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  • 1
    Electronic Resource
    Electronic Resource
    Preconditioning waveform relaxation iterations for differential systems (1996)
    Springer
    BIT 36 (1996), S. 54-76 
    Details
    ISSN: 1572-9125
    Keywords: Waveform relaxation ; splittings ; preconditioning ; overlapping ; error analysis ; parallel computing
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We discuss preconditioning and overlapping of waveform relaxation methods for sparse linear differential systems. It is demonstrated that these techniques significantly improve the speed of convergence of the waveform relaxation iterations resulting from application of various modes of block Gauss-Jacobi and block Gauss-Seidel methods to differential systems. Numerical results are presented for linear systems resulting from semi-discretization of the heat equation in one and two space variables. It turns out that overlapping is very effective for the system corresponding to the one-dimensional heat equation and preconditioning is very effective for the system corresponding to the two-dimensional case.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01740544
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    General linear methods with external stages of different orders (1996)
    Springer
    BIT 36 (1996), S. 688-712 
    Details
    ISSN: 1572-9125
    Keywords: General linear methods ; order conditions ; stage errors ; error estimation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The approach introduced recently by Albrecht to derive order conditions for Runge-Kutta formulas based on the theory of A-methods is also very powerful for the general linear methods. In this paper, using Albrecht's approach, we formulate the general theory of order conditions for a class of general linear methods where the components of the propagating vector of approximations to the solution have different orders. Using this theory we derive a class of diagonally implicit multistage integration methods (DIMSIMs) for which the global order is equal to the local order. We also derive a class of general linear methods with two nodal approximations of different orders which facilitate local error estimation. Our theory also applies to the class of two-step Runge-Kutta introduced recently by Jackiewicz and Tracogna.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01733788
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    Experiments with a variable-order type 1 DIMSIM code (1999)
    Springer
    Numerical algorithms 22 (1999), S. 237-261 
    Details
    ISSN: 1572-9265
    Keywords: DIMSIM methods ; Nordsieck representation ; local error estimation ; step size and order changing strategy ; 65L05
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract The issues related to the development of a new code for nonstiff ordinary differential equations are discussed. This code is based on the Nordsieck representation of type 1 DIMSIMs, implemented in a variable-step size variable-order mode. Numerical results demonstrate that the error estimation employed in the code is very reliable and that the step and order changing strategies are very robust. This code outperforms the Matlab ode45 code for moderate and stringent tolerances.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1019135630307
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    Paper (German National Licenses)
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  • 4
    Electronic Resource
    Electronic Resource
    Variable stepsize continuous two-step Runge-Kutta methods for ordinary differential equations (1996)
    Springer
    Numerical algorithms 12 (1996), S. 347-368 
    Details
    ISSN: 1572-9265
    Keywords: Continuous two-step Runge-Kutta method ; convergence ; order and stage order ; 65L05 ; 65L06
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract A general class of variable stepsize continuous two-step Runge-Kutta methods is investigated. These methods depend on stage values at two consecutive steps. The general convergence and order criteria are derived and examples of methods of orderp and stage orderq=p orq=p−1 are given forp≤5. Numerical examples are presented which demonstrate that high order and high stage order are preserved on nonuniform meshes with large variations in ratios between consecutive stepsizes.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02142812
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  • 5
    Electronic Resource
    Electronic Resource
    Nordsieck representation of DIMSIMs (1997)
    Springer
    Numerical algorithms 16 (1997), S. 209-230 
    Details
    ISSN: 1572-9265
    Keywords: DIMSIM methods ; Nordsieck representation ; variable step-size ; error accumulation ; error estimation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract A new representation for diagonally implicit multistage integration methods (DIMSIMs) is derived in which the vector of external stages directly approximates the Nordsieck vector. The methods in this formulation are zero-stable for any choice of variable mesh. They are also easy to implement since changing step-size corresponds to a simple rescaling of the vector of external approximations. The paper contains an analysis of local truncation error and of error accumulation in a variable step-size situation.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1019195215402
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  • 6
    Electronic Resource
    Electronic Resource
    Construction of two-step Runge-Kutta methods of high order for ordinary differential equations (1998)
    Springer
    Numerical algorithms 18 (1998), S. 51-70 
    Details
    ISSN: 1572-9265
    Keywords: two-step Runge-Kutta methods ; stability analysis ; least squares minimization ; 65L05
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract The construction of two-step Runge-Kutta methods of order p and stage order q=p with stability polynomial given in advance is described. This polynomial is chosen to have a large interval of absolute stability for explicit methods and to be A-stable and L-stable for implicit methods. After satisfying the order and stage order conditions the remaining free parameters are computed by minimizing the sum of squares of the difference between the stability function of the method and a given polynomial at a sufficiently large number of points in the complex plane.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1019157029031
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    Paper (German National Licenses)
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  • 7
    Electronic Resource
    Electronic Resource
    The performance of preconditioned waveform relaxation techniques for pseudospectral methods (1996)
    New York, NY [u.a.] : Wiley-Blackwell
    Numerical Methods for Partial Differential Equations 12 (1996), S. 245-263 
    Details
    ISSN: 0749-159X
    Keywords: Mathematics and Statistics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Waveform relaxation techniques for the pseudospectral solution of the heat conduction problem are discussed. The pseudospectral operator occurring in the equation is preconditioned by transformations in both space and time. In the spatial domain, domain stretching is used to more equally distribute the grid points across the domain, and hence improve the conditioning of the differential operator. Preconditioning in time is achieved either by an exponential or a polynomial transformation. Block Jacobi solutions of the systems are obtained and compared. The preconditioning in space by domain stretching determines the effectiveness of waveform relaxation in this case. Preconditioning in time is also effective in reducing the number of iterations required for convergence. The polynomial transformation is preferred, because it removes the requirement of a matrix exponential calculation in the time-stepping schemes and, at the same time, is no less effective than the direct exponential preconditioning. © 1996 John Wiley & Sons, Inc.
    Additional Material: 6 Ill.
    Type of Medium: Electronic Resource
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    Paper (German National Licenses)
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