Numerical solutions of Burgers' equation with random initial conditions using the Wiener chaos expansion and the Lax-Wendroff scheme

The work is concerned with efficient computation of statistical moments of solutions to Burgers' equation with random initial conditions. When the Lax-Wendroff scheme is expanded using the Wiener chaos expansion (WCE), it introduces an infinite system of deterministic equations with respect to non-random Hermite-Fourier coefficients. One of important properties of the system is that all the statistical moments of the solution can be computed using simple formulae that involve only the solution of the system. The stability, accuracy, and efficiency of the WCE approach to computing statistical moments have been numerically tested and compared to those for the Monte Carlo (MC) method. Strong evidence has been given that the WCE approach is as accurate as but substantially faster than the MC method, at least for certain classes of initial conditions.