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  • 1
    Digitale Medien
    Digitale Medien
    Classical numerical integrators for wave-packet dynamics (1996)
    College Park, Md. : American Institute of Physics (AIP)
    The Journal of Chemical Physics 104 (1996), S. 2349-2355 
    Details
    ISSN: 1089-7690
    Quelle: AIP Digital Archive
    Thema: Physik , Chemie und Pharmazie
    Notizen: Gray and Verosky have recently studied the reformulation of the N-state matrix representation of the time-dependent Schrödinger equation as an N-degrees of freedom classical Hamiltonian system. This opens the possibility of using in quantum dynamics numerical integrators originally devised for classical mechanics. When the Hamiltonian matrix is time-dependent, Gray and Verosky suggest the use of a Magnus approximation before reducing the quantum system to its classical format. We show that Magnus approximations are not necessary and suggest an alternative technique. With the new technique it is possible to obtain simple integrators of arbitrarily high orders of accuracy that can be applied to all matrix Schrödinger problems with a, possibly time-dependent, real Hamiltonian matrix. The connection between the new approach and high-order split-operator methods is studied. © 1996 American Institute of Physics.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1063/1.470930
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  • 2
    Digitale Medien
    Digitale Medien
    Soliton and antisoliton interactions in the "good'' Boussinesq equation (1988)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 29 (1988), S. 1964-1968 
    Details
    ISSN: 1089-7658
    Quelle: AIP Digital Archive
    Thema: Mathematik , Physik
    Notizen: The solitary-wave interaction mechanism for the good Boussinesq equation is investigated and found to be far more complicated than was previously thought. Three salient features are that solitary waves only exist for a finite range of velocities, that large solitons can turn into so-called antisolitons, and that it is possible for solitons to merge and split. Small solitons, however, appear to be stable. The existence of a potential well is linked to the different behaviors observed between small and large initial conditions.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1063/1.527850
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  • 3
    Digitale Medien
    Digitale Medien
    Convergence and order reduction of Runge-Kutta schemes applied to evolutionary problems in partial differential equations (1986)
    Springer
    Numerische Mathematik 50 (1986), S. 405-418 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65X02 ; 65M10 ; CR:G1.7 ; 65M20
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary We address the question of convergence of fully discrete Runge-Kutta approximations. We prove that, under certain conditions, the order in time of the fully discrete scheme equals the conventional order of the Runge-Kutta formula being used. However, these conditions, which are necessary for the result to hold, are not natural. As a result, in many problems the order in time will be strictly smaller than the conventional one, a phenomenon called order reduction. This phenomenon is extensively discussed, both analytically and numerically. As distinct from earlier contributions we here treat explicit Runge-Kutta schemes. Although our results are valid for both parabolic and hyperbolic problems, the examples we present are therefore taken from the hyperbolic field, as it is in this area that explicit discretizations are most appealing.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01396661
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  • 4
    Digitale Medien
    Digitale Medien
    Conservation of integrals and symplectic structure in the integration of differential equations by multistep methods (1992)
    Springer
    Numerische Mathematik 61 (1992), S. 281-290 
    Details
    ISSN: 0945-3245
    Schlagwort(e): 65L05
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary We consider the question of whether multistep methods inherit in some sense quadratic first integrals possessed by the differential system being integrated. We also investigate whether, in the integration of Hamiltonian systems, multistep methods conserve the symplectic structure of the phase space.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01385510
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  • 5
    Digitale Medien
    Digitale Medien
    Convergence of the Lambert-McLeod trajectory solver and of the celf method (1984)
    Springer
    Numerische Mathematik 45 (1984), S. 173-182 
    Details
    ISSN: 0945-3245
    Schlagwort(e): Primary 65L05 ; Secondary 65M10 ; CR: G 1.7
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary A trajectory problem is an initial value problemd y/dt=f(y),y(0)=η where the interest lies in obtaining the curve traced by the solution (the trajectory), rather than in finding the actual correspondanc between values of the parametert and points on that curve. We prove the convergence of the Lambert-McLeod scheme for the numerical integration of trajectory problems. We also study the CELF method, an explicit procedure for the integration in time of semidiscretizations of PDEs which has some useful conservation properties. The proofs rely on the concept of restricted stability introduced by Stetter. In order to show the convergence of the methods, an idea of Strang is also employed, whereby the numerical solution is compared with a suitable perturbation of the theoretical solution, rather than with the theoretical solution itself.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01389463
