Compatibility conditions for time-dependent partial differential equations and the rate of convergence of Chebyshev and Fourier spectral methods

Abstract

Compatibility conditions for partial differential equations (PDEs) are an infinite set of relations between the initial conditions, the PDE, and the boundary conditions which are necessary and sufficient for the solution to be C ∝, that is, infinitely differentiable, everywhere on the computational domain including the boundaries. Since the performance of Chebyshev spectral and spectral element methods is dramatically reduced when the solution is not C ∝, one would expect that the compatibility conditions would be a major theme in the spectral literature. Instead, it has been completely ignored. Therefore, we pursue three goals here. First, we present a proof of the compatibility conditions in a simplified form that does not require functional analysis. Second, we analyze the connection between the compatibility conditions and the rate of convergence of Chebyshev methods. Lastly, we describe strategies for slightly adjusting initial conditions so that the compatibility conditions are satisfied.