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Stable spectral methods for time-dependent problems and the preservation of structure

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Iserles, Arieh 


This paper is concerned with orthonormal systems in real intervals, given with zero Dirichlet boundary conditions. More speci cally, our interest is in systems with a skew-symmetric differentiation matrix (this excludes orthonor- mal polynomials). We consider a simple construction of such systems and pur- sue its rami cations. In general, given any C1(a; b) weight function such that w(a) = w(b) = 0, we can generate an orthonormal system with a skew-symmetric di erentiation matrix. Except for the case a = 􀀀1, b = +1, only few powers of that matrix are bounded and we establish a connection between properties of the weight function and boundedness. In particular, we examine in detail two weight functions: the Laguerre weight function x e􀀀x for x > 0 and > 0 and the ultraspherical weight function (1 􀀀 x2) , x 2 (􀀀1; 1), > 0, and estab- lish their properties. Both weights share a most welcome feature of separability, which allows for fast computation. The quality of approximation is highly sen- sitive to the choice of and we discuss how to choose optimally this parameter, depending on the number of zero boundary conditions.



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Foundations of Computational Mathematics

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Springer Verlag

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