an:07268836
Zbl 1451.65107
Ghoshal, Shyam Sundar; Jana, Animesh; Towers, John D.
Convergence of a Godunov scheme to an Audusse-Perthame adapted entropy solution for conservation laws with BV spatial flux
EN
Numer. Math. 146, No. 3, 629-659 (2020).
00454764
2020
j
65M06 65M12 35L65 35B44 35A01 65M08 76M20
Godunov scheme; convergence
Summary: In this article we consider the initial value problem for a scalar conservation law in one space dimension with a spatially discontinuous flux. There may be infinitely many flux discontinuities, and the set of discontinuities may have accumulation points. Thus the existence of traces cannot be assumed. In [\textit{E. Audusse} and \textit{B. Perthame}, Proc. R. Soc. Edinb., Sect. A, Math. 135, No. 2, 253--266 (2005; Zbl 1071.35079)] proved a uniqueness result that does not require the existence of traces, using adapted entropies. We generalize the Godunov-type scheme of \textit{Adimurthi} et al. [SIAM J. Numer. Anal. 42, No. 1, 179--208 (2004; Zbl 1081.65082)] for this problem with the following assumptions on the flux function, (i) the flux is BV in the spatial variable and (ii) the critical point of the flux is BV as a function of the space variable. We prove that the Godunov approximations converge to an adapted entropy solution, thus providing an existence result, and extending the convergence result of Adimurthi, Jaffr?? and Gowda.
Zbl 1071.35079; Zbl 1081.65082