MSC 82D15, 76A20, 76M25 DOI: 10.14529/mmp170313 LOCATION · msc 82d15, 76a20, 76m25 doi:...

MSC 82D15, 76A20, 76M25 DOI: 10.14529/mmp170313

COMPUTER SIMULATION OF THE VOLGA RIVER HYDROLOGICALREGIME: PROBLEM OF WATER-RETAINING DAM OPTIMALLOCATION

E.O. Agafonnikova, A.Yu. Klikunova, A.V. Khoperskov

Volgograd State University, Volgograd, Russian FederationE-mail: [email protected], [email protected], [email protected]

We investigate of a special dam optimal location at the Volga river in the area of theAkhtuba left sleeve beginning (7 km to the south of the Volga Hydroelectric Power Stationdam). We claim that a new water-retaining dam can resolve the key problem of the Volga-Akhtuba oodplain related to insucient water amount during spring oodings due to theoverregulation of the Lower Volga. Using a numerical integration of Saint-Vanant equationswe study the water dynamics across the northern part of the Volga-Akhtuba oodplaintaking into account its actual topography. As the result we found an amount of water VA

passing to the Akhtuba during spring period for a given water ow through the VolgaHydroelectric Power Station (so-called hydrograph which characterises the water ow perunit of time). By varying the location of the water-retaining dam xd, yd we obtained variousvalues of VA(xd, yd) as well as various ow spatial structure on the territory during the oodperiod. Gradient descent method provides the dam coordinated with the maximum value ofVA. Such approach to the dam location choice let us nd the best solution, that the valueVA increases by a factor of 2. Our analysis demonstrates a good potential of the numericalsimulations in the eld of hydraulic works.

Keywords: hydrodynamic simulation; Saint-Venant equations; numerical model;

optimization; hydrology.

Introduction. The unique landscape of 20000 km2 Volga-Akhtuba oodplain (VAF)depends on special features of the interuve hydrological regime. During the spring oodperiod the area between the Volga and Akhtuba rivers is heavily ooded [13], that ensuresa special composition of ora and fauna and possibility of agricultural use of the areaincluding the development of magnicent gardens and melon elds. The oodplain isalso the basis for the sh reproduction at the Lower Volga region [4]. Nowadays theoverregulation of the Volga-Kama basin hydrological regime by 22 Hydroelectric PowerPlants leads to the VAP degradation.

Various approaches to the problem solution have beed proposed. Let us point out theattempts to construct so-called optimal hydrograph Q(t) [57] which is close to the naturaland ensures preservation of the natural rate. Despite the progress in the construction ofthe mathematical and hydrological regime control problems and the territory as a whole[811], there is a great diculty of their practical implementation due to the conict ofvarious agents aspiration (energetics, environmental protection organizations, sh industry,agriculture, inhabitants, safety of reservoirs, etc.). Over the last decades the situationbecomes even more complicated due to changes in the Volga riverbed below the dammainly because of the violation of the spring ood natural process. In this study we discussthe possibility of the oodplain hydrological regime improvement by the construction ofthe water-retaining dam at the Volga riverbed close to the beginning of the Akhtuba leftsleeve which is located approximately 7 km below the Volga Hydroelectric Power Stationdam (Fig. 1). The main aim of the work is construction of a mathematical model for the

148 Bulletin of the South Ural State University. Ser. Mathematical Modelling, Programming& Computer Software (Bulletin SUSU MMCS), 2017, vol. 10, no. 3, pp. 148155

ÊÐÀÒÊÈÅ ÑÎÎÁÙÅÍÈß

Fig. 1. The map of the northern part of the Volga-Akhtuba oodplain with the topography.The insert shows the part of the Volga with the Akhtuba springhead

estimation of the dam optimal location which provides the larger water ow rate in theAkhtuba during the spring ood.

