Analysis of a Burgers equation with singular resonant source term and convergence of well-balanced schemes

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June 2012, 32(6): 1939-1964. doi: 10.3934/dcds.2012.32.1939

Analysis of a Burgers equation with singular resonant source term and convergence of well-balanced schemes

Boris Andreianov ^1, and Nicolas Seguin ^2,

1.	Laboratoire de Mathématiques CNRS UMR 6623, Université de Franche-Comté, 16 route de Gray, 25030 Besançon Cedex
2.	UPMC Univ Paris 06 and CNRS UMR 7598, Laboratoire J.-L. Lions, 75005 Paris, France

Received March 2011 Revised November 2011 Published February 2012

Abstract
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We define entropy weak solutions and establish well-posedness for the Cauchy problem for the formal equation $$\partial_t u(t,x) + \partial_x \frac{u^2}2(t,x) = - \lambda \, u(t,x)\,\delta_0(x),$$ which can be seen as two Burgers equations coupled in a non-conservative way through the interface located at $x=0$. This problem appears as an important auxiliary step in the theoretical and numerical study of the one-dimensional particle-in-fluid model developed by Lagoutière, Seguin and Takahashi [30].
The interpretation of the non-conservative product "$ u(t,x) \, \delta_0(x)$" follows the analysis of [30]; we can describe the associated interface coupling in terms of one-sided traces on the interface. Well-posedness is established using the tools of the theory of conservation laws with discontinuous flux ([4]).
For proving existence and for practical computation of solutions, we construct a finite volume scheme, which turns out to be a well-balanced scheme and which allows a simple and efficient treatment of the interface coupling. Numerical illustrations are given.

Keywords: Burgers equation, interface traces, well-posedness, non-conservative coupling, resonance, finite volume scheme, well-balanced scheme, convergence., fluid-particle interaction, adapted entropies, singular source term.

Mathematics Subject Classification: Primary: 35L65; Secondary: 35L81, 35R06, 65M1.

Citation: Boris Andreianov, Nicolas Seguin. Analysis of a Burgers equation with singular resonant source term and convergence of well-balanced schemes. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 1939-1964. doi: 10.3934/dcds.2012.32.1939

References:

