In this article we prove convergence of the Godunov scheme of [16] 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 contrast to the study appearing in [16], we do not restrict the flux to be unimodal. We allow for the case where the flux has degeneracies, i.e., the flux may vanish on some interval of state space. Since the flux is allowed to be degenerate, the corresponding singular map may not be invertible, and thus the convergence proof appearing in [16] does not pertain. We prove that the Godunov approximations nevertheless do converge in the presence of flux degeneracy, using an alternative method of proof. We additionally consider the case where the flux has the form described in [21]. For this case we prove convergence via yet another method. This method of proof provides a spatial variation bound on the solutions, which is of independent interest. We present numerical examples that illustrate the theory.

中文翻译：

---

具有BV空间通量的退化守恒律的Godunov格式的收敛性和Panov型通量的研究

在本文中，我们证明了[16]的Godunov方案在一个具有空间不连续通量的空间维中的标量守恒律的收敛性。可能存在无限多个通量不连续性，并且不连续性集合可能具有累积点。因此，不能假设存在痕迹。与[16]中出现的研究相反，我们不将通量限制为单峰。我们考虑了通量具有简并性的情况，即通量可能在状态空间的某个间隔上消失。由于允许通量退化，因此相应的奇异图可能不可逆，因此[16]中出现的收敛证明不适用。我们使用替代的证明方法证明了Godunov近似在存在通量简并的情况下仍会收敛。我们还考虑了磁通量具有[21]中所述形式的情况。对于这种情况，我们通过另一种方法证明了收敛性。这种证明方法提供了解决方案上的空间变化，这是独立引起关注的。我们提供了数值示例来说明该理论。