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 [6] Audusse and Perthame proved a uniqueness result that does not require the existence of traces, using adapted entropies. We generalize the Godunov-type scheme of Adimurthi, Jaffre and Gowda [2] 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, Jaffre and Gowda.

中文翻译：

---

具有 BV 空间通量的守恒定律的 Godunov 方案收敛到 Audusse-Perthame 自适应熵解

在本文中，我们考虑具有空间不连续通量的一维空间中标量守恒定律的初值问题。可能存在无限多的通量不连续点，并且不连续点集合可能具有累积点。因此，不能假设痕迹的存在。在 [6] 中，Audusse 和 Perthame 证明了一个唯一性结果，该结果不需要迹的存在，使用自适应熵。我们针对这个问题概括了 Adimurthi、Jaffre 和 Gowda [2] 的 Godunov 型方案，对通量函数做了以下假设，(i) 通量是空间变量中的 BV，(ii) 通量的临界点是BV 作为空间变量的函数。我们证明 Godunov 近似收敛到一个适应的熵解，从而提供一个存在结果，