1. The Hydrodynamic Model. At rst, we briey describe the hydrodynamic modelunderlying the base of our research. We use the Saint-Venant equations for the shallowwater dynamics at a given topography b(x, y) [1, 9, 12]:

∂H

∂t+

∂uH

∂x+

∂vH

∂y= σ(x, y, t) , (1)

∂(uH)

∂t+

∂(u2H)

∂x+

∂(uvH)

∂y= −gH

∂(H + b)

∂x+ 2vHΩE sin(θ) + f fric

x + fσx , (2)

∂(vH)

∂t+

∂(uvH)

∂x+

(v2H)

∂y= −gH

∂(H + b)

∂y− 2uHΩE sin(θ) + f fric

y + fσy , (3)

where H is the water depth, u and v are the velocity of x- and y-components (averagedvertically), σ is the source function, g is referred to the gravitational acceleration, ΩE isthe Earth's angular velocity, θ is the latitude, the values of fσx and fσy describe the waterimpulse associated with the sources σ. For the bottom friction force vector we use theChezy's model [1]:

f fricx = −u

2

√u2 + v2HΛ , f fric

y = −v

2

√u2 + v2HΛ, (4)

with the value of hydraulic friction Λ = 2gn2M/H4/3 and the Manning roughness

coecient nM .Value of nM is determined by the surface properties and generally it depends on the

coordinates. Moreover, for the unsteady ow regime the Manning coecient can vary withtime [1]. The hydrograph through the Volga Hydroelectric Power Station dam allows to setthe value of σ = dQ0

dS(where dS = dx dy is an elementary area). Under the VAP conditions

the value of nM for the Volga riverbed varies in the range 0,02 0,07 [1].For the numerical solution of equations (1) (3) we apply our combined Lagrangian

Eulerian method (cSPH-TVD) [13] which uses the benets of Smoothed Particle

Âåñòíèê ÞÓðÃÓ. Ñåðèÿ ≪Ìàòåìàòè÷åñêîå ìîäåëèðîâàíèåè ïðîãðàììèðîâàíèå≫ (Âåñòíèê ÞÓðÃÓ ÌÌÏ). 2017. Ò. 10, 3. Ñ. 148155

149


Hydrodynamics at dierent time steps and Total Variation Diminishing (see the detaileddescription in [1, 14]).

The most important positive characteristics of the cSPH-TVD approach are thefollowing: an adequate calculation of the dynamic boundaries between wet and dry beds in caseof non-stationary uxes through the strongly inhomogeneous bottom (even through thediscontinuous topography); calculation for subcritical (with Froude number Fr =

√u2 + v2/

√gH < 1) and

supercritical (Fr > 1) uxes without isolation of these zones; numerical scheme of CSPH-TVD is conservative, well-balanced and has the second orderof accuracy for smooth solutions the rst order accuracy approximation in the vicinity ofbreaks and fracture proles.

Both non steady solutions and strong heterogeneity of the topography require thespecic boundary conditions formulation. Ref. [15] is dedicated to the application of theboundary conditions for the same aims by using the conditions of "waterfall" type, whichwe adopted in the current study.

The software implementation by using parallel technologies on graphics acceleratorsis presented in [12]. All basic calculations were performed on the GPU NVIDIA TeslaK80 [12]. We use Digital Elevation Models (DEM) with spatial resolution ∆x = ∆y =50m and 25m, which is based on combination of several geodata: ASTER GDEM 2(Global DIGITAL Elevation Model), SRTM X-SAR (Shuttle Radar Topography Mission)and Sentinel-1 SAR data, topographic data for coastlines of the hydrological system, ourGPS / GLONASS measurements. To improve the model we use the sailing directions andspecial numerical hydrodynamical simulations allowing us to compare our results withobservational data.

2. The Optimal Dam Location. For a given Volga Hydroelectric Power Stationhydrograph Q0(t) we can calculate the water volume entering the Akhtuba during thespring ood (see Fig. 1):

VA =

B∫A

tQe∫tQs

H(x, y, t) (u · nx + v · ny) dt dℓ , (5)

where the unit vector n = (nx, ny) is a normal to the Akhtuba river section line (A,B)(see. Fig. 3 b), tQs and tQe are the water release at beginning and nal time, respectively(see. Fig. 2). Functions H(x, y, t), u(x, y, t) and v(x, y, t) are calculated by using thehydrodynamical model (1) (3). We set up the dam of length Ld at the Volga riverbedclose to the Akhtuba springhead at the point (xd, yd) (dam's center) which is perpendicularto the coastlines. The dam aects on the ow structure and on the value of VA. For a givenLd and Q0(t) we have function VA(xd, yd). We calculate the water-retaining dam optimallocation for the specic riverbed area SA according to the following condition:

V(max)A = max

(xd,yd)∈SA

VA(xd, yd) , (6)

SA is part of the Volga riverbed near the beginning of the Akhtuba (about 7 km downstreamand 5 km upstream).

