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

Abstract

The 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 blow-up 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.