Stable second-order scheme for integrating the Kuramoto-Sivanshinsky equation in polar coordinates using distributed approximating functionals.

We present an algorithm for the time integration of nonlinear partial differential equations. The algorithm uses distributed approximating functionals, which are based on an analytic approximation method, in order to achieve highly accurate spatial derivatives. The time integration is based on a second-order unconditionally A -stable Crank-Nicolson scheme with a Newton solver. We apply the integration scheme to the Kuramoto-Sivanshinsky equation in polar coordinates, which presents a significant computational challenge due to the stiffness introduced by the estimation of the spatial derivatives at the origin. We present several stationary and nonstationary solutions of the Kuramoto-Sivanshinsky equation and compare with previous numerical results as well as patterns observed in the combustion front of a circular burner. The numerical results of the proposed scheme reproduces several patterns--rotating two-cell, three-cell, hopping three-cell, stationary two-three-four- and five-cell, stationary 5/1,6/1,7/1,8/2 two-ring patterns, etc.--observed in physical experiments. The scheme is extremely robust and can produce long-term simulations consisting of several thousand frames. Although applied to a very specific problem, the approach of combining the framework of distributed approximating functionals with a Crank-Nicolson based time integration is generalizable to a large class of problems.