In Fig. 3 we show the V(max)A search procedure by using the gradient descent method:

r(k+1)d = r

(k)d + λ(k) grad(VA(r

(k)d )) . (7)



Fig. 2. The spring hydrograph Q0(t) of 2016 through the Volga Hydroelectric PowerStation dam adopted in the model (a). The hydrographs of Akhtuba QA at dierent dampositions (b): 1 the best dam position; 2 without dam; 3 the worst dam position

Finite dierence approximation for the gradient calculation is used in the following form:

grad(VA(rd)) ≃VA(xd + δx, yd)− VA(xd, yd)

δx;VA(xd, yd + δy)− VA(xd, yd)

δy

(8)

on the meshgrid xi+1 = xi + ∆x, yj+1 = yj + ∆y. Our test numerical simulationsdemonstrated that the relations δx = δy = 2∆x = 2∆y are reasonable choice.

Fig. 3. The results of numerical hydrodynamical simulations for the function VA(xD, yD)

and examples of the iterative procedure for the calculation of V(max)A in case of Ld = 300m

The symbols in Fig. 3 a indicate the positions of the iterative procedure (7), (8). Thechoice of λk parameter and the convergence of calculations are achieved by using thesteepest descent method.


151


3. Results and Discussion. By means of numerical simulations we have studied theproblem of the water-retaining dam location optimization at the Volga riverbed with theaim to increase the water amount at the left sleeve of the river Akhtuba. We summarizeour results as following. There are several dam locations providing the appearance of thewater volume local maximum VA in the Akhtuba during the spring ood. The positions(x

(max)dm , y

(max)dm ) (m = 1, 2, 3) are located about 6 km downstream from the Akhtuba's

beginning, that caused by the Volga riverbed structure at the area due to the large Volgawidth nearby the Akhtuba and large island (see Fig. 1). The best solution is V

(max)A =

75000m3. As a result, we can achieve a factor of 1,6 for maximum value of the hydrographQA(t) (see Fig. 2) and almost factor of 2 for VA from VA0 = 43000m3 up to VA0 = 75000m3

in comparison to the absence of the extra dam (VA0 = 43000m3).For a xed set of free parameters any deviation of the dam orientation from the

perpendicular relatively to the coastlines reduces the value VA. The optimum locationslightly depends on the dam size. It should be noted that there are some dam positions thatdetermine the value of VA which is smaller than in the case of absence of the water-retainingdam. Our analysis is robust to small scale perturbations of the digital topography. A smallvariation of the function b(x, y) conserve the approximate optimal solution (x

(max)d , y

(max)d ),

but the issue still requires further investigation.

Acknowledgements. We are thankful to the Ministry of Education and Scienceof the Russian Federation (project 2.852.2017/4.6). The study was supported by theSupercomputing Center of Lomonosov Moscow State University. EOA is thankful to theRFBR (grants 16-07-01037, 15-45-02655).

References

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2. Middelkoop H., Alabyan A.M., Babich D.B., Ivanov V.V. Post-Dam Channel and FloodplainAdjustment Along the Lower Volga River, Russia. Geomorphic Approaches to Integrated

Floodplain Management of Lowland Fluvial Systems in North America and Europe. Springer,2015, pp. 245264. DOI: 10.1007/978-1-4939-2380-9_10

3. Ladjel M. Lamination Method of Flood Wadis and Projection of the Laminated FloodHydrograph. Journal of Fundamental and Applied Sciences, 2016, vol. 8, no. 1, pp. 8391.DOI: 10.4314/jfas.v8i1.6

4. Gorski K., van den Bosch L.V., van de Wolfshaar K.E., Middelkoop H., Nagelkerke L.A.J.,Filippov O.V. et al. Post-Damming Flow Regime Development in a Large Lowland River(Volga, Russian Federation): Implications for Floodplain Inundation and Fisheries. RiverResearch and Applications, 2012, vol. 28, no. 8, pp. 11211134. DOI: 10.1002/rra.1499.