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  • 6
    Digitale Medien
    Digitale Medien
    An extension of the Lax-Richtmyer theory (1984)
    Springer
    Numerische Mathematik 44 (1984), S. 279-283 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS (MOS): 65M10 ; CR:5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary We give a rigorous proof of the validity of the Lax equivalence theorem when the true solution at timet and its approximation lie in different spaces, thus modelling the practical situation where the former is a function of the space variables and the latter is only defined at grid points.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01410111
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  • 7
    Digitale Medien
    Digitale Medien
    Regions of stability, equivalence theorems and the Courant-Friedrichs-Lewy condition (1986)
    Springer
    Numerische Mathematik 49 (1986), S. 319-329 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65M10 ; CR: G1.8
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary The celebrated CFL condition for discretizations of hyperbolic PDEs is shown to be equivalent to some results of Jeltsch and Nevanlinna concerning regions of stability ofk-step,m-stage linear methods for the integration of ODEs. We characterize the methods for the numerical integration of the model equation,u t=u x which are weakly stable when the mesh-ratio takes the maximum value allowed by the CFL condition. We provide new equivalence theorems between stability and convergence, which improve on the classical results.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01389633
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  • 8
    Digitale Medien
    Digitale Medien
    Nonlinear stability and convergence of finite-difference methods for the “good” Boussinesq equation (1990)
    Springer
    Numerische Mathematik 58 (1990), S. 215-229 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65M10 ; CR: G.1.8
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary The “good” Boussinesq equationu tt =−u xxxx +u xx +(u 2) xx has recently been found to possess an interesting soliton-interaction mechanism. In this paper we study the nonlinear stability and the convergence of some simple finite-difference schemes for the numerical solution of problems involving the “good” Boussinesq equation. Numerical experimentas are also reported.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01385620
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  • 9
    Digitale Medien
    Digitale Medien
    Equilibria of Runge-Kutta methods (1990)
    Springer
    Numerische Mathematik 58 (1990), S. 243-254 
    Details
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65L05 ; CR: G1.7
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary It is known that certain Runge-Kutta methods share the property that, in a constant-step implementation, if a solution trajectory converges to a bounded limit then it must be a fixed point of the underlying differential system. Such methods are calledregular. In the present paper we provide a recursive test to check whether given method is regular. Moreover, by examining solution trajectories of linear equations, we prove that the order of ans-stage regular method may not exceed 2[(s+2)/2] and that the maximal order of regular Runge-Kutta method with an irreducible stability function is 4.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF01385623
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  • 10
    Digitale Medien
    Digitale Medien
    Convergence of method of lines approximations to partial differential equations (1984)
    Springer
    Computing 33 (1984), S. 297-313 
    Details
    ISSN: 1436-5057
    Schlagwort(e): 65XX02 ; 65M10 ; 65M20 ; 1982 C.R. Categories: 5.17 ; Initial value problems ; partial differential equations ; method of lines ; stiff differential equations ; nonlinear problems ; convergence analysis
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Informatik
    Beschreibung / Inhaltsverzeichnis: Zusammenfassung Viele numerische Verfahren für Anfangswertprobleme für partielle Differentialgleichungen kann man als Linienmethoden interpretieren. Diese Arbeit behandelt solche Verfahren vom Einschriftt-Typ. Unser Ziel ist die Behandlung von Konvergenzfragen, insbesondere für nichtlineare Probleme. Unsere Hilfsmittel zum Nachweis der Stabilität entnehmen wir der stark entwickelten Theorie für nichtlineare steife gewöhnliche Differentialgleichungen. Wichtig sind hierbei die logarithmische Matrixnorm und der C-Stabilitätsbegriff. Eine nichtlineare parabolische Gleichung und die kubische Schrödingergleichung werden verwendet, um die Ideen zu illustrieren.
    Notizen: Abstract Many existing numerical schemes for evolutionary problems in partial differential equations (PDEs) can be viewed as method of lines (MOL) schemes. This paper treats the convergence of one-step MOL schemes. Our main purpose is to set up a general framework for a convergence analysis applicable to nonlinear problems. The stability materials for this framework are taken from the field of nonlinear stiff ODEs. In this connection, important concepts are the logarithmic matrix norm and C-stability. A nonlinear parabolic equation and the cubic Schrödinger equation are used for illustrating the ideas.
    Materialart: Digitale Medien
    URL: http://dx.doi.org/10.1007/BF02242274
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