Analysis of a Burgers equation with singular resonant source term and convergence of well-balanced schemes [Well-posedness of a singular balance law]

Abstract : 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 [LST08]. The interpretation of the non-conservative product ``$u(t,x) \delta_0(x)$'' follows the analysis of [LST08]; 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 ([AKR11]). 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.
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Boris Andreianov, Nicolas Seguin. Analysis of a Burgers equation with singular resonant source term and convergence of well-balanced schemes [Well-posedness of a singular balance law]. DCDS-A, 2012, 32 (6), pp. 1939-1964. 〈https://www.aimsciences.org/journals/displayPaperReference.jsp?paperID=6983〉. 〈10.3934/dcds.2012.32.1939〉. 〈hal-00576959〉

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