具有BV空间通量的退化守恒律方程Godstrom格式的收敛性及Panov型通量的研究
Convergence of a Godunov scheme for degenerate conservation laws with BV spatial flux and a study of Panov type fluxes
摘要：
本文证明了文[16]中关于一维标量守恒律的Goddom格式的收敛性。可能存在无限多个通量不连续性，并且不连续性的集合可能具有累积点。因此，痕迹的存在不能被假定。与文献[16]中的研究相反，我们没有将通量限制为单峰。我们允许通量具有简并性的情况，即，在状态空间的某个区间上，通量可能为零。由于允许通量退化，相应的奇异映射可能不可逆，因此出现在[16]中的收敛性证明不适用。我们证明，Goddom近似仍然收敛通量退化的存在下，使用另一种方法的证明。我们还考虑通量具有[21]中描述的形式的情况。在这种情况下，我们通过另一种方法证明收敛性。这种方法的证明提供了一个空间变化的解决方案，这是独立的利益。我们提出的数值例子来说明理论。
英文摘要：
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.
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