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  • Articles  (42)
  • 65L05  (42)
  • 1990-1994  (42)
  • 1950-1954
  • Mathematics  (42)
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  • Articles  (42)
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  • 1
    Electronic Resource
    Electronic Resource
    Boundary value methods for the solution of differential-algebraic equations (1993)
    Springer
    Numerische Mathematik 66 (1993), S. 411-421 
    Details
    ISSN: 0945-3245
    Keywords: 65L05
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Boundary value techniques for the solution of initial value problems of ODEs, despite their apparent higher cost, present some important advantages over initial value methods. Among them, there is the possibility to have greater accuracy, to control the global error, and to have an efficient parallel implementation. In this paper, the same techniques are applied to the solution of linear initial value problems of DAEs. We have considered three term numerical methods (Midpoint, Simpson, and an Adams type method) in order to obtain a block tridiagonal linear system as a discrete problem. Convergence results are stated in the case of constant coefficients, and numerical examples are given on linear time-varying problems.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01385705
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  • 2
    Electronic Resource
    Electronic Resource
    Symplectic phase flow approximation for the numerical integration of canonical systems (1992)
    Springer
    Numerische Mathematik 61 (1992), S. 501-521 
    Details
    ISSN: 0945-3245
    Keywords: 65L05 ; 65L07 ; 58F05 ; 70-08 ; 70F15 ; 70H15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary New methods are presented for the numerical integration of ordinary differential equations of the important family of Hamiltonian dynamical systems. These methods preserve the Poincaré invariants and, therefore, mimic relevant qualitative properties of the exact solutions. The methods are based on a Runge-Kutta-type ansatz for the generating function to realize the integration steps by canonical transformations. A fourth-order method is given and its implementation is discussed. Numerical results are presented for the Hénon-Heiles system, which describes the motion of a star in an axisymmetric galaxy.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01385523
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  • 3
    Electronic Resource
    Electronic Resource
    On symmetrizers for Gauss methods (1991)
    Springer
    Numerische Mathematik 60 (1991), S. 465-476 
    Details
    ISSN: 0945-3245
    Keywords: 65L05 ; 65L20
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In this paper the maximum attainable order of a special class of symmetrizers for Gauss methods is studied. In particular, it is shown that a symmetrizer of this type for thes-stage Gauss method can attain order 2s-1 only for 1 ≦s ≦ 3, and that these symmetrizers areL-stable. A classification of the maximum attainable order of symmetrizers for some higher stages is presented. AnL-stable symmetrizer is also shown to exist for each of the methods studied.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01385731
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  • 4
    Electronic Resource
    Electronic Resource
    Some remarks on an ODE-solver of Kirchgraber (1992)
    Springer
    Numerische Mathematik 61 (1992), S. 261-264 
    Details
    ISSN: 0945-3245
    Keywords: 65L05
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Kirchgraber derived in 1988 an integration procedure (called the LIPS-code) for long-term prediction of the solutions of equations which are perturbations of systems having only periodic solutions. His basic idea is to use the Poincaré map to define a new system which can be integrated with large step-size; the method is specially successful when the period is close to the unperturbed one. Obviously the size of the perturbation modifies the period and therefore affects the precision of the algorithm. In this paper we propose a double modification of Kirchgraber's code: to use a first-order approximation of the perturbed period instead of the unperturbed one, and a scheme specially designed for integration of orbits instead of the Runge-Kutta method. We show that this new code permits a spectacular improvement in accuracy and computation time.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01385507
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  • 5
    Electronic Resource
    Electronic Resource
    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
    Keywords: 65L05
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01385510
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  • 6
    Electronic Resource
    Electronic Resource
    A supraconvergent scheme for the Korteweg-de Vries equation (1992)
    Springer
    Numerische Mathematik 61 (1992), S. 291-310 
    Details
    ISSN: 0945-3245
    Keywords: 65L05 ; 65L10
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A finite-difference method for the integration of the Korteweg-de Vries equation on irregular grids is analyzed. Under periodic boundary conditions, the method is shown to be supraconvergent in the sense that, though being inconsistent, it is second order convergent. However, such a convergence only takes place on grids with an odd number of points per period. When a grid with an even number of points is used, the inconsistency of the method leads to divergence. Numerical results backing the analysis are presented.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01385511
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  • 7
    Electronic Resource
    Electronic Resource
    Power bounded prolongations and Picard-Lindelöf iteration (1990)
    Springer
    Numerische Mathematik 58 (1990), S. 479-501 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS) ; 65L05 ; CR: G1.7
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The possibility of balancing the iteration and discretization errors in iterative solution of large systems of initial value problems is discussed. The main result answers the question affirmatively by stating that in the convergence process, in a model situation, the decay rate of iteration errors is essentially independent of the step size refinement process.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01385637
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  • 8
    Electronic Resource
    Electronic Resource
    On the discretization of a partial differential equation in the neighborhood of a periodic orbit (1993)
    Springer
    Numerische Mathematik 65 (1993), S. 143-175 
    Details
    ISSN: 0945-3245
    Keywords: 35A40 ; 35B40 ; 65L05 ; 65L20 ; 58F22
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Sumamry This article is concerned with the comparison of the dynamic of a partial differential equation and its time discretization. We restrict our attention to the neighborhood of a hyperbolic periodic orbit. We show that the discretization possesses an invariant closed curve near the periodic orbit and that the trajectories of the semigroups defined by the partial differential equations and its approximation are close in a sense to be precised provided that different data are allowed. This answers partly an open problem posed in [4]. Examples of application to dissipative partial equations are provided.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01385746
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  • 9
    Electronic Resource
    Electronic Resource
    On optimal triangular meshes for minimizing the gradient error (1991)
    Springer
    Numerische Mathematik 59 (1991), S. 321-348 
    Details
    ISSN: 0945-3245
    Keywords: 65D05 ; 65L05
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Construction of optimal triangular meshes for controlling the errors in gradient estimation for piecewise linear interpolation of data functions in the plane is discussed. Using an appropriate linear coordinate transformation, rigorously optimal meshes for controlling the error in quadratic data functions are constructed. It is shown that the transformation can be generated as a curvilinear coordinate transformation for anyC data function with nonsingular Hessian matrix. Using this transformation, a construction of nearly optimal meshes for general data functions is described and the error equilibration properties of these meshes discussed. In particular, it is shown that equilibration of errors is not a sufficient condition for optimality. A comparison of meshes generated under several different criteria is made, and their equilibrating properties illustrated.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01385784
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  • 10
    Electronic Resource
    Electronic Resource
    The Drazin inverse in multibody system dynamics (1993)
    Springer
    Numerische Mathematik 64 (1993), S. 521-539 
    Details
    ISSN: 0945-3245
    Keywords: 65L05 ; 70-08
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The numerical analysis of multibody system dynamics is based on the equations of motion as differential-algebraic systems. A thorough analysis of the linearized equations and their solution theory leads to an equivalent system of ordinary differential equations which gives deeper insight into the derivation of integration schemes and into the stabilization approaches. The main tool is the Drazin inverse, a generalized matrix inverse, which preserves the eigenvalues. The results are illustrated by a realistic truck model. Finally, the approach is extended to the nonlinear index 2 formulation.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01388703
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