Application of orthogonal wavelets for the stochastic wavelet-Galerkin solution of the Kraichnan-Orszag system

The estimation of statistical momenta of the stochastic systems dx/dt = f(x, {ξ_i}_i), where {ξ_i}_i is the set of random parameters, is an important problem in computations. The direct solution consists in integration of evolution equations followed by the Monte-Carlo averaging. Recently a method of estimation statistical momenta of such systems, the so-called intrusive method, based on the expansion x(ξ, t) = Σ_m c_m(t)Φ_m(ξ) was used to make the original system into Galerkin system of equations with known basic functions Φ_m(ξ). The most of the research was based on the polynomial basic functions Φ_m(ξ), that is why called a polynomial chaos expansion (R.Ghanem and P.Spanos, 1991). In our contribution we use the expansion with respect to the orthogonal wavelets, instead of orthogonal polynomials. Developing the ideas of Wiener-Haar expansion (Le Maitre et al., 2004), that uses wavelets for studying certain domains of the random parameters with better resolution than other, we construct an expansion using the set of the orthogonal Daubechies wavelets (DAUB4, DAUB6, ...) with compact support. Being sensitive to the derivatives higher than one, our expansion provides better possibilities for estimation of statistical momenta of random solutions of differential equations. An example of the Kraichnan-Orszag system (S.A.Orszag and L.R.Bissonnette, 1967) is presented.