the behavior of the velocities while the figures on the right show the movement of the water. The figures begin showing the second time step (t = 0.001s) and continue with a plot every of the results every second after (t = 1s, t = 2s, t = 3s, t = 4s, t = 5s).

Figures ([[#img-19|19]]) to ([[#img-21|21]]) show the result of the computed total amount of conservative quantities over all the time steps. In all these figures the blue line represents the total amount of mass while the red-dotted line represents the total amount of momentum.

<div id='img-7'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Review_838453312870-fig09.png|570px|Results for QUAD in the conservative case.]]
|[[Image:Review_838453312870-fig10.png|570px|]]
|- style="text-align: center; font-size: 75%;"
| (7) Results for QUAD in the conservative case.
|-
| colspan="2"|[[Image:Review_838453312870-fig11.png|570px|]]
|}

<div id='img-8'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Review_838453312870-fig12.png|570px|Results for QUAD in the conservative case.]]
|[[Image:Review_838453312870-fig13.png|570px|]]
|- style="text-align: center; font-size: 75%;"
| (8) Results for QUAD in the conservative case.
|-
| colspan="2"|[[Image:Review_838453312870-fig14.png|570px|]]
|}

<div id='img-9'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Review_838453312870-fig15.png|570px|Results for QUAD in the non-conservative case.]]
|[[Image:Review_838453312870-fig16.png|570px|]]
|- style="text-align: center; font-size: 75%;"
| (9) Results for QUAD in the non-conservative case.
|-
| colspan="2"|[[Image:Review_838453312870-fig17.png|570px|]]
|}

<div id='img-10'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Review_838453312870-fig18.png|570px|Results for QUAD in the non-conservative case.]]
|[[Image:Review_838453312870-fig19.png|570px|]]
|- style="text-align: center; font-size: 75%;"
| (10) Results for QUAD in the non-conservative case.
|-
| colspan="2"|[[Image:Review_838453312870-fig20.png|570px|]]
|}

<div id='img-11'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Review_838453312870-fig21.png|570px|Results for DOME in the conservative case.]]
|[[Image:Review_838453312870-fig22.png|570px|]]
|- style="text-align: center; font-size: 75%;"
| (11) Results for DOME in the conservative case.
|-
| colspan="2"|[[Image:Review_838453312870-fig23.png|570px|]]
|}

<div id='img-12'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Review_838453312870-fig24.png|570px|Results for DOME in the conservative case.]]
|[[Image:Review_838453312870-fig25.png|570px|]]
|- style="text-align: center; font-size: 75%;"
| (12) Results for DOME in the conservative case.
|-
| colspan="2"|[[Image:Review_838453312870-fig26.png|570px|]]
|}

<div id='img-13'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Review_838453312870-fig27.png|570px|Results for DOME in the non-conservative case.]]
|[[Image:Review_838453312870-fig28.png|570px|]]
|- style="text-align: center; font-size: 75%;"
| (13) Results for DOME in the non-conservative case.
|-
| colspan="2"|[[Image:Review_838453312870-fig29.png|570px|]]
|}

<div id='img-14'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Review_838453312870-fig30.png|570px|Results for DOME in the non-conservative case.]]
|[[Image:Review_838453312870-fig31.png|570px|]]
|- style="text-align: center; font-size: 75%;"
| (14) Results for DOME in the non-conservative case.
|-
| colspan="2"|[[Image:Review_838453312870-fig32.png|570px|]]
|}

<div id='img-15'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Review_838453312870-fig33.png|570px|Results for ZIRA in the conservative case.]]
|[[Image:Review_838453312870-fig34.png|570px|]]
|- style="text-align: center; font-size: 75%;"
| (15) Results for ZIRA in the conservative case.
|-
| colspan="2"|[[Image:Review_838453312870-fig35.png|570px|]]
|}

<div id='img-16'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Review_838453312870-fig36.png|570px|Results for ZIRA in the conservative case.]]
|[[Image:Review_838453312870-fig37.png|570px|]]
|- style="text-align: center; font-size: 75%;"
| (16) Results for ZIRA in the conservative case.
|-
| colspan="2"|[[Image:Review_838453312870-fig38.png|570px|]]
|}

<div id='img-17'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Review_838453312870-fig39.png|570px|Results for ZIRA in the non-conservative case.]]
|[[Image:Review_838453312870-fig40.png|570px|]]
|- style="text-align: center; font-size: 75%;"
| (17) Results for ZIRA in the non-conservative case.
|-
| colspan="2"|[[Image:Review_838453312870-fig41.png|570px|]]
|}

<div id='img-18'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Review_838453312870-fig42.png|570px|Results for ZIRA in the non-conservative case.]]
|[[Image:Review_838453312870-fig43.png|570px|]]
|- style="text-align: center; font-size: 75%;"
| (18) Results for ZIRA in the non-conservative case.
|-
| colspan="2"|[[Image:Review_838453312870-fig44.png|570px|]]
|}

<div id='img-19'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Review_838453312870-fig45.png|570px|G<sub>q</sub> and G<sub>v</sub> for QUAD.]]
|[[Image:Review_838453312870-fig46.png|570px|]]
|- style="text-align: center; font-size: 75%;"
| (19) <math>G_q</math> and <math>G_v</math> for QUAD.
|}

<div id='img-20'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Review_838453312870-fig47.png|570px|G<sub>q</sub> and G<sub>v</sub> for DOME.]]
|[[Image:Review_838453312870-fig48.png|570px|]]
|- style="text-align: center; font-size: 75%;"
| (20) <math>G_q</math> and <math>G_v</math> for DOME.
|}

<div id='img-21'></div>
{| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
|-
|[[Image:Review_838453312870-fig49.png|570px|G<sub>q</sub> and G<sub>v</sub> for ZIRA.]]
|[[Image:Review_838453312870-fig50.png|570px|]]
|- style="text-align: center; font-size: 75%;"
| (21) <math>G_q</math> and <math>G_v</math> for ZIRA.
|}

It must be pointed out that, in figures ([[#img-19|19]]) to ([[#img-21|21]]), the conservation of mass and momentum can be appreciated, since the losses and gains of these conservative quantities, over all the computed time, is small (around <math display="inline">10^{-5}</math>).

==4 Conclusions and future work==

Figures ([[#img-19|19]]) to ([[#img-21|21]]) show that, with the criteria used to select <math display="inline">\Delta t</math>, neither spurious oscillations nor instabilities are observed in the tests. Even when these schemes can be applied to non-rectangular regions with ease, the quality of the grid is an important issue that must be taken into account; it is convenient to have quality grids to obtain better results, in all the tests of the present work quality grids were used.

It is important to remark that, in the tests, the boundary conditions where selected to be reflective and the tests show that, both proposed Generalized Finite Difference-Volume hybrid schemes, show a remarkable ability to produce stable results in the selected regions, even in the irregular cases; the numerical tests show that the conservation laws (mass and momentum) are fulfilled, correctly reflecting the expected behavior of a water body. As future work, these schemes need to be implemented in more realistic scenarios, which include inflows, outflows and non-slip boundary conditions.

==Acknowledgments==

We want to thank AULA CIMNE-Morelia and Finnish Government Scholarship Pool KM-17-10397 for the financial support for this work. We are grateful to the University of Helsinki for giving all the necessary work materials and the space to work at the Department of Physics in Helsinki, Finland.

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