[1]	D. Amadori, L. Gosse and G. Guerra, Godunov-type approximation for a general resonant balance law with large data, J. Differ. Equ., 198 (2004), 233-274. doi: 10.1016/j.jde.2003.10.004.
[2]	B. Andreianov, P. Goatin and N. Seguin, Finite volume schemes for locally constrained conservation laws, Numer. Math., 115 (2010), 609-645. doi: 10.1007/s00211-009-0286-7.
[3]	B. Andreianov, K. H. Karlsen and N. H. Risebro, On vanishing viscosity approximation of conservation laws with discontinuous flux, Netw. Heterog. Media, 5 (2010), 617-633.
[4]	B. Andreianov, K. H. Karlsen and N. H. Risebro, A theory of $L^1$-dissipative solvers for scalar conservation laws with discontinuous flux, Arch. Ration. Mech. Anal., 201 (2011), 27-86. doi: 10.1007/s00205-010-0389-4.
[5]	B. Andreianov, F. Lagoutière, N. Seguin and T. Takahashi, Small solids in an inviscid fluid, Netw. Heter. Media, 5 (2010), 385-404.
[6]	B. Andreianov, F. Lagoutière, N. Seguin and T. Takahashi, Well-posedness for a one-dimensional fluid-particle interaction model,, In preparation., ().
[7]	Adimurthi, S. Mishra and G. D. Veerappa Gowda, Optimal entropy solutions for conservation laws with discontinuous flux-functions, J. Hyperbolic Differ. Equ., 2 (2005), 783-837. doi: 10.1142/S0219891605000622.
[8]	E. Audusse and B. Perthame, Uniqueness for scalar conservation laws with discontinuous flux via adapted entropies, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 253-265.
[9]	F. Bachmann, "Équations Hyperboliques Scalaires à Flux Discontinu,'' Ph.D thesis, Université de Provence, France, 2005.
[10]	B. Boutin, F. Coquel and E. Godlewski, Dafermos regularization for interface coupling of conservation laws, in "Hyperbolic Problems: Theory, Numerics, Applications,'' Springer, Berlin, (2008), 567-575. doi: 10.1007/978-3-540-75712-2_55.
[11]	R. Bürger, A. García, K. H. Karlsen and J. D. Towers, A family of numerical schemes for kinematic flows with discontinuous flux, J. Engrg. Math., 60 (2008), 387-425.
[12]	P. Baiti and H. K. Jenssen, Well-posedness for a class of $2\times2$ conservation laws with $L^\infty$ data, J. Differ. Equ., 140 (1997), 161-185. doi: 10.1006/jdeq.1997.3308.
[13]	R. Bürger, K. H. Karlsen and J. D. Towers, An Engquist-Osher-type scheme for conservation laws with discontinuous flux adapted to flux connections, SIAM J. Numer. Anal., 47 (2009), 1684-1712.
[14]	B. Boutin, "Couplage de Lois de Conservation Scalaires par une Régularisation à la Dafermos,'' Master's thesis, Université Pierre et Marie Curie-Paris 6, 2006.
[15]	R. Botchorishvili, B. Perthame and A. Vasseur, Equilibrium schemes for scalar conservation laws with stiff sources, Math. Comp., 72 (2003), 131-157. doi: 10.1090/S0025-5718-01-01371-0.
[16]	R. M. Colombo and P. Goatin, A well posed conservation law with a variable unilateral constraint, J. Differ. Equ., 234 (2007), 654-675. doi: 10.1016/j.jde.2006.10.014.
[17]	A. Chinnayya, A.-Y. LeRoux and N. Seguin, A well-balanced numerical scheme for the approximation of the shallow-water equations with topography: The resonance phenomenon, Int. J. Finite Volumes, 1 (2004), 33 pp.
[18]	M. G. Crandall and L. Tartar, Some relations between nonexpansive and order preserving mappings, Proc. AMS, 78 (1980), 385-390.
[19]	R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, in "Handbook of Numerical Analysis,'' Vol. VII, North-Holland, Amsterdam, (2000), 713-1020.
[20]	L. Gosse and A.-Y. LeRoux, Un schéma-équilibre adapté aux lois de conservation scalaires non-homogènes, C. R. Acad. Sci. Paris Sér. I Math., 323 (1996), 543-546.
[21]	J. M. Greenberg and A.-Y. LeRoux, A well-balanced scheme for the numerical processing of source terms in hyperbolic equations, SIAM J. Numer. Anal., 33 (1996), 1-16. doi: 10.1137/0733001.
[22]	P. Goatin and P. G. LeFloch, The Riemann problem for a class of resonant hyperbolic systems of balance laws, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 881-902.
[23]	L. Gosse, Localization effects and measure source terms in numerical schemes for balance laws, Math. Comp., 71 (2002), 553-582. doi: 10.1090/S0025-5718-01-01354-0.
[24]	E. Godlewski and P.-A. Raviart, "Hyperbolic Systems of Conservation Laws,'' Mathématiques & Applications (Paris) [Mathematics and Applications], 3/4, Ellipses, Paris, 1991.
[25]	L. Gosse and G. Toscani, Space localization and well-balanced schemes for discrete kinetic models in diffusive regimes, SIAM J. Numer. Anal., 41 (2003), 641-658. doi: 10.1137/S0036142901399392.
[26]	G. Guerra, Well-posedness for a scalar conservation law with singular nonconservative source, J. Differ. Equ., 206 (2004), 438-469. doi: 10.1016/j.jde.2004.04.008.
[27]	E. Isaacson and B. Temple, Convergence of the $2\times 2$ Godunov method for a general resonant nonlinear balance law, SIAM J. Appl. Math., 55 (1995), 625-640. doi: 10.1137/S0036139992240711.
[28]	S. N. Kružkov, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81 (123) (1970), 228-255.
[29]	A.-Y. LeRoux, Riemann solvers for some hyperbolic problems with a source term, In "Actes du 30ème Congrès d'Analyse Numérique: CANum '98'' (Arles, 1998), ESAIM Proc., 6, Soc. Math. Appl. Indust., Paris, (1999), 75-90.
[30]	F. Lagoutière, N. Seguin and T. Takahashi, A simple 1D model of inviscid fluid-solid interaction, J. Differ. Equ., 245 (2008), 3503-3544.
[31]	R. J. LeVeque and B. Temple, Stability of Godunov's method for a class of $2\times 2$ systems of conservation laws, Trans. Amer. Math. Soc., 288 (1985), 115-123. doi: 10.2307/2000429.
[32]	P. G. LeFloch and A. E. Tzavaras, Representation of weak limits and definition of nonconservative products, SIAM J. Math. Anal., 30 (1999), 1309-1342. doi: 10.1137/S0036141098341794.
[33]	E. Yu. Panov, Existence of strong traces for quasi-solutions of multidimensional conservation laws, J. Hyperbolic Differ. Equ., 4 (2007), 729-770. doi: 10.1142/S0219891607001343.
[34]	N. Seguin and J. Vovelle, Analysis and approximation of a scalar conservation law with a flux function with discontinuous coefficients, Math. Models Methods Appl. Sci., 13 (2003), 221-257. doi: 10.1142/S0218202503002477.
[35]	B. Temple, Global solution of the Cauchy problem for a class of $2\times 2$ nonstrictly hyperbolic conservation laws, Adv. in Appl. Math., 3 (1982), 335-375. doi: 10.1016/S0196-8858(82)80010-9.
[36]	A. Vasseur, Strong traces for solutions of multidimensional scalar conservation laws, Arch. Ration. Mech. Anal., 160 (2001), 181-193. doi: 10.1007/s002050100157.
[37]	A. Vasseur, Well-posedness of scalar conservation laws with singular sources, Methods Appl. Anal., 9 (2002), 291-312.
[38]	J. Vovelle, Convergence of finite volume monotone schemes for scalar conservation laws on bounded domains, Numer. Math., 90 (2002), 563-596. doi: 10.1007/s002110100307.

