This article concerns a scalar multidimensional conservation law where the flux is of Panov type and may contain spatial discontinuities. We define a notion of entropy solution and prove that entropy solutions are unique. We propose a Godunov-type finite volume scheme and prove that the Godunov approximations converge to an entropy solution, thus establishing existence of entropy solutions. We also show that our numerical scheme converges at an optimal rate of $\mathcal{O}(\sqrt{\D t}).$ To the best of our knowledge, convergence of the Godunov type methods in multi-dimension and error estimates of the numerical scheme in one as well as in several dimensions are the first of it's kind for conservation laws with discontinuous flux. We present numerical examples that illustrate the theory.
翻译:本条涉及一个卡路里多维保护法, 通量为帕诺夫型, 可能包含空间不连续性。 我们定义了一种 entropy 溶液的概念, 并证明 entropy 溶液是独一无二的。 我们提出一个Godunov 型的有限量计划, 并证明Godunov 近似会与 entropy 溶液相融合, 从而证明存在 entropy 溶液。 我们还表明, 我们的数值方案会以 $\ mathcal{O} (\ sqrt ~D t}) 的最佳速率相融合 。 根据我们所知, 将 Godunov 型方法在多元化和误差估计数法中, 在一个和几个方面是第一个, 是 具有不连续通量性 的 保护法 。 我们提出数字例子来说明这个理论 。