Measure‐valued weak solutions for conservation laws with discontinuous flux are proposed and explicit formulae have been derived. We propose convergent discontinuous flux‐based numerical schemes for the class of hyperbolic systems that admit nonclassical ‐shocks, by extending the theory of discontinuous flux for nonlinear conservation laws to scalar transport equation with a discontinuous coefficient. The article also discusses the concentration phenomenon of solutions along the line of discontinuity, for scalar transport equations with a discontinuous coefficient. The existence of the solutions for transport equation is shown using the vanishing viscosity approach and the asymptotic behavior of the solutions is also established. The performance of the numerical schemes for both scalar conservation laws and systems to capture the ‐shocks effectively is displayed through various numerical experiments.

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不连续通量守恒律的集中解及其在双曲系统数值解中的应用

提出了具有不连续通量的守恒律的量值弱解，并推导了显式。对于允许非经典的双曲系统，我们提出了基于收敛的基于不连续通量的数值方案通过将非线性守恒定律的不连续通量理论扩展到具有不连续系数的标量输运方程来进行震荡。本文还讨论了具有不连续系数的标量输运方程沿着不连续线的解的集中现象。使用消失粘度法显示了输运方程解的存在性，并且还建立了溶液的渐近行为。标量守恒定律和系统有效地捕获冲击的数值方案的性能通过各种数值实验得以展示。