November  2013, 33(11&12): 5327-5345. doi: 10.3934/dcds.2013.33.5327

An interface problem: The two-layer shallow water equations

1. 

Laboratoire de Mathématiques et applications, Univ. de Poitiers, Teleport 2 - BP 30179, Boulevard Marie et Pierre Curie, 86962, Futuroscope Chasseneuil Cedex, France

2. 

The Institute for Scientific Computing and Applied Mathematics, Indiana University, 831 East Third Street, Bloomington, Indiana 47405

Received  September 2011 Revised  April 2012 Published  May 2013

The aim of this article is to study a model of two superposed layers of fluid governed by the shallow water equations in space dimension one. Under some suitable hypotheses the governing equations are hyperbolic. We introduce suitable boundary conditions and establish a result of existence and uniqueness of smooth solutions for a limited time for this model.
Citation: Madalina Petcu, Roger Temam. An interface problem: The two-layer shallow water equations. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 5327-5345. doi: 10.3934/dcds.2013.33.5327
References:
[1]

E. Audusse, A multilayer Saint-Venant system: Derivation and numerical validation, Discrete Contin. Dyn. Syst. Ser. B, 5 (2005), 189-214. doi: 10.3934/dcdsb.2005.5.189.

[2]

A. Bousquet, M. Petcu, M.-C. Shiue, R. Temam and J. Tribbia, Boundary conditions for limited area models, Commun. Comput. Phys., 14 (2013), no. 3, 664-702. doi: 10.4208/cicp.070312.061112a.

[3]

S. Benzoni-Gavage and D. Serre, "Multidimensional Hyperbolic Partial Differential Equations. First-Order Systems and Applications," Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2007.

[4]

F. Bouchut and V. Zeitlin, A robust well-balanced scheme for multi-layer shallow water equations, Discrete Cont. Dyn. Syst. Ser. B, 13 (2010), 739-758. doi: 10.3934/dcdsb.2010.13.739.

[5]

G.-Q. Chen and P. LeFloch, Existence theory for the isentropic Euler equations, Arch. Rational Mech. Anal., 166 (2003), 81-98. doi: 10.1007/s00205-002-0229-2.

[6]

Q. Chen, M.-C. Shiue and R. Temam, The barotropic mode for the primitive equations, Special issue in memory of David Gottlieb, Journal of Scientific Computing, SpringerLink, 2009. doi: 10.1007/s10915-009-9343-8.

[7]

Q. Chen, M.-C. Shiue, R. Temam and J. Tribbia, Numerical approximation of the inviscid 3D Primitive equations in a limited domain, Math. Mod. and Num. Anal., (M2AN), 46 (2012), no. 3, 619-646. doi: 10.1051/m2an/2011058.

[8]

M. J. P. Cullen, Analysis of the semi-geostrophic shallow water equations, Phys. D, 237 (2008), 1461-1465. doi: 10.1016/j.physd.2008.03.014.

[9]

R. J. DiPerna, Convergence of the viscosity method for isentropic gas dynamics, Comm. Math. Phys., 91 (1983), 1-30. doi: 10.1007/BF01206047.

[10]

R. J. DiPerna, Convergence of approximate solutions to conservation laws, Arch. Rational Mech. Anal., 82 (1983), 27-70. doi: 10.1007/BF00251724.

[11]

B. Engquist and A. Majda, Absorbing boundary conditions for numerical simulation of waves, Proc. Nat. Acad. Sci. USA, 74 (1977), 1765-1766. doi: 10.1073/pnas.74.5.1765.

[12]

B. Engquist and L. Halpern, Far field boundary conditions for computation over long time, Appl. Numer. Math., 4 (1988), 21-45. doi: 10.1016/S0168-9274(88)80004-7.

[13]

D. Givoli and B. Neta, High-order nonre ecting boundary conditions for the dispersive shallow water equations, J. Comput. Appl. Math., 158 (2003), 49-60. Selected Papers from the Conference on Computational and Math- Ematical Methods for Science and Engineering (Alicante, 2002). doi: 10.1016/S0377-0427(03)00462-X.

