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  • ddc:000  (4)
  • error bounds  (2)
  • Mathematics Subject Classification (1991):65L05, 65L70, 65M12, 65M20  (1)
  • 1
    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)
    Fulltext
  • 2
    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
  • 3
    Electronic Resource
    Electronic Resource
    Error Bounds for Exponential Operator Splittings (2000)
    Springer
    BIT 40 (2000), S. 735-744 
    Details
    ISSN: 1572-9125
    Keywords: Strang splitting ; Trotter product ; Schrödinger equation ; error bounds
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1022396519656
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    Paper (German National Licenses)
    Fulltext
  • 4
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    Adaptive, Fast and Oblivious Convolution in Evolution Equations with Memory (2006)
    Details
    Publication Date: 2016-06-09
    Description: To approximate convolutions which occur in evolution equations with memory terms, a variable-stepsize algorithm is presented for which advancing $N$ steps requires only $O(N\log N)$ operations and $O(\log N)$ active memory, in place of $O(N^2)$ operations and $O(N)$ memory for a direct implementation. A basic feature of the fast algorithm is the reduction, via contour integral representations, to differential equations which are solved numerically with adaptive step sizes. Rather than the kernel itself, its Laplace transform is used in the algorithm. The algorithm is illustrated on three examples: a blow-up example originating from a Schrödinger equation with concentrated nonlinearity, chemical reactions with inhibited diffusion, and viscoelasticity with a fractional order constitutive law.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/pdf
    Format: application/postscript
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    OPUS
  • 5
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    MEXX - Numerical Software for the Integration of Constrained Mechanical Systems. (1992)
    Details
    Publication Date: 2020-11-16
    Description: MEXX (short for MEXanical systems eXtrapolation integrator) is a Fortran code for time integration of constrained mechanical systems. MEXX is suited for direct integration of the equations of motion in descriptor form. It is based on extrapolation of a time stepping method that is explicit in the differential equations and linearly implicit in the nonlinear constraints. It only requires the solution of well--structured systems of linear equations which can be solved with a computational work growing linearly with the number of bodies, in the case of multibody systems with few closed kinematic loops. Position and velocity constraints are enforced throughout the integration interval, whereas acceleration constraints need not be formulated. MEXX has options for time--continuous solution representation (useful for graphics) and for the location of events such as impacts. The present article describes MEXX and its underlying concepts.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/pdf
    Format: application/postscript
    Library Location Call Number Volume/Issue/Year Availability
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    OPUS
  • 6
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    Fast and oblivious convolution quadrature (2005)
    Details
    Publication Date: 2016-06-09
    Description: We give an algorithm to compute $N$ steps of a convolution quadrature approximation to a continuous temporal convolution using only $O(N\, \log N)$ multiplications and $O(\log N)$ active memory. The method does not require evaluations of the convolution kernel, but instead $O(\log N)$ evaluations of its Laplace transform, which is assumed sectorial. The algorithm can be used for the stable numerical solution with quasi-optimal complexity of linear and nonlinear integral and integro-differential equations of convolution type. In a numerical example we apply it to solve a subdiffusion equation with transparent boundary conditions.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/postscript
    Format: application/pdf
    Library Location Call Number Volume/Issue/Year Availability
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    OPUS
  • 7
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    Fast Runge-Kutta approximation of inhomogeneous parabolic equations (2005)
    Details
    Publication Date: 2014-02-26
    Description: The result after $N$ steps of an implicit Runge-Kutta time discretization of an inhomogeneous linear parabolic differential equation is computed, up to accuracy $\varepsilon$, by solving only $$O\Big(\log N\, \log \frac1\varepsilon \Big) $$ linear systems of equations. We derive, analyse, and numerically illustrate this fast algorithm.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/pdf
    Format: application/postscript
    Library Location Call Number Volume/Issue/Year Availability
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    OPUS
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