show all references

References:

[1]	D. Amadori, L. Gosse and G. Guerra, Godunov-type approximation for a general resonant balance law with large data, J. Differ. Equ., 198 (2004), 233-274. doi: 10.1016/j.jde.2003.10.004.
[2]	B. Andreianov, P. Goatin and N. Seguin, Finite volume schemes for locally constrained conservation laws, Numer. Math., 115 (2010), 609-645. doi: 10.1007/s00211-009-0286-7.
[3]	B. Andreianov, K. H. Karlsen and N. H. Risebro, On vanishing viscosity approximation of conservation laws with discontinuous flux, Netw. Heterog. Media, 5 (2010), 617-633.
[4]	B. Andreianov, K. H. Karlsen and N. H. Risebro, A theory of $L^1$-dissipative solvers for scalar conservation laws with discontinuous flux, Arch. Ration. Mech. Anal., 201 (2011), 27-86. doi: 10.1007/s00205-010-0389-4.
[5]	B. Andreianov, F. Lagoutière, N. Seguin and T. Takahashi, Small solids in an inviscid fluid, Netw. Heter. Media, 5 (2010), 385-404.
[6]	B. Andreianov, F. Lagoutière, N. Seguin and T. Takahashi, Well-posedness for a one-dimensional fluid-particle interaction model,, In preparation., ().
[7]	Adimurthi, S. Mishra and G. D. Veerappa Gowda, Optimal entropy solutions for conservation laws with discontinuous flux-functions, J. Hyperbolic Differ. Equ., 2 (2005), 783-837. doi: 10.1142/S0219891605000622.
[8]	E. Audusse and B. Perthame, Uniqueness for scalar conservation laws with discontinuous flux via adapted entropies, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 253-265.
[9]	F. Bachmann, "Équations Hyperboliques Scalaires à Flux Discontinu,'' Ph.D thesis, Université de Provence, France, 2005.
[10]	B. Boutin, F. Coquel and E. Godlewski, Dafermos regularization for interface coupling of conservation laws, in "Hyperbolic Problems: Theory, Numerics, Applications,'' Springer, Berlin, (2008), 567-575. doi: 10.1007/978-3-540-75712-2_55.
[11]	R. Bürger, A. García, K. H. Karlsen and J. D. Towers, A family of numerical schemes for kinematic flows with discontinuous flux, J. Engrg. Math., 60 (2008), 387-425.
[12]	P. Baiti and H. K. Jenssen, Well-posedness for a class of $2\times2$ conservation laws with $L^\infty$ data, J. Differ. Equ., 140 (1997), 161-185. doi: 10.1006/jdeq.1997.3308.
[13]	R. Bürger, K. H. Karlsen and J. D. Towers, An Engquist-Osher-type scheme for conservation laws with discontinuous flux adapted to flux connections, SIAM J. Numer. Anal., 47 (2009), 1684-1712.
[14]	B. Boutin, "Couplage de Lois de Conservation Scalaires par une Régularisation à la Dafermos,'' Master's thesis, Université Pierre et Marie Curie-Paris 6, 2006.
[15]	R. Botchorishvili, B. Perthame and A. Vasseur, Equilibrium schemes for scalar conservation laws with stiff sources, Math. Comp., 72 (2003), 131-157. doi: 10.1090/S0025-5718-01-01371-0.
[16]	R. M. Colombo and P. Goatin, A well posed conservation law with a variable unilateral constraint, J. Differ. Equ., 234 (2007), 654-675. doi: 10.1016/j.jde.2006.10.014.
[17]	A. Chinnayya, A.-Y. LeRoux and N. Seguin, A well-balanced numerical scheme for the approximation of the shallow-water equations with topography: The resonance phenomenon, Int. J. Finite Volumes, 1 (2004), 33 pp.
[18]	M. G. Crandall and L. Tartar, Some relations between nonexpansive and order preserving mappings, Proc. AMS, 78 (1980), 385-390.
[19]	R. Eymard, T. Gallouët and R. Herbin, Finite volume methods, in "Handbook of Numerical Analysis,'' Vol. VII, North-Holland, Amsterdam, (2000), 713-1020.
