A simple, flexible and generic deterministic approach to uncertainty quantifications in non linear problems: application to fluid flow problems

This paper deals with the computation of some statistics of the solutions of linear and non linear PDEs by mean of a method that is simple and flexible. A particular emphasis is given on non linear hyperbolic type equations such as the Burger equation and the Euler equations. Given a PDE and starting from a description of the solution in term of a space variable and a (family) of random variables that may be correlated, the solution is numerically described by its conditional expectancies of point values or cell averages. This is done via a tessellation of the random space as in finite volume methods for the space variables. Then, using these conditional expectancies and the geometrical description of the tessellation, a piecewise polynomial approximation in the random variables is computed using a reconstruction method that is standard for high order finite volume space, except that the measure is no longer the standard Lebesgue measure but the probability measure. Starting from a given scheme for the deterministic version of the PDE, we use this reconstruction to formulate a scheme on the numerical approximation of the solution. This method enables maximum flexibility in term of the PDE and the probability measure. In particular, the scheme is non intrusive, can handle any type of probability measure, even with Dirac terms. The method is illustrated on ODEs, elliptic and hyperbolic problems, linear and non linear