[14]

O. Guès, Problème mixte hyperbolique quasi-linéaire caractéristique. (French) [The characteristic quasilinear hyperbolic mixed problem], Comm. Partial Differential Equations, 15 (1990), 595-645. doi: 10.1080/03605309908820701.

[15]

R. L. Higdon, Absorbing boundary conditions for difference approximations to the multidimensional wave equation, Math. Comp., 47 (1986), 437-459. doi: 10.2307/2008166.

[16]

R. L. Higdon, Numerical absorbing boundary conditions for the wave equation, Math. Comput., 49 (1987), 65-90. doi: 10.1090/S0025-5718-1987-0890254-1.

[17]

A. Huang, M. Petcu and R. Temam, The one-dimensional supercritical shallow-water equations with topography, Annals of the University of Bucharest, Mathematical Series 2 (LX), no. 1 (2011), 63-82.

[18]

R. G. Keys, Absorbing boundary conditions for acoustic media, Geophysics, 50 (1985), 892-902. doi: 10.1190/1.1441969.

[19]

P.-L. Lions, B. Perthame and P. E. Souganidis, Weak stability of isentropic gas dynamics for $\gamma=5/3$, in "Progress in Elliptic and Parabolic Partial Differential Equations" (Capri, 1994), 350 of Pitman Res. Notes Math. Ser., Longman, Harlow, (1996), 184-192.

[20]

P.-L. Lions, B. Perthame and P. E. Souganidis, Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates, Comm. Pure Appl. Math., 49 (1996), 599-638. doi: 10.1002/(SICI)1097-0312(199606)49:6<599::AID-CPA2>3.0.CO;2-5.

[21]

P.-L. Lions, B. Perthame and E. Tadmor, Kinetic formulation of the isentropic gas dynamics and $p$-systems, Comm. Math. Phys., 163 (1994), 415-431. doi: 10.1007/BF02102014.

[22]

T. T. Li and W. C. Yu, "Boundary Value Problems for Quasilinear Hyperbolic Systems," Duke University Mathematics Series, V. Duke University, Mathematics Department, Durham, NC, 1985.

[23]

A. J. Majda and A. L. Bertozzi, "Vorticity and Incompressible Flow," Cambridge Texts in Applied Mathematics, Cambridge University Press, New York, 2002.

[24]

A. J. Majda and S. Osher, Initial-boundary value problems for hyperbolic equations with uniformly characteristic boundary, Comm. Pure Appl. Math., 28 (1975), 607-675. doi: 10.1002/cpa.3160280504.

[25]

A. McDonald, Transparent boundary conditions for the shallow water equa- tions: Testing in a nested environment, Mon. Wea. Rev., 131 (2003), 698-705.

[26]

J. Nycander and K. Döös, Open boundary conditions for barotropic waves, Journal of Geophysical Research, 108 (2003).

[27]

J. Nycander, A. McC. Hogg and L. M. Frankcombe, Open boundary conditions for nonlinear channel flow, Ocean Modelling, 24 (2008), 108-121.

[28]

I. M. Navon, B. Neta and M. Y. Hussaini, A perfectly matched layer approach to the linearized shallow water equations models, Monthly Weather Review, 132 (2004), 1369-1378. doi: 10.1175/1520-0493(2004)132<1369:APMLAT>2.0.CO;2.

[29]

J. Oliger and A. Sundström, Theoretical and practical aspects of some initial boundary value problems in fluid dynamics, SIAM J. Applied Math., 35 (1978), 419-446. doi: 10.1137/0135035.

[30]

P. Orenga, Un théorème d'existence de solutions d'un problème de shallow water, Arch. Rational Mech. Anal., 130 (1995), 183-204. doi: 10.1007/BF00375155.

[31]

D. Pritchard and L. Dickinson, The near-shore behaviour of shallow-water waves with localized initial conditions, J. Fluid Mech., 591 (2007), 413-436. doi: 10.1017/S002211200700835X.

[32]

M. Petcu and R. Temam, The one-dimensional shallow water equations with transparent boundary conditions, Mathematical Methods in the Applied Sciences, (2011). doi: 10.1002/mma.1482.

