March  2016, 11(1): 1-27. doi: 10.3934/nhm.2016.11.1

A combined finite volume - finite element scheme for a dispersive shallow water system

1. 

Inria, EPC ANGE, 2 rue Simone Iff, F75012 Paris, France, France, France, France

Received  June 2015 Revised  October 2015 Published  January 2016

We propose a variational framework for the resolution of a non-hydrostatic Saint-Venant type model with bottom topography. This model is a shallow water type approximation of the incompressible Euler system with free surface and slightly differs from the Green-Nagdhi model, see [13] for more details about the model derivation.
    The numerical approximation relies on a prediction-correction type scheme initially introduced by Chorin-Temam [17] to treat the incompressibility in the Navier-Stokes equations. The hyperbolic part of the system is approximated using a kinetic finite volume solver and the correction step implies to solve a mixed problem where the velocity and the pressure are defined in compatible finite element spaces.
    The resolution of the incompressibility constraint leads to an elliptic problem involving the non-hydrostatic part of the pressure. This step uses a variational formulation of a shallow water version of the incompressibility condition.
    Several numerical experiments are performed to confirm the relevance of our approach.
Citation: Nora Aïssiouene, Marie-Odile Bristeau, Edwige Godlewski, Jacques Sainte-Marie. A combined finite volume - finite element scheme for a dispersive shallow water system. Networks & Heterogeneous Media, 2016, 11 (1) : 1-27. doi: 10.3934/nhm.2016.11.1
References:
[1]

N. Aïssiouene, M. O. Bristeau, E. Godlewski and J. Sainte-Marie, A robust and stable numerical scheme for a depth-averaged Euler system,, Submitted., ().   Google Scholar

[2]

B. Alvarez-Samaniego and D. Lannes, Large time existence for 3D water-waves and asymptotics,, Invent. Math., 171 (2008), 485.  doi: 10.1007/s00222-007-0088-4.  Google Scholar

[3]

B. Alvarez-Samaniego and D. Lannes, A Nash-Moser theorem for singular evolution equations. Application to the Serre and Green-Naghdi equations,, Indiana Univ. Math. J., 57 (2008), 97.  doi: 10.1512/iumj.2008.57.3200.  Google Scholar

[4]

E. Audusse, F. Bouchut, M.-O. Bristeau, R. Klein and B. Perthame, A fast and stable well-balanced scheme with hydrostatic reconstruction for Shallow Water flows,, SIAM J. Sci. Comput., 25 (2004), 2050.  doi: 10.1137/S1064827503431090.  Google Scholar

[5]

E. Audusse, F. Bouchut, M.-O. Bristeau and J. Sainte-Marie, Kinetic entropy inequality and hydrostatic reconstruction scheme for the Saint-Venant system, 2014,, URL , ().   Google Scholar

[6]

J.-L. Bona, T.-B. Benjamin and J.-J. Mahony, Model equations for long waves in nonlinear dispersive systems,, Philos. Trans. Royal Soc. London Series A, 272 (1972), 47.  doi: 10.1098/rsta.1972.0032.  Google Scholar

[7]

P. Bonneton, E. Barthelemy, F. Chazel, R. Cienfuegos, D. Lannes, F. Marche and M. Tissier, Recent advances in Serre-Green Naghdi modelling for wave transformation, breaking and runup processes,, European Journal of Mechanics - B/Fluids, 30 (2011), 589.  doi: 10.1016/j.euromechflu.2011.02.005.  Google Scholar

[8]

F. Bouchut, An introduction to finite volume methods for hyperbolic conservation laws,, ESAIM Proc., 15 (2005), 1.   Google Scholar

[9]

F. Bouchut, Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well-Balanced Schemes for Sources,, Birkhäuser, (2004).  doi: 10.1007/b93802.  Google Scholar

[10]

F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers,, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge, 8 (1974), 129.   Google Scholar

[11]

M.-O. Bristeau and B. Coussin, Boundary Conditions for the Shallow Water Equations Solved by Kinetic Schemes,, Rapport de recherche RR-4282, (2001).   Google Scholar

