Higher-order incremental unknowns, hierarchical basis, and nonlinear dissipative evolutionary equations Original Research Article

The use of (second-order) incremental unknowns to study the long-term dynamic behavior of nonlinear dissipative evolutionary equations when finite-difference approximations of such equations are considered was first proposed by Temam (1990); in this work, we introduce higher-order incremental unknowns to perform such a study. We bring forward the basis which allows to recover these new incremental unknowns and we find out wavelet-like bases which are decomposed in two parts: the first part, constituted of large (significant) values, follows a well-ordered (linear) law; whereas, the second part, constituted of small (negligible) values, follows a chaotic law. On the other hand, we consider numerical schemes retaining the inherent property of dissipativity first introduced by Foias and Titi (1991) to put into action the higher-order incremental unknowns methodology in order to study the long-term dynamic behavior of the one-dimensional Kuramoto-Sivashinsky equation, for which numerical simulations concerning the strange hyperbolic branch are reported, and of a two-dimensional Burgers-like equation. In both cases, a turbulent zone is traversed before reaching the steady-state solution.