Accurately Model the Kuramoto--Sivashinsky Dynamics with Holistic Discretization

Abstract

We analyze the nonlinear Kuramoto--Sivashinsky equation to develop accurate discretizations modeling its dynamics on coarse grids. The analysis is based upon center manifold theory, so we are assured that the discretization accurately models the dynamics and may be constructed systematically. The theory is applied after dividing the physical domain into small elements by introducing isolating internal boundaries which are later removed. Comprehensive numerical solutions and simulations show that the holistic discretizations excellently reproduce the steady states and the dynamics of the Kuramoto--Sivashinsky equation. The Kuramoto--Sivashinsky equation is used as an example to show how holistic discretization may be successfully applied to fourth-order, nonlinear, spatio-temporal dynamical systems. This novel center manifold approach is holistic in the sense that it treats the dynamical equations as a whole, not just as the sum of separate terms.