5. Voronin A.A., Eliseeva M.V., Khrapov S.S., Pisarev A.V., Khoperskov A.V. [The RegimenControl Task in The Eco-Economic System "Volzhskaya Hydroelectric Power Station theVolga-Akhtuba Floodplain". II. Synthesis of Control System]. Control Sciences, 2012, no. 6,pp. 1925. (in Russian)

6. Chen D., Li R., Chen Q., Cai D. Deriving Optimal Daily Reservoir Operation Scheme withConsideration of Downstream Ecological Hydrograph Through a Time-Nested Approach.Water Resources Management, 2015, vol. 29, no. 9, pp. 33713386. DOI: 10.1007/s11269-015-1005-z



7. Haghighi A.T., Kløve B. Development of Monthly Optimal Flow Regimes forAllocated Environmental Flow Considering Natural Flow Regimes and SeveralSurface Water Protection Targets. Ecological Engineering, 2015, vol. 82, pp. 390399.DOI: 10.1016/j.ecoleng.2015.05.035

8. Borsche R., Klar A. Flooding in Urban Drainage Systems: Coupling Hyperbolic ConservationLaws for Sewer Systems and Surface Flow. International Journal for Numerical Methods in

Fluids, 2014, vol. 76, pp. 789810. DOI: 10.1002/d.3957

9. Voronin A.A., Vasilchenko A.A., Pisarev A.V., Khrapov S.S., Radchenko Yu.E. [DesigningMechanisms of the Hydrological Regime Management of the Volga-Akhtuba FloodplainBased on Geoinformation and Hydrodynamic Modelling]. Science Journal of Volgograd StateUniversity. Mathematics. Physics, 2016, no. 1 (32), pp. 2437. DOI: 10.15688/jvolsu1.2016.1.3(in Russian)

10. Izem N., Seaid M., Wakrim M. A Discontinuous Galerkin Method for Two-Layer ShallowWater Equations. Mathematics and Computers in Simulation, 2016, vol. 120, pp. 1223.DOI: 10.1016/j.matcom.2015.04.009

11. Bulatov O.V., Elizarova T.G. Regularized Shallow Water Equations for Numerical Simulationof Flows with a Moving Shoreline. Computational Mathematics and Mathematical Physics,2016, vol. 56, no. 4, pp. 661679. DOI: 10.1134/S0965542516040047

12. D'yakonova, T.A., Khoperskov A.V., Khrapov S.S. Numerical Model of Shallow Water: theUse of GPUs NVIDIA CUDA. Communications in Computer and Information Science, 2017,vol. 687, pp. 132145.

13. Khrapov S.S., Khoperskov A.V., Kuz'min N.M., Pisarev A.V., Kobelev I.A. [A NumericalScheme for Simulating the Dynamics of Surface Water on the Basis of the Combined SPH-TVD Approach]. Numerical Methods and Programming, 2011, vol. 12, pp. 282297. (inRussian)

14. Khrapov S.S., Kuzmin N.M., Butenko M.A. [The Comparison of Accuracy and Convergencefor the CSPHTVD Method and Some Eulerian Schemes for Solving Gas-DynamicEquations]. Science Journal of Volgograd State University. Mathematics. Physics, 2016, no. 6(37), pp. 166173. DOI: 10.15688/jvolsu1.2016.6.15 (in Russian)

15. D'yakonova, T.A., Khrapov S.S., Khoperskov A.V. [The Problem of Boundary Conditions forthe Shallow Water Equations]. Vestnik Udmurtskogo universiteta. Matematika. Mekhanika.