[12]

M.-O. Bristeau, N. Goutal and J. Sainte-Marie, Numerical simulations of a non-hydrostatic Shallow Water model,, Computers & Fluids, 47 (2011), 51.  doi: 10.1016/j.compfluid.2011.02.013.  Google Scholar

[13]

M. O. Bristeau, A. Mangeney, J. Sainte-Marie and N. Seguin, An energy-consistent depth-averaged Euler system: Derivation and properties,, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 961.  doi: 10.3934/dcdsb.2015.20.961.  Google Scholar

[14]

M.-O. Bristeau and J. Sainte-Marie, Derivation of a non-hydrostatic shallow water model; Comparison with Saint-Venant and Boussinesq systems,, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 733.  doi: 10.3934/dcdsb.2008.10.733.  Google Scholar

[15]

R. Camassa, D. Holm and J. Hyman, A new integrable shallow water equation,, Adv. Appl. Math., 31 (1994), 1.  doi: 10.1016/S0065-2156(08)70254-0.  Google Scholar

[16]

F. Chazel, D. Lannes and F. Marche, Numerical simulation of strongly nonlinear and dispersive waves using a Green-Naghdi model,, J. Sci. Comput., 48 (2011), 105.  doi: 10.1007/s10915-010-9395-9.  Google Scholar

[17]

A. J. Chorin, Numerical solution of the Navier-Stokes equations,, Math. Comp., 22 (1968), 745.  doi: 10.1090/S0025-5718-1968-0242392-2.  Google Scholar

[18]

M.-W. Dingemans, Wave Propagation Over Uneven Bottoms,, Advanced Series on Ocean Engineering - World Scientific, (1997).  doi: 10.1142/9789812796042.  Google Scholar

[19]

M. Dingemans, Comparison of Computations with Boussinesq-like Models and Laboratory Measurements,, Technical Report H1684-12, (1994), 1684.   Google Scholar

[20]

A. Duran and F. Marche, Discontinuous-Galerkin discretization of a new class of Green-Naghdi equations,, Communications in Computational Physics, 17 (2015), 721.  doi: 10.4208/cicp.150414.101014a.  Google Scholar

[21]

W. E and J.-G. Liu, Projection method I: Convergence and numerical boundary layers,, SIAM J. Numer. Anal., 32 (1995), 1017.  doi: 10.1137/0732047.  Google Scholar

[22]

A. Ern and S. Meunier, A posteriori error analysis of euler-galerkin approximations to coupled elliptic-parabolic problems,, ESAIM Math. Model. Numer. Anal., 43 (2009), 353.  doi: 10.1051/m2an:2008048.  Google Scholar

[23]

E. Godlewski and P.-A. Raviart, Numerical Approximations of Hyperbolic Systems of Conservation Laws,, Applied Mathematical Sciences, (1996).  doi: 10.1007/978-1-4612-0713-9.  Google Scholar

[24]

A. Green and P. Naghdi, A derivation of equations for wave propagation in water of variable depth,, J. Fluid Mech., 78 (1976), 237.  doi: 10.1017/S0022112076002425.  Google Scholar

[25]

P. Gresho and S. Chan, Semi-consistent mass matrix techniques for solving the incompressible Navier-Stokes equations,, First Int. Conf. on Comput. Methods in Flow Analysis, ().   Google Scholar

[26]

J.-L. Guermond, Some implementations of projection methods for Navier-Stokes equations,, ESAIM: Mathematical Modelling and Numerical Analysis, 30 (1996), 637.   Google Scholar

[27]

J.-L. Guermond and J. Shen, On the error estimates for the rotational pressure-correction projection methods,, Math. Comput., 73 (2004), 1719.  doi: 10.1090/S0025-5718-03-01621-1.  Google Scholar

[28]