[33]

J. Rauch, Symmetric positive systems with boundary characteristic of constant multiplicity, Trans. Amer. Math. Soc., 291 (1985), 167-187. doi: 10.1090/S0002-9947-1985-0797053-4.

[34]

J. Rauch and F. Massey, Differentiability of solutions to hyperbolic initial boundary value problems, Trans. Amer. Math. Soc., 189 (1974), 303-318. doi: 10.2307/1996861.

[35]

A. Rousseau, R. Temam and J. Tribbia, The 3D Primitive Equations in the absence of viscosity: Boundary Conditions and well-posedness in the linearized case, J. Math. Pures Appl., 89 (2008), 297-319. doi: 10.1016/j.matpur.2007.12.001.

[36]

A. Rousseau, R. Temam and J. Tribbia, Boundary value problems for the inviscid primitive equations in limited domains, in computational methods for the atmosphere and the oceans, Special Volume of the Handbook of Numerical Analysis, XIV, Roger M. Temam and Joseph J. Tribbia, Guest Editors, P. G. Ciarlet Editor, Elsevier, Amsterdam, (2008).

[37]

R. Salmon, Numerical solution of the two-layer shallow water equation with bottom topography, Journal of Marine Research, 60 (2002), 605-638. doi: 10.1357/002224002762324194.

[38]

M-C. Shiue, J. Laminie, R. Temam and J. Tribbia, Boundary value problems for the shallow water equations with topography, Journal of Geophysical Research, Oceans, 116 (2011). doi: 10.1029/2010JC006315.

[39]

R. Temam and J. Tribbia, Open boundary conditions for the primitive and Boussinesq equations, J. Atmospheric Sci. 60 (2003), 2647-2660. doi: 10.1175/1520-0469(2003)060<2647:OBCFTP>2.0.CO;2.

[40]

B. Whitham, "Linear and Nonlinear Waves," Pure and Applied Mathematics, (New York). A Wiley-Interscience Publication. John Wiley & Sons Inc., New York 1999. doi: 10.1002/9781118032954.

[41]

T. Warner, R. Peterson and R. Treadon, A tutorial on lateral boundary conditions as a basic and potentially serious limitation to regional numerical weather prediction, Bull. Amer. Meteor. Soc., 78 (1997), 2599-2617. doi: 10.1175/1520-0477(1997)078<2599:ATOLBC>2.0.CO;2.

[42]

T. Yanagisawa, The initial boundary value problem for equations of ideal magneto-hydrodynamics, Hakkaido Math. Jour., 16 (1987), 295-314.

show all references

References:
[1]

E. Audusse, A multilayer Saint-Venant system: Derivation and numerical validation, Discrete Contin. Dyn. Syst. Ser. B, 5 (2005), 189-214. doi: 10.3934/dcdsb.2005.5.189.

[2]

A. Bousquet, M. Petcu, M.-C. Shiue, R. Temam and J. Tribbia, Boundary conditions for limited area models, Commun. Comput. Phys., 14 (2013), no. 3, 664-702. doi: 10.4208/cicp.070312.061112a.

[3]

S. Benzoni-Gavage and D. Serre, "Multidimensional Hyperbolic Partial Differential Equations. First-Order Systems and Applications," Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2007.

[4]

F. Bouchut and V. Zeitlin, A robust well-balanced scheme for multi-layer shallow water equations, Discrete Cont. Dyn. Syst. Ser. B, 13 (2010), 739-758. doi: 10.3934/dcdsb.2010.13.739.

[5]

G.-Q. Chen and P. LeFloch, Existence theory for the isentropic Euler equations, Arch. Rational Mech. Anal., 166 (2003), 81-98. doi: 10.1007/s00205-002-0229-2.

[6]

Q. Chen, M.-C. Shiue and R. Temam, The barotropic mode for the primitive equations, Special issue in memory of David Gottlieb, Journal of Scientific Computing, SpringerLink, 2009. doi: 10.1007/s10915-009-9343-8.

