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

Previous Article
The Cauchy problem at a node with buffer
DCDS Home
This Issue
Next Article
A profinite group invariant for hyperbolic toral automorphisms

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
Related Papers

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 & Continuous Dynamical Systems - A, 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. 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. 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.
[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. 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.
[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. 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.
[9]	F. Bachmann, "Équations Hyperboliques Scalaires à Flux Discontinu,'', Ph.D thesis, (2005).
[10]	B. Boutin, F. Coquel and E. Godlewski, Dafermos regularization for interface coupling of conservation laws,, in, (2008), 567. 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.
[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. 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.
[14]	B. Boutin, "Couplage de Lois de Conservation Scalaires par une Régularisation à la Dafermos,'', Master's thesis, (2006).
[15]	R. Botchorishvili, B. Perthame and A. Vasseur, Equilibrium schemes for scalar conservation laws with stiff sources,, Math. Comp., 72 (2003), 131. 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. 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).
[18]	M. G. Crandall and L. Tartar, Some relations between nonexpansive and order preserving mappings,, Proc. AMS, 78 (1980), 385.
[19]	R. Eymard, T. Gallouët and R. Herbin, Finite volume methods,, in, (2000), 713.
[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.
[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. 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.
[23]	L. Gosse, Localization effects and measure source terms in numerical schemes for balance laws,, Math. Comp., 71 (2002), 553. 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, (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. doi: 10.1137/S0036142901399392.
[26]	G. Guerra, Well-posedness for a scalar conservation law with singular nonconservative source,, J. Differ. Equ., 206 (2004), 438. 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. doi: 10.1137/S0036139992240711.
[28]	S. N. Kružkov, First order quasilinear equations with several independent variables,, Mat. Sb. (N.S.), 81 (123) (1970), 228.
[29]	A.-Y. LeRoux, Riemann solvers for some hyperbolic problems with a source term,, In, 6 (1999), 75.
[30]	F. Lagoutière, N. Seguin and T. Takahashi, A simple 1D model of inviscid fluid-solid interaction,, J. Differ. Equ., 245 (2008), 3503.
[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. 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. 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. 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. 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. 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. doi: 10.1007/s002050100157.
[37]	A. Vasseur, Well-posedness of scalar conservation laws with singular sources,, Methods Appl. Anal., 9 (2002), 291.
[38]	J. Vovelle, Convergence of finite volume monotone schemes for scalar conservation laws on bounded domains,, Numer. Math., 90 (2002), 563. 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. 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. 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.
[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. 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.
[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. 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.
[9]	F. Bachmann, "Équations Hyperboliques Scalaires à Flux Discontinu,'', Ph.D thesis, (2005).
[10]	B. Boutin, F. Coquel and E. Godlewski, Dafermos regularization for interface coupling of conservation laws,, in, (2008), 567. 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.
[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. 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.
[14]	B. Boutin, "Couplage de Lois de Conservation Scalaires par une Régularisation à la Dafermos,'', Master's thesis, (2006).
[15]	R. Botchorishvili, B. Perthame and A. Vasseur, Equilibrium schemes for scalar conservation laws with stiff sources,, Math. Comp., 72 (2003), 131. 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. 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).
[18]	M. G. Crandall and L. Tartar, Some relations between nonexpansive and order preserving mappings,, Proc. AMS, 78 (1980), 385.
[19]	R. Eymard, T. Gallouët and R. Herbin, Finite volume methods,, in, (2000), 713.
[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.
[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. 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.
[23]	L. Gosse, Localization effects and measure source terms in numerical schemes for balance laws,, Math. Comp., 71 (2002), 553. 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, (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. doi: 10.1137/S0036142901399392.
[26]	G. Guerra, Well-posedness for a scalar conservation law with singular nonconservative source,, J. Differ. Equ., 206 (2004), 438. 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. doi: 10.1137/S0036139992240711.
[28]	S. N. Kružkov, First order quasilinear equations with several independent variables,, Mat. Sb. (N.S.), 81 (123) (1970), 228.
[29]	A.-Y. LeRoux, Riemann solvers for some hyperbolic problems with a source term,, In, 6 (1999), 75.
[30]	F. Lagoutière, N. Seguin and T. Takahashi, A simple 1D model of inviscid fluid-solid interaction,, J. Differ. Equ., 245 (2008), 3503.
[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. 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. 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. 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. 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. 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. doi: 10.1007/s002050100157.
[37]	A. Vasseur, Well-posedness of scalar conservation laws with singular sources,, Methods Appl. Anal., 9 (2002), 291.
[38]	J. Vovelle, Convergence of finite volume monotone schemes for scalar conservation laws on bounded domains,, Numer. Math., 90 (2002), 563. doi: 10.1007/s002110100307.