H. Johnston and J.-G. Liu, Accurate, stable and efficient Navier-Stokes solvers based on explicit treatment of the pressure term,, Journal of Computational Physics, 199 (2004), 221.  doi: 10.1016/j.jcp.2004.02.009.  Google Scholar

[29]

D. Lannes and P. Bonneton, Derivation of asymptotic two-dimensional time-dependent equations for surface water wave propagation,, Physics of Fluids, 21 (2009).  doi: 10.1063/1.3053183.  Google Scholar

[30]

O. Le Métayer, S. Gavrilyuk and S. Hank, A numerical scheme for the Green-Naghdi model,, J. Comput. Phys., 229 (2010), 2034.  doi: 10.1016/j.jcp.2009.11.021.  Google Scholar

[31]

R.-J. LeVeque, Finite Volume Methods for Hyperbolic Problems,, Cambridge University Press, (2002).  doi: 10.1017/CBO9780511791253.  Google Scholar

[32]

O. Nwogu, Alternative form of Boussinesq equations for nearshore wave propagation,, Journal of Waterway, 119 (1993), 618.  doi: 10.1061/(ASCE)0733-950X(1993)119:6(618).  Google Scholar

[33]

D. Peregrine, Long waves on a beach,, J. Fluid Mech., 27 (1967), 815.  doi: 10.1017/S0022112067002605.  Google Scholar

[34]

O. Pironneau, Méthodes Des Éléments Finis Pour Les Fluides.,, Masson, (1988).   Google Scholar

[35]

R. Rannacher, On Chorin's projection method for the incompressible Navier-Stokes equations,, in The Navier-Stokes Equations II - Theory and Numerical Methods (eds. G. Heywood John, (1530), 167.  doi: 10.1007/BFb0090341.  Google Scholar

[36]

J. Shen, Pseudo-compressibility methods for the unsteady incompressible Navier-Stokes equations,, 11th AIAA Computational Fluid Dynamic Conference, ().   Google Scholar

[37]

J. Shen, On error estimates of the penalty method for unsteady Navier-Stokes equations,, SIAM J. Numer. Anal., 32 (1995), 386.  doi: 10.1137/0732016.  Google Scholar

[38]

W. C. Thacker, Some exact solutions to the non-linear shallow water wave equations,, J. Fluid Mech., 107 (1981), 499.  doi: 10.1017/S0022112081001882.  Google Scholar

show all references

References:
[1]

N. Aïssiouene, M. O. Bristeau, E. Godlewski and J. Sainte-Marie, A robust and stable numerical scheme for a depth-averaged Euler system,, Submitted., ().   Google Scholar

[2]

B. Alvarez-Samaniego and D. Lannes, Large time existence for 3D water-waves and asymptotics,, Invent. Math., 171 (2008), 485.  doi: 10.1007/s00222-007-0088-4.  Google Scholar

[3]

B. Alvarez-Samaniego and D. Lannes, A Nash-Moser theorem for singular evolution equations. Application to the Serre and Green-Naghdi equations,, Indiana Univ. Math. J., 57 (2008), 97.  doi: 10.1512/iumj.2008.57.3200.  Google Scholar

[4]

E. Audusse, F. Bouchut, M.-O. Bristeau, R. Klein and B. Perthame, A fast and stable well-balanced scheme with hydrostatic reconstruction for Shallow Water flows,, SIAM J. Sci. Comput., 25 (2004), 2050.  doi: 10.1137/S1064827503431090.  Google Scholar

[5]

E. Audusse, F. Bouchut, M.-O. Bristeau and J. Sainte-Marie, Kinetic entropy inequality and hydrostatic reconstruction scheme for the Saint-Venant system, 2014,, URL , ().   Google Scholar

[6]

J.-L. Bona, T.-B. Benjamin and J.-J. Mahony, Model equations for long waves in nonlinear dispersive systems,, Philos. Trans. Royal Soc. London Series A, 272 (1972), 47.  doi: 10.1098/rsta.1972.0032.  Google Scholar

[7]

