Numerical Stability in the Presence of Variable Coefficients

Abstract

The main concern of this paper is with the stable discretisation of linear partial differential equations of evolution with time-varying coefficients. We commence by demonstrating that an approximation of the first derivative by a skew-symmetric matrix is fundamental in ensuring stability for many differential equations of evolution. This motivates our detailed study of skew-symmetric differentiation matrices for univariate finite-difference methods.We prove that, in order to sustain a skew-symmetric differentiation matrix of order p ≥ 2, a grid must satisfy 2p − 3 polynomial conditions. Moreover, once it satisfies these conditions, it supports a banded skew-symmetric differentiation matrix of this order and of the bandwidth 2p − 1 which can be derived in a constructive manner.Some applications require not just skew-symmetry but also that the growth in the elements of the differentiation matrix is at most linear in the number of unknowns. This is always true for our tridiagonal matrices of order 2 but need not be true otherwise, a subject which we explore further.Another subject which we examine is the existence and practical construction of grids that support skew-symmetric differentiation matrices of a given order. We resolve this issue completely for order-two methods.We conclude the paper with a list of open problems and their discussion.0 Communicated by Peter Olver 0

The Joy and Pain of Skew Symmetry

“…The challenge in this case is to replace a gridx, say, which has been produced by an h-adaptive procedure, by a nearby grid which is compatible with suitable order conditions. A considerable discussion in Hairer and Iserles [8] notwithstanding, this is a subject requiring much further research.…”

“…This growth is incompatible with the conditions of Theorem 1, and this restricts their usability-cf. further discussion in Hairer and Iserles [8], where we also present much additional material on the construction of grids compatible with high-order skewsymmetric differentiation matrices.…”

Banded, stable, skew-symmetric differentiation matrices of high order

“…The idea is to modify the coefficients in these parts to achieve skew symmetry and retain high order, stability and the banded structure. It follows from the results of [2,1] that this is not possible for an equidistant grid. We therefore modify the grid, but we do this only close to the endpoints of the interval.…”

“…Since the first derivative is a skew-symmetric operator, this is a natural assumption in the spirit of geometric numerical integration. It has several interesting implications for the discretisation of partial differential equations (see [1]).…”

Orthogonal Systems with a Skew-Symmetric Differentiation Matrix

“…The matrix in (1.1) is skew symmetric, yet it is a sobering thought that this second-order approximation of the derivative is as good as it gets: no skew-symmetric finite-difference differentiation matrix on a uniform grid may exceed order 2 [17]. It is possible (but not easy) to obtain higher-order differentiation matrices of this kind carefully choosing specific non-uniform grids [15,16], but this is far from easy for high orders and, at any rate, finite differences are not the approach of choice in this paper.…”