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Numerical stability in the presence of variable coefficients

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Ernst Hairer and Arieh Iserles

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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.

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Key Words. Stable discretisation, linear PDEs, skew-symmetric differentiation matrices, order conditions, banded differentiation matrices.

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