P. Bonneton, E. Barthelemy, F. Chazel, R. Cienfuegos, D. Lannes, F. Marche and M. Tissier, Recent advances in Serre-Green Naghdi modelling for wave transformation, breaking and runup processes,, European Journal of Mechanics - B/Fluids, 30 (2011), 589.  doi: 10.1016/j.euromechflu.2011.02.005.  Google Scholar

[8]

F. Bouchut, An introduction to finite volume methods for hyperbolic conservation laws,, ESAIM Proc., 15 (2005), 1.   Google Scholar

[9]

F. Bouchut, Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well-Balanced Schemes for Sources,, Birkhäuser, (2004).  doi: 10.1007/b93802.  Google Scholar

[10]

F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers,, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge, 8 (1974), 129.   Google Scholar

[11]

M.-O. Bristeau and B. Coussin, Boundary Conditions for the Shallow Water Equations Solved by Kinetic Schemes,, Rapport de recherche RR-4282, (2001).   Google Scholar

[12]

M.-O. Bristeau, N. Goutal and J. Sainte-Marie, Numerical simulations of a non-hydrostatic Shallow Water model,, Computers & Fluids, 47 (2011), 51.  doi: 10.1016/j.compfluid.2011.02.013.  Google Scholar

[13]

M. O. Bristeau, A. Mangeney, J. Sainte-Marie and N. Seguin, An energy-consistent depth-averaged Euler system: Derivation and properties,, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 961.  doi: 10.3934/dcdsb.2015.20.961.  Google Scholar

[14]

M.-O. Bristeau and J. Sainte-Marie, Derivation of a non-hydrostatic shallow water model; Comparison with Saint-Venant and Boussinesq systems,, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 733.  doi: 10.3934/dcdsb.2008.10.733.  Google Scholar

[15]

R. Camassa, D. Holm and J. Hyman, A new integrable shallow water equation,, Adv. Appl. Math., 31 (1994), 1.  doi: 10.1016/S0065-2156(08)70254-0.  Google Scholar

[16]

F. Chazel, D. Lannes and F. Marche, Numerical simulation of strongly nonlinear and dispersive waves using a Green-Naghdi model,, J. Sci. Comput., 48 (2011), 105.  doi: 10.1007/s10915-010-9395-9.  Google Scholar

[17]

A. J. Chorin, Numerical solution of the Navier-Stokes equations,, Math. Comp., 22 (1968), 745.  doi: 10.1090/S0025-5718-1968-0242392-2.  Google Scholar

[18]

M.-W. Dingemans, Wave Propagation Over Uneven Bottoms,, Advanced Series on Ocean Engineering - World Scientific, (1997).  doi: 10.1142/9789812796042.  Google Scholar

[19]

M. Dingemans, Comparison of Computations with Boussinesq-like Models and Laboratory Measurements,, Technical Report H1684-12, (1994), 1684.   Google Scholar

[20]

A. Duran and F. Marche, Discontinuous-Galerkin discretization of a new class of Green-Naghdi equations,, Communications in Computational Physics, 17 (2015), 721.  doi: 10.4208/cicp.150414.101014a.  Google Scholar

[21]

W. E and J.-G. Liu, Projection method I: Convergence and numerical boundary layers,, SIAM J. Numer. Anal., 32 (1995), 1017.  doi: 10.1137/0732047.  Google Scholar

[22]

A. Ern and S. Meunier, A posteriori error analysis of euler-galerkin approximations to coupled elliptic-parabolic problems,, ESAIM Math. Model. Numer. Anal., 43 (2009), 353.  doi: 10.1051/m2an:2008048.  Google Scholar

[23]

E. Godlewski and P.-A. Raviart, Numerical Approximations of Hyperbolic Systems of Conservation Laws,, Applied Mathematical Sciences, (1996).  doi: 10.1007/978-1-4612-0713-9.  Google Scholar

[24]

A. Green and P. Naghdi, A derivation of equations for wave propagation in water of variable depth,, J. Fluid Mech., 78 (1976), 237.  doi: 10.1017/S0022112076002425.  Google Scholar