Komp'yuternye nauki [The Bulletin of Udmurt University. Mathematics. Mechanics.Computer Science], 2016, vol. 26, no. 3, pp. 401417. DOI: 10.20537/vm160309 (in Russian)

Received April 10, 2017

MSC 82D15, 76A20, 76M25 DOI: 10.14529/mmp170313

ÊÎÌÏÜÞÒÅÐÍÎÅ ÌÎÄÅËÈÐÎÂÀÍÈÅ ÃÈÄÐÎËÎÃÈ×ÅÑÊÎÃÎÐÅÆÈÌÀ ÂÎËÃÈ: ÇÀÄÀ×À ÎÏÒÈÌÀËÜÍÎÃÎ ÐÀÑÏÎËÎÆÅÍÈßÂÎÄÎÍÀÏÎÐÍÎÉ ÄÀÌÁÛ

Å.Î. Àãàôîííèêîâà, À.Þ. Êëèêóíîâà, À.Â. Õîïåðñêîâ

Âîëãîãðàäñêèé ãîñóäàðñòâåííûé óíèâåðñèòåò, ã. Âîëãîãðàä

Ðàññìîòðåíà çàäà÷à îïòèìàëüíîãî ðàñïîëîæåíèÿ ñïåöèàëüíîé äàìáû â ðóñëå Âîë-

ãè â çîíå íà÷àëà åå ëåâîãî ðóêàâà Àõòóáà (7 êì þæíåå ïëîòèíû Âîëæñêîé ÃÝÑ). Íîâàÿ


153


âîäîïîäïîðíàÿ äàìáà ñïîñîáíà ðåøèòü êëþ÷åâóþ ïðîáëåìó Âîëãî-Àõòóáèíñêîé ïîé-ìû, ñâÿçàííóþ ñ íåäîñòàòî÷íûì îáúåìîì âîäû â ïåðèîä âåñåííåãî ïàâîäêà èç-çà çà-ðåãóëèðîâàííîñòè Íèæíåé Âîëãè. Â îñíîâå ìàòåìàòè÷åñêîé ìîäåëè ðàñ÷åòà äèíàìèêèâîäû ëåæèò ÷èñëåííîå èíòåãðèðîâàíèå íåñòàöèîíàðíûõ óðàâíåíèé Ñåí-Âåíàíà äëÿðåàëüíîãî ðåëüåôà ìåñòíîñòè ñåâåðíîé ÷àñòè Âîëãî-Àõòóáèíñêîé ïîéìû. Ðåçóëüòà-òîì òàêîãî ìîäåëèðîâàíèÿ ÿâëÿåòñÿ ðàñ÷åò îáúåìà âîäû VA, ïðîõîäÿùèé çà âåñåííèéïåðèîä â Àõòóáó äëÿ îïðåäåëåííîãî ïîòîêà âîäû ÷åðåç ãèäðîýëåêòðîñòàíöèþ (òàê íà-çûâàåìûé ãèäðîãðàô õàðàêòåðèçóåò îáúåì âîäû, ïðîòåêàþùåé â åäèíèöó âðåìåíè).Âàðüèðóÿ ïîëîæåíèå âîäîïîäïîðíîé äàìáû xd, yd, ìû ïîëó÷àåì ðàçëè÷íóþ ñòðóêòóðóòå÷åíèÿ íà òåððèòîðèè â ïåðèîä ïàâîäêà è ðàçëè÷íûå çíà÷åíèÿ VA(xd, yd). Èñïîëü-çîâàíèå ìåòîäà ãðàäèåíòíîãî ñïóñêà äàåò êîîðäèíàòû äàìáû ñ ìàêñèìàëüíûì çíà÷å-íèåì VA. Òàêîé ïîäõîä ê âûáîðó ïîëîæåíèÿ äàìáû ïîçâîëÿåò ðàññ÷èòàòü íàèáîëååîïòèìàëüíîå ðåøåíèå, òàê ÷òî âåëè÷èíà VA óâåëè÷èâàåòñÿ ïðèìåðíî â 2 ðàçà. Íàøåèññëåäîâàíèå äåìîíñòðèðóåò áîëüøèå âîçìîæíîñòè ìàòåìàòè÷åñêîãî ìîäåëèðîâàíèÿïðè ïðîåêòèðîâàíèè òàêîãî ðîäà ãèäðîñîîðóæåíèé.

Êëþ÷åâûå ñëîâà: ãèäðîäèíàìè÷åñêîå ìîäåëèðîâàíèå; óðàâíåíèå Ñåí-Âåíàíà; ÷èñ-

ëåííàÿ ìîäåëü; îïòèìèçàöèÿ; ãèäðîëîãèÿ.

