High-order skew-symmetric differentiation matrix on symmetric grid

Hairer and Iserles (2016) presented a detailed study of skew-symmetric matrix approximation to a first derivative which is proved to be fundamental in ensuring stability of discretisation for evolutional partial differential equations with variable coefficients. An open problem is proposed in that paper which concerns about the existence and construction of the perturbed grid that supports high-order skew-symmetric differentiation matrix for a given grid and only the case p=2 for this problem have been solved. This paper is an attempt to solve the problem for any p⩾3. We focus ourselves on the symmetric grid and prove the existence of the perturbed grid for arbitrarily high order p and give in detail the construction of the perturbed grid. Numerical experiments are carried out to illustrate our theory.