Green's function-based time stepping for the Kuramoto-Sivashinsky initial-boundary value problem

TL;DR

This paper introduces a Green's function-based time-stepping method for the Kuramoto-Sivashinsky initial-boundary value problem, effectively handling fixed boundary conditions without numerical differentiation, enabling detailed analysis across various viscosities.

Contribution

It develops a novel Green's function approach that eliminates numerical differentiation in time-stepping for the Kuramoto-Sivashinsky equation with fixed boundaries, extending analysis capabilities.

Findings

Effective handling of boundary layers with Green's functions

Elimination of numerical differentiation improves stability

Applicable across a wide viscosity range

Abstract

Both theoretical and numerical studies of the Kuramoto-Sivashinsky equation have mostly considered periodic boundary conditions. In this setting, the Fourier decomposition of the solution is central to theoretical ideas, such as renormalization group arguments, as well as to numerical solution, allowing for the construction of accurate and efficient time-steppers using standard pseudo-spectral methods. In contrast, fixed boundary conditions induce boundary layers and necessitate the use of non-uniform grids, usually generated by orthogonal polynomials. On such bases, numerical differentiation is ill-conditioned and can potentially lead to a catastrophic blow-up of round-off error. In this paper, we use ideas recently explored by Viswanath (J. Comput. Phys. 251(2013), pp. 414-431) to completely eliminate numerical differentiation and linear solving from the time-stepping algorithm. We use the Green's function-based method to investigate elements of the Kuramoto-Sivashinsky dynamics over a range of five decades of the viscosity.