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  • 1
    Electronic Resource
    Electronic Resource
    Hopf bifurcation of reaction-diffusion and Navier-Stokes equations under discretization (1998)
    Springer
    Numerische Mathematik 81 (1998), S. 53-84 
    Details
    ISSN: 0945-3245
    Keywords: Mathematics Subject Classification (1991):65M12, 65M15, 35B32, 58F14
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary. The long-time behaviour of numerical approximations to the solutions of a semilinear parabolic equation undergoing a Hopf bifurcation is studied in this paper. The framework includes reaction-diffusion and incompressible Navier-Stokes equations. It is shown that the phase portrait of a supercritical Hopf bifurcation is correctly represented by Runge-Kutta time discretization. In particular, the bifurcation point and the Hopf orbits are approximated with higher order. A basic tool in the analysis is the reduction of the dynamics to a two-dimensional center manifold. A large portion of the paper is therefore concerned with studying center manifolds of the discretization.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002110050384
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    A Gautschi-type method for oscillatory second-order differential equations (1999)
    Springer
    Numerische Mathematik 83 (1999), S. 403-426 
    Details
    ISSN: 0945-3245
    Keywords: Mathematics Subject Classification (1991):65L05, 65L70, 65M12, 65M20
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002110050456
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    Exponential Integrators for Quantum-Classical Molecular Dynamics (1999)
    Springer
    BIT 39 (1999), S. 620-645 
    Details
    ISSN: 1572-9125
    Keywords: Numerical integrator ; oscillatory solutions ; Schrödinger equation ; quantum-classical coupling ; error bounds ; stability
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1022335122807
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    Paper (German National Licenses)
    Fulltext
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