[25]

P. Gresho and S. Chan, Semi-consistent mass matrix techniques for solving the incompressible Navier-Stokes equations,, First Int. Conf. on Comput. Methods in Flow Analysis, ().   Google Scholar

[26]

J.-L. Guermond, Some implementations of projection methods for Navier-Stokes equations,, ESAIM: Mathematical Modelling and Numerical Analysis, 30 (1996), 637.   Google Scholar

[27]

J.-L. Guermond and J. Shen, On the error estimates for the rotational pressure-correction projection methods,, Math. Comput., 73 (2004), 1719.  doi: 10.1090/S0025-5718-03-01621-1.  Google Scholar

[28]

H. Johnston and J.-G. Liu, Accurate, stable and efficient Navier-Stokes solvers based on explicit treatment of the pressure term,, Journal of Computational Physics, 199 (2004), 221.  doi: 10.1016/j.jcp.2004.02.009.  Google Scholar

[29]

D. Lannes and P. Bonneton, Derivation of asymptotic two-dimensional time-dependent equations for surface water wave propagation,, Physics of Fluids, 21 (2009).  doi: 10.1063/1.3053183.  Google Scholar

[30]

O. Le Métayer, S. Gavrilyuk and S. Hank, A numerical scheme for the Green-Naghdi model,, J. Comput. Phys., 229 (2010), 2034.  doi: 10.1016/j.jcp.2009.11.021.  Google Scholar

[31]

R.-J. LeVeque, Finite Volume Methods for Hyperbolic Problems,, Cambridge University Press, (2002).  doi: 10.1017/CBO9780511791253.  Google Scholar

[32]

O. Nwogu, Alternative form of Boussinesq equations for nearshore wave propagation,, Journal of Waterway, 119 (1993), 618.  doi: 10.1061/(ASCE)0733-950X(1993)119:6(618).  Google Scholar

[33]

D. Peregrine, Long waves on a beach,, J. Fluid Mech., 27 (1967), 815.  doi: 10.1017/S0022112067002605.  Google Scholar

[34]

O. Pironneau, Méthodes Des Éléments Finis Pour Les Fluides.,, Masson, (1988).   Google Scholar

[35]

R. Rannacher, On Chorin's projection method for the incompressible Navier-Stokes equations,, in The Navier-Stokes Equations II - Theory and Numerical Methods (eds. G. Heywood John, (1530), 167.  doi: 10.1007/BFb0090341.  Google Scholar

[36]

J. Shen, Pseudo-compressibility methods for the unsteady incompressible Navier-Stokes equations,, 11th AIAA Computational Fluid Dynamic Conference, ().   Google Scholar

[37]

J. Shen, On error estimates of the penalty method for unsteady Navier-Stokes equations,, SIAM J. Numer. Anal., 32 (1995), 386.  doi: 10.1137/0732016.  Google Scholar

[38]

W. C. Thacker, Some exact solutions to the non-linear shallow water wave equations,, J. Fluid Mech., 107 (1981), 499.  doi: 10.1017/S0022112081001882.  Google Scholar

[1]

Marie-Odile Bristeau, Anne Mangeney, Jacques Sainte-Marie, Nicolas Seguin. An energy-consistent depth-averaged Euler system: Derivation and properties. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 961-988. doi: 10.3934/dcdsb.2015.20.961

[2]

Yi Zhou, Zhen Lei. Logarithmically improved criteria for Euler and Navier-Stokes equations. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2715-2719. doi: 10.3934/cpaa.2013.12.2715

[3]

Michele Coti Zelati. Remarks on the approximation of the Navier-Stokes equations via the implicit Euler scheme. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2829-2838. doi: 10.3934/cpaa.2013.12.2829

[4]

Carlo Morosi, Livio Pizzocchero. On the constants in a Kato inequality for the Euler and Navier-Stokes equations. Communications on Pure & Applied Analysis, 2012, 11 (2) : 557-586. doi: 10.3934/cpaa.2012.11.557

