Abstract
We investigate the use of Krylov subspace methods to solve linear, oscillatory ODEs. When we apply a Krylov subspace method to a properly formulated equation, we retain the asymptotic accuracy of the asymptotic expansion whilst converging to the exact solution. We demonstrate the effectiveness of this method by computing error and Mathieu functions.
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References
M. Abramowitz and I.A. Stegun. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. US Government Printing Office, Washington, DC, 1964
Z. Battles and L. Trefethen. An extension of matlab to continuous functions and operators. SIAM J. Sci. Comput., 25:1743–1770, 2004
B. Datta and Y. Saad. Arnoldi methods for large sylvester-like observer matrix equations and an associated algorithm for partial pole assignment. Lin. Algebra Appl., 154–156:225–244, 1991
D. Huybrechs and S. Olver. Highly oscillatory quadrature. ed. B. Engquist et al. Highly oscillatory quadrature: Computation, Theory and Applications. Cambridge University Press, Cambridge, 2008
A. Iserles. On the method of neumann series for highly oscillatory equations. Bit Numer. Math., 44(3):473–488, 2004
D. Levin. Procedures for computing one and two-dimensional integrals of functions with rapid irregular oscillations. Math. Comp., 38(158):531–538, 1982
S. Olver. Gmres for the differentiation operator. SIAM J. Numer. Anal., 47:3359–3373, 2009
S. Olver. Fast, numerically stable computation of oscillatory integrals with stationary points. BIT, 50:149–171, 2010
S. Olver. Shifted gmres for oscillatory integrals. Numer. Math., 114:607–628, 2010
Y. Saad and M. Schultz. Gmres: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput., 7:856–869, 1986
L. Trefethen and D. Bau III. Numer. Lin. Algebra. SIAM, 1997
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Olver, S. (2011). GMRES for Oscillatory Matrix-Valued Differential Equations. In: Hesthaven, J., Rønquist, E. (eds) Spectral and High Order Methods for Partial Differential Equations. Lecture Notes in Computational Science and Engineering, vol 76. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15337-2_24
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DOI: https://doi.org/10.1007/978-3-642-15337-2_24
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