THE LOSS OF ACCURACY OF STOCHASTIC COLLOCATION METHOD IN SOLVING NONLINEAR DIFFERENTIAL EQUATIONS WITH RANDOM INPUT DATA

Abstract

n this paper we show how stochastic collocation method (SCM) could fail to con- verge for nonlinear differential equations with random coefficients. First, we consider Navier-Stokes equation with uncertain viscosity and derive error estimates for stochastic collocation discretization. Our analysis gives some indicators on how the nonlinearity negatively affects the accuracy of the method. The stochastic collocation method is then applied to noisy Lorenz system. Simulation re- sults demonstrate that the solution of a nonlinear equation could be highly irregular on the random data and in such cases, stochastic collocation method cannot capture the correct solution.