[7]

Q. Chen, M.-C. Shiue, R. Temam and J. Tribbia, Numerical approximation of the inviscid 3D Primitive equations in a limited domain, Math. Mod. and Num. Anal., (M2AN), 46 (2012), no. 3, 619-646. doi: 10.1051/m2an/2011058.

[8]

M. J. P. Cullen, Analysis of the semi-geostrophic shallow water equations, Phys. D, 237 (2008), 1461-1465. doi: 10.1016/j.physd.2008.03.014.

[9]

R. J. DiPerna, Convergence of the viscosity method for isentropic gas dynamics, Comm. Math. Phys., 91 (1983), 1-30. doi: 10.1007/BF01206047.

[10]

R. J. DiPerna, Convergence of approximate solutions to conservation laws, Arch. Rational Mech. Anal., 82 (1983), 27-70. doi: 10.1007/BF00251724.

[11]

B. Engquist and A. Majda, Absorbing boundary conditions for numerical simulation of waves, Proc. Nat. Acad. Sci. USA, 74 (1977), 1765-1766. doi: 10.1073/pnas.74.5.1765.

[12]

B. Engquist and L. Halpern, Far field boundary conditions for computation over long time, Appl. Numer. Math., 4 (1988), 21-45. doi: 10.1016/S0168-9274(88)80004-7.

[13]

D. Givoli and B. Neta, High-order nonre ecting boundary conditions for the dispersive shallow water equations, J. Comput. Appl. Math., 158 (2003), 49-60. Selected Papers from the Conference on Computational and Math- Ematical Methods for Science and Engineering (Alicante, 2002). doi: 10.1016/S0377-0427(03)00462-X.

[14]

O. Guès, Problème mixte hyperbolique quasi-linéaire caractéristique. (French) [The characteristic quasilinear hyperbolic mixed problem], Comm. Partial Differential Equations, 15 (1990), 595-645. doi: 10.1080/03605309908820701.

[15]

R. L. Higdon, Absorbing boundary conditions for difference approximations to the multidimensional wave equation, Math. Comp., 47 (1986), 437-459. doi: 10.2307/2008166.

[16]

R. L. Higdon, Numerical absorbing boundary conditions for the wave equation, Math. Comput., 49 (1987), 65-90. doi: 10.1090/S0025-5718-1987-0890254-1.

[17]

A. Huang, M. Petcu and R. Temam, The one-dimensional supercritical shallow-water equations with topography, Annals of the University of Bucharest, Mathematical Series 2 (LX), no. 1 (2011), 63-82.

[18]

R. G. Keys, Absorbing boundary conditions for acoustic media, Geophysics, 50 (1985), 892-902. doi: 10.1190/1.1441969.

[19]

P.-L. Lions, B. Perthame and P. E. Souganidis, Weak stability of isentropic gas dynamics for $\gamma=5/3$, in "Progress in Elliptic and Parabolic Partial Differential Equations" (Capri, 1994), 350 of Pitman Res. Notes Math. Ser., Longman, Harlow, (1996), 184-192.

[20]

P.-L. Lions, B. Perthame and P. E. Souganidis, Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates, Comm. Pure Appl. Math., 49 (1996), 599-638. doi: 10.1002/(SICI)1097-0312(199606)49:6<599::AID-CPA2>3.0.CO;2-5.

[21]

P.-L. Lions, B. Perthame and E. Tadmor, Kinetic formulation of the isentropic gas dynamics and $p$-systems, Comm. Math. Phys., 163 (1994), 415-431. doi: 10.1007/BF02102014.

[22]

T. T. Li and W. C. Yu, "Boundary Value Problems for Quasilinear Hyperbolic Systems," Duke University Mathematics Series, V. Duke University, Mathematics Department, Durham, NC, 1985.

[23]

A. J. Majda and A. L. Bertozzi, "Vorticity and Incompressible Flow," Cambridge Texts in Applied Mathematics, Cambridge University Press, New York, 2002.

[24]

A. J. Majda and S. Osher, Initial-boundary value problems for hyperbolic equations with uniformly characteristic boundary, Comm. Pure Appl. Math., 28 (1975), 607-675. doi: 10.1002/cpa.3160280504.