[1]	François Bouchut, Vladimir Zeitlin. A robust well-balanced scheme for multi-layer shallow water equations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (4) : 739-758. doi: 10.3934/dcdsb.2010.13.739
[2]	Paola Goatin, Sheila Scialanga. Well-posedness and finite volume approximations of the LWR traffic flow model with non-local velocity. Networks & Heterogeneous Media, 2016, 11 (1) : 107-121. doi: 10.3934/nhm.2016.11.107
[3]	Tristan Roy. Adapted linear-nonlinear decomposition and global well-posedness for solutions to the defocusing cubic wave equation on $\mathbb{R}^{3}$. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1307-1323. doi: 10.3934/dcds.2009.24.1307
[4]	Yuri Trakhinin. On well-posedness of the plasma-vacuum interface problem: the case of non-elliptic interface symbol. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1371-1399. doi: 10.3934/cpaa.2016.15.1371
[5]	Rajesh Kumar, Jitendra Kumar, Gerald Warnecke. Convergence analysis of a finite volume scheme for solving non-linear aggregation-breakage population balance equations. Kinetic & Related Models, 2014, 7 (4) : 713-737. doi: 10.3934/krm.2014.7.713
[6]	Mohamed Alahyane, Abdelilah Hakim, Amine Laghrib, Said Raghay. Fluid image registration using a finite volume scheme of the incompressible Navier Stokes equation. Inverse Problems & Imaging, 2018, 12 (5) : 1055-1081. doi: 10.3934/ipi.2018044
[7]	Markus Grasmair. Well-posedness and convergence rates for sparse regularization with sublinear $l^q$ penalty term. Inverse Problems & Imaging, 2009, 3 (3) : 383-387. doi: 10.3934/ipi.2009.3.383
[8]	Laurent Gosse. Well-balanced schemes using elementary solutions for linear models of the Boltzmann equation in one space dimension. Kinetic & Related Models, 2012, 5 (2) : 283-323. doi: 10.3934/krm.2012.5.283
[9]	Huafei Di, Yadong Shang, Xiaoxiao Zheng. Global well-posedness for a fourth order pseudo-parabolic equation with memory and source terms. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 781-801. doi: 10.3934/dcdsb.2016.21.781
[10]	Gabriela Marinoschi. Well posedness of a time-difference scheme for a degenerate fast diffusion problem. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 435-454. doi: 10.3934/dcdsb.2010.13.435
[11]	Young-Sam Kwon. On the well-posedness of entropy solutions for conservation laws with source terms. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 933-949. doi: 10.3934/dcds.2009.25.933
[12]	George Avalos, Roberto Triggiani. Semigroup well-posedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 417-447. doi: 10.3934/dcdss.2009.2.417
[13]	Boris Kolev. Local well-posedness of the EPDiff equation: A survey. Journal of Geometric Mechanics, 2017, 9 (2) : 167-189. doi: 10.3934/jgm.2017007
[14]	Jerry Bona, Nikolay Tzvetkov. Sharp well-posedness results for the BBM equation. Discrete & Continuous Dynamical Systems - A, 2009, 23 (4) : 1241-1252. doi: 10.3934/dcds.2009.23.1241
[15]	Nils Strunk. Well-posedness for the supercritical gKdV equation. Communications on Pure & Applied Analysis, 2014, 13 (2) : 527-542. doi: 10.3934/cpaa.2014.13.527
[16]	A. Alexandrou Himonas, Curtis Holliman. On well-posedness of the Degasperis-Procesi equation. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 469-488. doi: 10.3934/dcds.2011.31.469
[17]	George Avalos, Pelin G. Geredeli, Justin T. Webster. Semigroup well-posedness of a linearized, compressible fluid with an elastic boundary. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1267-1295. doi: 10.3934/dcdsb.2018151
[18]	Stefano Bosia. Well-posedness and long term behavior of a simplified Ericksen-Leslie non-autonomous system for nematic liquid crystal flows. Communications on Pure & Applied Analysis, 2012, 11 (2) : 407-441. doi: 10.3934/cpaa.2012.11.407
[19]	Jan Prüss, Vicente Vergara, Rico Zacher. Well-posedness and long-time behaviour for the non-isothermal Cahn-Hilliard equation with memory. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 625-647. doi: 10.3934/dcds.2010.26.625
[20]	Keyan Wang. Global well-posedness for a transport equation with non-local velocity and critical diffusion. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1203-1210. doi: 10.3934/cpaa.2008.7.1203

2017 Impact Factor: 1.179

American Institute of Mathematical Sciences