An investigation of uncertainty due to stochastically varying geometry: An initial study

We study hyperbolic problems with uncertain stochastically varying geometries. Our aim is to investigate how the stochastically varying uncertainty in the geometry affects the solution of the partial differential equation in terms of the mean and variance of the solution. The problem considered is the two dimensional advection equation on a general domain, which is transformed using curvilinear coordinates to a unit square. The numerical solution is computed using a high order finite difference formulation on summation-by-parts form with weakly imposed boundary conditions. The statistics of the solution are computed nonintrusively using quadrature rules given by the probability density function of the random variable. We prove that the continuous problem is strongly well-posed and that the semi-discrete problem is strongly stable. Numerical calculations using the method of manufactured solution verify the accuracy of the scheme and the statistical properties of the solution are discussed.