On Green’s function-based time stepping for semilinear initial–boundary value problems

Abstract

a b s t r a c tThe Fourier spectral approach is central to the analysis of semilinear initial-boundary value problems when considered with periodic boundary conditions. It aids theoretical arguments, such as renormalisation group analysis, as well as numerical simulation through standard pseudo-spectral methods. Fixed boundary conditions, in contrast, require the use of non-uniform grids, usually generated by bases of orthogonal polynomials. On such bases, numerical differentiation is ill-conditioned and can potentially lead to a catastrophic blowup of round-off error. In this paper, we apply ideas explored by Viswanath (2013) in the context of Navier-Stokes solvers to completely eliminate numerical differentiation and linear solving from the time-stepping algorithm in favour of numerical quadrature. We propose a concrete quadrature rule and test its stability, scalability and spectral accuracy on a Kuramoto-Sivashinsky template problem over a range of five orders of magnitude of the viscosity.