[25]

A. McDonald, Transparent boundary conditions for the shallow water equa- tions: Testing in a nested environment, Mon. Wea. Rev., 131 (2003), 698-705.

[26]

J. Nycander and K. Döös, Open boundary conditions for barotropic waves, Journal of Geophysical Research, 108 (2003).

[27]

J. Nycander, A. McC. Hogg and L. M. Frankcombe, Open boundary conditions for nonlinear channel flow, Ocean Modelling, 24 (2008), 108-121.

[28]

I. M. Navon, B. Neta and M. Y. Hussaini, A perfectly matched layer approach to the linearized shallow water equations models, Monthly Weather Review, 132 (2004), 1369-1378. doi: 10.1175/1520-0493(2004)132<1369:APMLAT>2.0.CO;2.

[29]

J. Oliger and A. Sundström, Theoretical and practical aspects of some initial boundary value problems in fluid dynamics, SIAM J. Applied Math., 35 (1978), 419-446. doi: 10.1137/0135035.

[30]

P. Orenga, Un théorème d'existence de solutions d'un problème de shallow water, Arch. Rational Mech. Anal., 130 (1995), 183-204. doi: 10.1007/BF00375155.

[31]

D. Pritchard and L. Dickinson, The near-shore behaviour of shallow-water waves with localized initial conditions, J. Fluid Mech., 591 (2007), 413-436. doi: 10.1017/S002211200700835X.

[32]

M. Petcu and R. Temam, The one-dimensional shallow water equations with transparent boundary conditions, Mathematical Methods in the Applied Sciences, (2011). doi: 10.1002/mma.1482.

[33]

J. Rauch, Symmetric positive systems with boundary characteristic of constant multiplicity, Trans. Amer. Math. Soc., 291 (1985), 167-187. doi: 10.1090/S0002-9947-1985-0797053-4.

[34]

J. Rauch and F. Massey, Differentiability of solutions to hyperbolic initial boundary value problems, Trans. Amer. Math. Soc., 189 (1974), 303-318. doi: 10.2307/1996861.

[35]

A. Rousseau, R. Temam and J. Tribbia, The 3D Primitive Equations in the absence of viscosity: Boundary Conditions and well-posedness in the linearized case, J. Math. Pures Appl., 89 (2008), 297-319. doi: 10.1016/j.matpur.2007.12.001.

[36]

A. Rousseau, R. Temam and J. Tribbia, Boundary value problems for the inviscid primitive equations in limited domains, in computational methods for the atmosphere and the oceans, Special Volume of the Handbook of Numerical Analysis, XIV, Roger M. Temam and Joseph J. Tribbia, Guest Editors, P. G. Ciarlet Editor, Elsevier, Amsterdam, (2008).

[37]

R. Salmon, Numerical solution of the two-layer shallow water equation with bottom topography, Journal of Marine Research, 60 (2002), 605-638. doi: 10.1357/002224002762324194.

[38]

M-C. Shiue, J. Laminie, R. Temam and J. Tribbia, Boundary value problems for the shallow water equations with topography, Journal of Geophysical Research, Oceans, 116 (2011). doi: 10.1029/2010JC006315.

[39]

R. Temam and J. Tribbia, Open boundary conditions for the primitive and Boussinesq equations, J. Atmospheric Sci. 60 (2003), 2647-2660. doi: 10.1175/1520-0469(2003)060<2647:OBCFTP>2.0.CO;2.

[40]

B. Whitham, "Linear and Nonlinear Waves," Pure and Applied Mathematics, (New York). A Wiley-Interscience Publication. John Wiley & Sons Inc., New York 1999. doi: 10.1002/9781118032954.

[41]

T. Warner, R. Peterson and R. Treadon, A tutorial on lateral boundary conditions as a basic and potentially serious limitation to regional numerical weather prediction, Bull. Amer. Meteor. Soc., 78 (1997), 2599-2617. doi: 10.1175/1520-0477(1997)078<2599:ATOLBC>2.0.CO;2.

[42]

T. Yanagisawa, The initial boundary value problem for equations of ideal magneto-hydrodynamics, Hakkaido Math. Jour., 16 (1987), 295-314.

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