| Abstract: |
| It is known that the stochastic Galerkin method applied to hyperbolic systems such as the compressible Euler equations subject to random inputs may lead to an enlarged system which is not necessarily hyperbolic. In addition, such a method usually relies on the positivity of some macroscopic quantities (e.g. sound speed), which may break down when solution presents severe discontinuities. We introduce a stochastic Galerkin method for the compressible Euler equations based on a kinetic formulation. The method solves the Boltzmann equation efficiently for a large range of Knudsen numbers and reduces to an approximated (regularized) solver for the Euler equations when the Knudsen number is small. Furthermore, the method does not need to evaluate any macroscopic quantities nor require their values to be positive, hence is especially suited for problems involving discontinuities. Joint work with Shi Jin and Ruiwen Shu. |
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