Abstract
In this paper, we explore orthogonal systems in mathrm {L}_2({mathbb R}) which give rise to a real skew-symmetric, tridiagonal, irreducible differentiation matrix. Such systems are important since they are stable by design and, if necessary, preserve Euclidean energy for a variety of time-dependent partial differential equations. We prove that there is a one-to-one correspondence between such an orthonormal system {varphi _n}_{nin {mathbb Z}_+} and a sequence of polynomials {p_n}_{nin {mathbb Z}_+} orthonormal with respect to a symmetric probability measure mathrm{d}mu (xi ) = w(xi ){mathrm {d}}xi . If mathrm{d}mu is supported by the real line, this system is dense in mathrm {L}_2({mathbb R}); otherwise, it is dense in a Paley–Wiener space of band-limited functions. The path leading from mathrm{d}mu to {varphi _n}_{nin {mathbb Z}_+} is constructive, and we provide detailed algorithms to this end. We also prove that the only such orthogonal system consisting of a polynomial sequence multiplied by a weight function is the Hermite functions. The paper is accompanied by a number of examples illustrating our argument.
Highlights
The broad theme underlying this paper is the important benefits accrued in the semidiscretisation of time-dependent partial differential equations once space derivatives are approximated in a skew-symmetric manner
We explore a general mechanism to generate orthogonal systems with skew-symmetric differentiation matrices since such systems can be implemented in the context of spectral methods
Not least in quantum control, call for a long-time solution of (1.3) and modern time-stepping methods can achieve the stability required to achieve this if there exists a discretisation yielding a skew-symmetric differentiation matrix
Summary
The broad theme underlying this paper is the important benefits accrued in the semidiscretisation of time-dependent partial differential equations once space derivatives are approximated in a skew-symmetric (or skew-Hermitian in the complex case) manner. D = −D, and A being positive definite, it follows at once that d u 2/ dt ≤ 0: the numerical solution is dissipative (and, incidentally, stable!) Likewise, semidiscretising (1.2) with finite differences, we have u = Du + f (u), t ≥ 0, u(0) = u0, where fm(u) = f (um)—we can again prove, identical to the above argument, that d u 2/ dt ≤ 0 once D is skew-symmetric. It is possible to construct higher-order skew-symmetric differentiation matrices on special grids, but this is far from easy and large orders become fairly complicated [10,11]. This complexity makes them less suitable for efficient computation
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