[5]

Eric Blayo, Antoine Rousseau. About interface conditions for coupling hydrostatic and nonhydrostatic Navier-Stokes flows. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1565-1574. doi: 10.3934/dcdss.2016063

[6]

Marcel Oliver. The Lagrangian averaged Euler equations as the short-time inviscid limit of the Navier–Stokes equations with Besov class data in $\mathbb{R}^2$. Communications on Pure & Applied Analysis, 2002, 1 (2) : 221-235. doi: 10.3934/cpaa.2002.1.221

[7]

Hamid Bellout, Jiří Neustupa, Patrick Penel. On a $\nu$-continuous family of strong solutions to the Euler or Navier-Stokes equations with the Navier-Type boundary condition. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1353-1373. doi: 10.3934/dcds.2010.27.1353

[8]

Yoshihiro Shibata. Local well-posedness of free surface problems for the Navier-Stokes equations in a general domain. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 315-342. doi: 10.3934/dcdss.2016.9.315

[9]

Hi Jun Choe, Do Wan Kim, Yongsik Kim. Meshfree method for the non-stationary incompressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 17-39. doi: 10.3934/dcdsb.2006.6.17

[10]

Dongho Chae, Kyungkeun Kang, Jihoon Lee. Notes on the asymptotically self-similar singularities in the Euler and the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1181-1193. doi: 10.3934/dcds.2009.25.1181

[11]

Erik Endres, Helge Kristian Jenssen, Mark Williams. Singularly perturbed ODEs and profiles for stationary symmetric Euler and Navier-Stokes shocks. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 133-169. doi: 10.3934/dcds.2010.27.133

[12]

Jian Su, Yinnian He. The almost unconditional convergence of the Euler implicit/explicit scheme for the three dimensional nonstationary Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3421-3438. doi: 10.3934/dcdsb.2017173

[13]

Leonardi Filippo. A projection method for the computation of admissible measure valued solutions of the incompressible Euler equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 941-961. doi: 10.3934/dcdss.2018056

[14]

Marie-Odile Bristeau, Jacques Sainte-Marie. Derivation of a non-hydrostatic shallow water model; Comparison with Saint-Venant and Boussinesq systems. Discrete & Continuous Dynamical Systems - B, 2008, 10 (4) : 733-759. doi: 10.3934/dcdsb.2008.10.733

[15]

Yinnian He, Yanping Lin, Weiwei Sun. Stabilized finite element method for the non-stationary Navier-Stokes problem. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 41-68. doi: 10.3934/dcdsb.2006.6.41

[16]

Joel Avrin. Global existence and regularity for the Lagrangian averaged Navier-Stokes equations with initial data in $H^{1//2}$. Communications on Pure & Applied Analysis, 2004, 3 (3) : 353-366. doi: 10.3934/cpaa.2004.3.353

[17]

Daniel Coutand, J. Peirce, Steve Shkoller. Global well-posedness of weak solutions for the Lagrangian averaged Navier-Stokes equations on bounded domains. Communications on Pure & Applied Analysis, 2002, 1 (1) : 35-50. doi: 10.3934/cpaa.2002.1.35

[18]

Vena Pearl Bongolan-walsh, David Cheban, Jinqiao Duan. Recurrent motions in the nonautonomous Navier-Stokes system. Discrete & Continuous Dynamical Systems - B, 2003, 3 (2) : 255-262. doi: 10.3934/dcdsb.2003.3.255

[19]

Bum Ja Jin, Kyungkeun Kang. Caccioppoli type inequality for non-Newtonian Stokes system and a local energy inequality of non-Newtonian Navier-Stokes equations without pressure. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4815-4834. doi: 10.3934/dcds.2017207

[20]

Hantaek Bae. Solvability of the free boundary value problem of the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 769-801. doi: 10.3934/dcds.2011.29.769

2018 Impact Factor: 0.871

Metrics

  • PDF downloads (14)
  • HTML views (0)
  • Cited by (2)

[Back to Top]