Ëèòåðàòóðà

1. Khrapov, S. The Numerical Simulation of Shallow Water: Estimation of the RoughnessCoecient on the Flood Stage / S. Khrapov, A. Pisarev, I. Kobelev, A. Zhumaliev,E. Agafonnikova, A. Losev, A. Khoperskov // Advances in Mechanical Engineering. 2013. V. 5. P. 111.

2. Middelkoop, H. Post-Dam Channel and Floodplain Adjustment Along the Lower Volga River,Russia / H. Middelkoop, A.M. Alabyan, D.B. Babich, V.V. Ivanov // Geomorphic Approachesto Integrated Floodplain Management of Lowland Fluvial Systems in North America andEurope. 2015. P. 245264.

3. Ladjel, M. Lamination Method of Flood Wadis and Projection of the Laminated FloodHydrograph / M. Ladjel // Journal of Fundamental and Applied Sciences. 2016. V. 8, 1. P. 8391.

4. Gorski, K. Post-Damming Flow Regime Development in a Large Lowland River (Volga,Russian Federation): Implications for Floodplain Inundation and Fisheries / K. Gorski,L.V. van den Bosch, K.E. van de Wolfshaar et al. // River Research and Applications. 2012. V. 28, 8. P. 11211134.

5. Çàäà÷à óïðàâëåíèÿ ãèäðîëîãè÷åñêèì ðåæèìîì â ýêîëîãî-ýêîíîìè÷åñêîé ñèñòåìå≪Âîëæñêàÿ ÃÝÑ Âîëãî-Àõòóáèíñêàÿ ïîéìà≫. ×. 2. Ìîäåëèðîâàíèå äèíàìèêè ïîâåðõ-íîñòíûõ âîä â ïåðèîä âåñåííåãî ïàâîäêà / À.À. Âîðîíèí, Ì.Â. Åëèñååâà, Ñ.Ñ. Õðàïîâ,À.Â. Ïèñàðåâ, À.Â. Õîïåðñêîâ // Ïðîáëåìû óïðàâëåíèÿ. 2012. Ò. 5, 6. Ñ. 1925.

6. Chen, D. Deriving Optimal Daily Reservoir Operation Scheme with Consideration ofDownstream Ecological Hydrograph Through a Time-Nested Approach / D. Chen, R. Li,Q. Chen, D. Cai // Water Resources Management. 2015. V. 29, 9. P. 33713386.

7. Haghighi, A.T. Development of Monthly Optimal Flow Regimes for Allocated EnvironmentalFlow Considering Natural Flow Regimes and Several Surface Water Protection Targets /A.T. Haghighi, B. Kløve // Ecological Engineering. 2015. V. 82. P. 390-399.

8. Borsche, R. Flooding in Urban Drainage Systems: Coupling Hyperbolic Conservation Lawsfor Sewer Systems and Surface Flow / R. Borsche, A. Klar // International Journal forNumerical Methods in Fluids. 2014. V. 76. P. 789810.



9. Âîðîíèí, À.À. Ïðîåêòèðîâàíèå ìåõàíèçìîâ óïðàâëåíèÿ ãèäðîëîãè÷åñêèì ðåæèìîìÂîëãî-Àõòóáèíñêîé ïîéìû íà îñíîâå ãåîèíôîðìàöèîííîãî è ãèäðîäèíàìè÷åñêîãî ìî-äåëèðîâàíèÿ / À.À. Âîðîíèí, À.À. Âàñèëü÷åíêî, À.Â. Ïèñàðåâ, Ñ.Ñ. Õðàïîâ, Þ.Å. Ðàä-÷åíêî // Âåñòíèê Âîëãîãðàäñêîãî ãîñóäàðñòâåííîãî óíèâåðñèòåòà. Ñåðèÿ 1: Ìàòåìàòèêà.Ôèçèêà. 2016. 1 (32). Ñ. 2437.

10. Izem, N. A Discontinuous Galerkin Method for Two-Layer Shallow Water Equations /N. Izem, M. Seaid, M. Wakrim // Mathematics and Computers in Simulation. 2016. V. 120. P. 1223.

11. Bulatov, O.V. Regularized Shallow Water Equations for Numerical Simulation of Flows witha Moving Shoreline / O.V. Bulatov, T.G. Elizarova // Computational Mathematics andMathematical Physics. 2016. V. 56, 4. P. 661679.

