Exponential Runge-Kutta methods for the Schrödinger equation

• Autores: Guillaume Dujardin
• Localización: Applied numerical mathematics, ISSN-e 0168-9274, Vol. 59, Nº. 8, 2009, págs. 1839-1857
• Idioma: inglés
• DOI: 10.1016/j.apnum.2009.02.002
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• Resumen

  • We consider exponential Runge�Kutta methods of collocation type, and use them to solve linear and semi-linear Schrödinger Cauchy problems on the d-dimensional torus. We prove that in both cases (linear and non-linear) and with suitable assumptions, s-stage methods are of order s and we give sufficient conditions to achieve orders s+1 and s+2. We show and explain the effects of resonant time steps that occur when solving linear Schrödinger problems on a finite time interval with such methods. This work is inspired by [M. Hochbruck, A. Ostermann, Exponential Runge�Kutta methods for parabolic problems, Appl. Numer. Math. 53 (2�4) (2005) 323-339], where exponential Runge�Kutta methods of collocation type are applied to parabolic Cauchy problems. We compare our results with those obtained for parabolic problems and provide numerical experiments for illustration.