[20]	L. Gosse and A.-Y. LeRoux, Un schéma-équilibre adapté aux lois de conservation scalaires non-homogènes, C. R. Acad. Sci. Paris Sér. I Math., 323 (1996), 543-546.
[21]	J. M. Greenberg and A.-Y. LeRoux, A well-balanced scheme for the numerical processing of source terms in hyperbolic equations, SIAM J. Numer. Anal., 33 (1996), 1-16. doi: 10.1137/0733001.
[22]	P. Goatin and P. G. LeFloch, The Riemann problem for a class of resonant hyperbolic systems of balance laws, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 881-902.
[23]	L. Gosse, Localization effects and measure source terms in numerical schemes for balance laws, Math. Comp., 71 (2002), 553-582. doi: 10.1090/S0025-5718-01-01354-0.
[24]	E. Godlewski and P.-A. Raviart, "Hyperbolic Systems of Conservation Laws,'' Mathématiques & Applications (Paris) [Mathematics and Applications], 3/4, Ellipses, Paris, 1991.
[25]	L. Gosse and G. Toscani, Space localization and well-balanced schemes for discrete kinetic models in diffusive regimes, SIAM J. Numer. Anal., 41 (2003), 641-658. doi: 10.1137/S0036142901399392.
[26]	G. Guerra, Well-posedness for a scalar conservation law with singular nonconservative source, J. Differ. Equ., 206 (2004), 438-469. doi: 10.1016/j.jde.2004.04.008.
[27]	E. Isaacson and B. Temple, Convergence of the $2\times 2$ Godunov method for a general resonant nonlinear balance law, SIAM J. Appl. Math., 55 (1995), 625-640. doi: 10.1137/S0036139992240711.
[28]	S. N. Kružkov, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81 (123) (1970), 228-255.
[29]	A.-Y. LeRoux, Riemann solvers for some hyperbolic problems with a source term, In "Actes du 30ème Congrès d'Analyse Numérique: CANum '98'' (Arles, 1998), ESAIM Proc., 6, Soc. Math. Appl. Indust., Paris, (1999), 75-90.
[30]	F. Lagoutière, N. Seguin and T. Takahashi, A simple 1D model of inviscid fluid-solid interaction, J. Differ. Equ., 245 (2008), 3503-3544.
[31]	R. J. LeVeque and B. Temple, Stability of Godunov's method for a class of $2\times 2$ systems of conservation laws, Trans. Amer. Math. Soc., 288 (1985), 115-123. doi: 10.2307/2000429.
[32]	P. G. LeFloch and A. E. Tzavaras, Representation of weak limits and definition of nonconservative products, SIAM J. Math. Anal., 30 (1999), 1309-1342. doi: 10.1137/S0036141098341794.
[33]	E. Yu. Panov, Existence of strong traces for quasi-solutions of multidimensional conservation laws, J. Hyperbolic Differ. Equ., 4 (2007), 729-770. doi: 10.1142/S0219891607001343.
[34]	N. Seguin and J. Vovelle, Analysis and approximation of a scalar conservation law with a flux function with discontinuous coefficients, Math. Models Methods Appl. Sci., 13 (2003), 221-257. doi: 10.1142/S0218202503002477.
[35]	B. Temple, Global solution of the Cauchy problem for a class of $2\times 2$ nonstrictly hyperbolic conservation laws, Adv. in Appl. Math., 3 (1982), 335-375. doi: 10.1016/S0196-8858(82)80010-9.
[36]	A. Vasseur, Strong traces for solutions of multidimensional scalar conservation laws, Arch. Ration. Mech. Anal., 160 (2001), 181-193. doi: 10.1007/s002050100157.
[37]	A. Vasseur, Well-posedness of scalar conservation laws with singular sources, Methods Appl. Anal., 9 (2002), 291-312.
[38]	J. Vovelle, Convergence of finite volume monotone schemes for scalar conservation laws on bounded domains, Numer. Math., 90 (2002), 563-596. doi: 10.1007/s002110100307.

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