Recent advances in 2+1-dimensional simulations of the pattern-forming Kuramoto–Sivashinsky equation

We present an overview of recent advances in numerical simulations of the 2+1-dimensional Kuramoto–Sivashinsky equation, describing the flame-front deformation in a combustion experiment. Algorithmic development includes a second-order unconditionally A-stable Crank–Nicolson scheme, using distributed approximating functionals (DAFs) for well-tempered, highly accurate, representation of the physical quantity and its derivatives. The simulator reproduces a multitude of patterns observed in experiments-in-the-wild, including rotating 2-cell, 3-cell, hopping 3-cell, stationary 2, 3, 4, 5-cell, stationary 5/1, 6/1, 7/1, 8/2 two-ring patterns, etc. The numerical observation of hopping flame patterns – characterized by non-uniform rotations of a ring of cells, in which individual cells make abrupt changes in their angular positions while they rotate around the ring – is the first outside of physical experiments. We show modal decomposition analysis of the simulated patterns, via the singular value decomposition (SVD), which exposes the spatio-temporal behavior in which the overall temporal dynamics is similar to that of equivalent experimental states. Symmetry-based arguments are used to derive normal form equations for the temporal behavior, and a bifurcation analysis of the associated normal form equations quantifies the complexity of hopping patterns. Conditions for their existence and their stability are also derived from the bifurcation analysis. Further, we study the effects of thermal noise in a stochastic formulation of the Kuramoto–Sivashinsky equation. Numerical integration reveals that the presence of noise increases the propensity of dynamic cellular states, which seems to explain the generic behavior of related laboratory experiments. Most importantly, we also report on observations of certain dynamic states, homoclinic intermittent states, previously only observed in physical experiments.