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  • error bounds  (2)
  • Mathematics Subject Classification (1991):65L05, 65L70, 65M12, 65M20  (1)
  • 1
    Digitale Medien
    Digitale Medien
    A Gautschi-type method for oscillatory second-order differential equations (1999)
    Springer
    Numerische Mathematik 83 (1999), S. 403-426 
    Details
    ISSN: 0945-3245
    Schlagwort(e): Mathematics Subject Classification (1991):65L05, 65L70, 65M12, 65M20
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary. We study a numerical method for second-order differential equations in which high-frequency oscillations are generated by a linear part. For example, semilinear wave equations are of this type. The numerical scheme is based on the requirement that it solves linear problems with constant inhomogeneity exactly. We prove that the method admits second-order error bounds which are independent of the product of the step size with the frequencies. Our analysis also provides new insight into the m ollified impulse method of García-Archilla, Sanz-Serna, and Skeel. We include results of numerical experiments with the sine-Gordon equation.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/s002110050456
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  • 2
    Digitale Medien
    Digitale Medien
    Exponential Integrators for Quantum-Classical Molecular Dynamics (1999)
    Springer
    BIT 39 (1999), S. 620-645 
    Details
    ISSN: 1572-9125
    Schlagwort(e): Numerical integrator ; oscillatory solutions ; Schrödinger equation ; quantum-classical coupling ; error bounds ; stability
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract We study time integration methods for equations of mixed quantum-classical molecular dynamics in which Newtonian equations of motion and Schrödinger equations are nonlinearly coupled. Such systems exhibit different time scales in the classical and the quantum evolution, and the solutions are typically highly oscillatory. The numerical methods use the exponential of the quantum Hamiltonian whose product with a state vector is approximated using Lanczos' method. This allows time steps that are much larger than the inverse of the highest frequencies. We describe various integration schemes and analyze their error behaviour, without assuming smoothness of the solution. As preparation and as a problem of independent interest, we study also integration methods for Schrödinger equations with time-dependent Hamiltonian.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1023/A:1022335122807
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  • 3
    Digitale Medien
    Digitale Medien
    Error Bounds for Exponential Operator Splittings (2000)
    Springer
    BIT 40 (2000), S. 735-744 
    Details
    ISSN: 1572-9125
    Schlagwort(e): Strang splitting ; Trotter product ; Schrödinger equation ; error bounds
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Abstract Error bounds for the Strang splitting in the presence of unbounded operators are derived in a general setting and are applied to evolutionary Schrödinger equations and their pseudo-spectral space discretization.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1023/A:1022396519656
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