12. D'yakonova, T.A. Numerical Model of Shallow Water: the Use of GPUs NVIDIA CUDA/ T.A. D'yakonova, A.V. Khoperskov, S.S. Khrapov // Communications in Computer andInformation Science. 2017. V. 687. P. 132145

13. Õðàïîâ, Ñ.Ñ. ×èñëåííàÿ ñõåìà äëÿ ìîäåëèðîâàíèÿ äèíàìèêè ïîâåðõíîñòíûõ âîä íà îñíî-âå êîìáèíèðîâàííîãî SPH-TVD ïîäõîäà / Ñ.Ñ. Õðàïîâ, À.Â. Õîïåðñêîâ, Í.Ì. Êóçüìèí,À.Â. Ïèñàðåâ, È.À. Êîáåëåâ // Âû÷èñëèòåëüíûå ìåòîäû è ïðîãðàììèðîâàíèå. 2011. Ò. 12, 2. Ñ. 282297.

14. Õðàïîâ, Ñ.Ñ. Ñðàâíåíèå òî÷íîñòè è ñõîäèìîñòè äëÿ ìåòîäà CSPH TVD è íåêîòîðûõýéëåðîâûõ ñõåì äëÿ ðåøåíèÿ óðàâíåíèÿ ãàçîäèíàìèêè / Ñ.Ñ. Õðàïîâ, Í.Ì. Êóçüìèí,Ì.À. Áóòåíêî // Âåñòíèê Âîëãîãðàäñêîãî ãîñóäàðñòâåííîãî óíèâåðñèòåòà. Ñåðèÿ 1. Ìà-òåìàòèêà. Ôèçèêà. 2016. 6 (37). Ñ. 166173.

15. Äüÿêîíîâà, Ò.À. Ïðîáëåìà ãðàíè÷íûõ óñëîâèé äëÿ óðàâíåíèé ìåëêîé âîäû / Ò.À. Äüÿêî-íîâà, Ñ.Ñ. Õðàïîâ, À.Â. Õîïåðñêîâ // Âåñòíèê Óäìóðòñêîãî óíèâåðñèòåòà. Ìàòåìàòèêà.Ìåõàíèêà. Êîìïüþòåðíûå íàóêè. 2016, Ò. 26, 3. Ñ. 401417.

Ðàáîòà âûïîëíåíà â ðàìêàõ ãîñçàäàíèÿ Ìèíèñòåðñòâà îáðàçîâàíèÿ è íàóêè ÐÔ(ïðîåêò 2.852.2017/Ï×).

Åêàòåðèíà Îëåãîâíà Àãàôîííèêîâà, ñòàðøèé ïðåïîäàâàòåëü, êàôåäðà ≪Èíôîð-ìàöèîííûå ñèñòåìû è êîìïüþòåðíîå ìîäåëèðîâàíèå≫, Âîëãîãðàäñêèé ãîñóäàðñòâåí-íûé óíèâåðñèòåò (ã. Âîëãîãðàä, Ðîññèéñêàÿ Ôåäåðàöèÿ), [email protected].

Àííà Þðüåâíà Êëèêóíîâà, ñòóäåíò, êàôåäðà ≪Èíôîðìàöèîííûå ñèñòåìû è êîì-ïüþòåðíîå ìîäåëèðîâàíèå≫, Âîëãîãðàäñêèé ãîñóäàðñòâåííûé óíèâåðñèòåò (ã. Âîëãî-ãðàä, Ðîññèéñêàÿ Ôåäåðàöèÿ), [email protected].

Àëåêñàíäð Âàëåíòèíîâè÷ Õîïåðñêîâ, äîêòîð ôèçèêî-ìàòåìàòè÷åñêèõ íàóê, ïðî-ôåññîð, çàâåäóþùèé êàôåäðîé ≪Èíôîðìàöèîííûå ñèñòåìû è êîìïüþòåðíîå ìîäå-ëèðîâàíèå≫, Âîëãîãðàäñêèé ãîñóäàðñòâåííûé óíèâåðñèòåò (ã. Âîëãîãðàä, ÐîññèéñêàÿÔåäåðàöèÿ), [email protected].

Ïîñòóïèëà â ðåäàêöèþ 10 àïðåëÿ 2017 ã.


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