基于广义黎曼问题的Glimm方法的一个版本
A version of the Glimm method based on generalized Riemann problems
AI总结：
本文提出了一种基于广义黎曼问题的Glimm随机选择方法的推广，用于逼近拟线性双曲守恒律系统的熵解，并证明了其非线性稳定性，从而得到了带源项拟线性双曲系统的全局存在性结果。
摘要：
我们介绍了Glimm随机选择方法的一种推广，该方法为我们提供了拟线性双曲守恒律系统的熵解的近似。方程的流函数和源项可能依赖于未知量以及时间和空间变量。该方法基于广义黎曼问题的局部近似解，这些解构成了我们方案中的构建块，并使我们能够自然地考虑流和源项的影响。为了建立这些逼近的非线性稳定性，我们研究了广义波模式之间的非线性相互作用。这一分析使我们得到了带源项拟线性双曲系统的全局存在性结果，并且例如适用于一般几何中的可压缩欧拉方程以及定义在洛伦兹流形上的双曲系统。
英文摘要：
We introduce a generalization of Glimm's random choice method, which provides us with an approximation of entropy solutions to quasilinear hyperbolic system of balance laws. The flux-function and the source term of the equations may depend on the unknown as well as on the time and space variables. The method is based on local approximate solutions of the generalized Riemann problem, which form building blocks in our scheme and allow us to take into account naturally the effects of the flux and source terms. To establish the nonlinear stability of these approximations, we investigate nonlinear interactions between generalized wave patterns. This analysis leads us to a global existence result for quasilinear hyperbolic systems with source-term, and applies, for instance, to the compressible Euler equations in general geometries and to hyperbolic systems posed on a Lorentzian manifold.
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