A spectral MUSCL scheme for gPC-Galerkin method to uncertain hyperbolic equations

This paper deals with a problem that we found when directly implementing the scalar MUSCL scheme to spectral coefficient equations of the gPC-Galerkin method. The order degradation at extrema of spectral coefficients and non-uniform reconstruction process lead to a distortion of standard deviation at some extrema, especially where the peak consists of several extrema of coefficients. From the perspective of probability space, a spectral MUSCL scheme is proposed based on the generalized minmod limiter to avoid the defects mentioned above when applying the scalar MUSCL. In this paper, we present some properties of the new scheme related to total variation and demonstrate them by the uncertain linear scalar conservation law. Finally, we employ this scheme to compressible Euler equations, and a good correlation of standard deviation is obtained. The details of implementation of the gPC-Galerkin method for Euler equations are also discussed, and the approximate Jacobian matrix is adopted.