# American Institute of Mathematical Sciences

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

 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

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.
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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [5] B. Andreianov, F. Lagoutière, N. Seguin and T. Takahashi, Small solids in an inviscid fluid,, Netw. Heter. Media, 5 (2010), 385.   Google Scholar [6] B. Andreianov, F. Lagoutière, N. Seguin and T. Takahashi, Well-posedness for a one-dimensional fluid-particle interaction model,, In preparation., ().   Google Scholar [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.  Google Scholar [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.   Google Scholar [9] F. Bachmann, "Équations Hyperboliques Scalaires à Flux Discontinu,'', Ph.D thesis, (2005).   Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.   Google Scholar [14] B. Boutin, "Couplage de Lois de Conservation Scalaires par une Régularisation à la Dafermos,'', Master's thesis, (2006).   Google Scholar [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.  Google Scholar [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.  Google Scholar [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).   Google Scholar [18] M. G. Crandall and L. Tartar, Some relations between nonexpansive and order preserving mappings,, Proc. AMS, 78 (1980), 385.   Google Scholar [19] R. Eymard, T. Gallouët and R. Herbin, Finite volume methods,, in, (2000), 713.   Google Scholar [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.   Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [24] E. Godlewski and P.-A. Raviart, "Hyperbolic Systems of Conservation Laws,'' Mathématiques & Applications (Paris) [Mathematics and Applications], 3/4,, Ellipses, (1991).   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [28] S. N. Kružkov, First order quasilinear equations with several independent variables,, Mat. Sb. (N.S.), 81 (123) (1970), 228.   Google Scholar [29] A.-Y. LeRoux, Riemann solvers for some hyperbolic problems with a source term,, In, 6 (1999), 75.   Google Scholar [30] F. Lagoutière, N. Seguin and T. Takahashi, A simple 1D model of inviscid fluid-solid interaction,, J. Differ. Equ., 245 (2008), 3503.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [36] A. Vasseur, Strong traces for solutions of multidimensional scalar conservation laws,, Arch. Ration. Mech. Anal., 160 (2001), 181.  doi: 10.1007/s002050100157.  Google Scholar [37] A. Vasseur, Well-posedness of scalar conservation laws with singular sources,, Methods Appl. Anal., 9 (2002), 291.   Google Scholar [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.  Google Scholar

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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [5] B. Andreianov, F. Lagoutière, N. Seguin and T. Takahashi, Small solids in an inviscid fluid,, Netw. Heter. Media, 5 (2010), 385.   Google Scholar [6] B. Andreianov, F. Lagoutière, N. Seguin and T. Takahashi, Well-posedness for a one-dimensional fluid-particle interaction model,, In preparation., ().   Google Scholar [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.  Google Scholar [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.   Google Scholar [9] F. Bachmann, "Équations Hyperboliques Scalaires à Flux Discontinu,'', Ph.D thesis, (2005).   Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [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.   Google Scholar [14] B. Boutin, "Couplage de Lois de Conservation Scalaires par une Régularisation à la Dafermos,'', Master's thesis, (2006).   Google Scholar [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.  Google Scholar [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.  Google Scholar [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).   Google Scholar [18] M. G. Crandall and L. Tartar, Some relations between nonexpansive and order preserving mappings,, Proc. AMS, 78 (1980), 385.   Google Scholar [19] R. Eymard, T. Gallouët and R. Herbin, Finite volume methods,, in, (2000), 713.   Google Scholar [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.   Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [24] E. Godlewski and P.-A. Raviart, "Hyperbolic Systems of Conservation Laws,'' Mathématiques & Applications (Paris) [Mathematics and Applications], 3/4,, Ellipses, (1991).   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [28] S. N. Kružkov, First order quasilinear equations with several independent variables,, Mat. Sb. (N.S.), 81 (123) (1970), 228.   Google Scholar [29] A.-Y. LeRoux, Riemann solvers for some hyperbolic problems with a source term,, In, 6 (1999), 75.   Google Scholar [30] F. Lagoutière, N. Seguin and T. Takahashi, A simple 1D model of inviscid fluid-solid interaction,, J. Differ. Equ., 245 (2008), 3503.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [36] A. Vasseur, Strong traces for solutions of multidimensional scalar conservation laws,, Arch. Ration. Mech. Anal., 160 (2001), 181.  doi: 10.1007/s002050100157.  Google Scholar [37] A. Vasseur, Well-posedness of scalar conservation laws with singular sources,, Methods Appl. Anal., 9 (2002), 291.   Google Scholar [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.  Google Scholar
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