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  • Articles  (5)
  • 41A21  (2)
  • Runge-Kutta methods  (2)
  • 34A45
  • rationality
  • Springer  (5)
  • Mathematics  (5)
  • Computer Science  (3)
  • 1
    Electronic Resource
    Electronic Resource
    Runge-Kutta Methods for the Numerical Solution of Stiff Semilinear Systems (2000)
    Springer
    BIT 40 (2000), S. 611-639 
    Details
    ISSN: 1572-9125
    Keywords: Runge-Kutta methods ; stability ; convergence ; stiff problems
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper studies the stability and convergence properties of general Runge-Kutta methods when they are applied to stiff semilinear systems y′(t) = J(t)y(t) + g(t, y(t)) with the stiffness contained in the variable coefficient linear part. We consider two assumptions on the relative variation of the matrix J(t) and show that for each of them there is a family of implicit Runge-Kutta methods that is suitable for the numerical integration of the corresponding stiff semilinear systems, i.e. the methods of the family are stable, convergent and the stage equations possess a unique solution. The conditions on the coefficients of a method to belong to these families turn out to be essentially weaker than the usual algebraic stability condition which appears in connection with the B-stability and convergence for stiff nonlinear systems. Thus there are important RK methods which are not algebraically stable but, according to our theory, they are suitable for the numerical integration of semilinear problems. This paper also extends previous results of Burrage, Hundsdorfer and Verwer on the optimal convergence of implicit Runge-Kutta methods for stiff semilinear systems with a constant coefficients linear part.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1022332200092
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  • 2
    Electronic Resource
    Electronic Resource
    Optimal Convergence Results for Runge-Kutta Discretizations of Linear Nonautonomous Parabolic Problems (1999)
    Springer
    BIT 39 (1999), S. 79-95 
    Details
    ISSN: 1572-9125
    Keywords: Runge-Kutta methods ; abstract parabolic equations ; time-dependent problems ; error bounds ; optimal order of convergence ; variable stepsizes
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We analyze the convergence properties of implicit Runge-Kutta methods for the time-discretization of linear nonautonomous parabolic problems. We work in an abstract Banach space setting that covers standard parabolic initial-boundary value problems with time-dependent smooth coefficients. The Runge-Kutta methods are only assumed to be A(θ)-stable, and variable stepsizes are allowed in the analysis. As the main result, we derive optimal convergence rates for Runge-Kutta discretizations of temporally smooth solutions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1022369208270
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  • 3
    Electronic Resource
    Electronic Resource
    On the convergence of multipoint Padé-type approximants and quadrature formulas associated with the unit circle (1996)
    Springer
    Numerical algorithms 13 (1996), S. 321-344 
    Details
    ISSN: 1572-9265
    Keywords: multipoint Padé-type approximation ; orthogonal rational functions ; quadrature formula ; 41A21 ; 30E05 ; 41A55
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We study the convergence of rational interpolants with prescribed poles on the unit circle to the Herglotz-Riesz transform of a complex measure supported on [−π, π]. As a consequence, quadrature formulas arise which integrate exactly certain rational functions. Estimates of the rate of convergence of these quadrature formulas are also included.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02207699
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  • 4
    Electronic Resource
    Electronic Resource
    Approximated solutions in rational form for systems of differential equations (1999)
    Springer
    Numerical algorithms 21 (1999), S. 185-203 
    Details
    ISSN: 1572-9265
    Keywords: systems of differential equations ; analytic solutions ; matrix Padé approximation ; rationality ; minimum degrees ; uniqueness ; partial differential equations ; 41A21 ; 34A45 ; 35A35
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract In this paper we present a technique to study the existence of rational solutions for systems of differential equations — for an ordinary differential equation, in particular. The method is relatively straightforward; it is based on a rationality characterisation that involves matrix Padé approximants. It is important to note that, when the solution is rational, we use formal power series “without taking into account” their circle of convergence; at the end of this paper we justify this. We expound the theory for systems of linear first-order ordinary differential equations in the general case. However, the main ideas are applied in numerical resolution of partial differential equations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1019109429882
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  • 5
    Electronic Resource
    Electronic Resource
    Matrix Padé Approximation of rational functions (1997)
    Springer
    Numerical algorithms 15 (1997), S. 167-192 
    Details
    ISSN: 1572-9265
    Keywords: Matrix Padé Approximation ; rationality ; minimum degrees ; uniqueness ; 65 ; 41 ; 15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract In many applications it is of major interest to decide whether a given formal power series with matrix-valued coefficients of arbitrary dimensions results from a matrix-valued rational function. As the main result of this paper we provide an answer to this question in terms of Matrix Padé Approximants of the given power series. Furthermore, given a matrix rational function, the “smallest” degrees of the matrix polynomials which represent it are not necessarily unique. Therefore we study a certain minimality-type, that is, minimum degrees. We aim to obtain all the minimum degrees for the polynomials which represent the function as equivalents. In addition, given that the rational representation of the function for the same pair of degrees need not be unique, we have obtained conditions to study the uniqueness of said representation. All the results obtained are presented graphically in tables setting out the above information. They lead to a number of properties concerning special structures, staired blocks, in the Padé Table.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1019193820428
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