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  • 1
    Electronic Resource
    Electronic Resource
    Convergence properties of the Runge-Kutta-Chebyshev method (1990)
    Springer
    Numerische Mathematik 57 (1990), S. 157-178 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS):65M20, 65M10 ; CR:G1.8
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The Runge-Kutta-Chebyshev method is ans-stage Runge-Kutta method designed for the explicit integration of stiff systems of ordinary differential equations originating from spatial discretization of parabolic partial differential equations (method of lines). The method possesses an extended real stability interval with a length β proportional tos 2. The method can be applied withs arbitrarily large, which is an attractive feature due to the proportionality of β withs 2. The involved stability property here is internal stability. Internal stability has to do with the propagation of errors over the stages within one single integration step. This internal stability property plays an important role in our examination of full convergence properties of a class of 1st and 2nd order schemes. Full convergence means convergence of the fully discrete solution to the solution of the partial differential equation upon simultaneous space-time grid refinement. For a model class of linear problems we prove convergence under the sole condition that the necessary time-step restriction for stability is satisfied. These error bounds are valid for anys and independent of the stiffness of the problem. Numerical examples are given to illustrate the theoretical results.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01386405
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  • 2
    Electronic Resource
    Electronic Resource
    Splitting Methods for Three-Dimensional Transport Models with Interaction Terms (1997)
    Springer
    Journal of scientific computing 12 (1997), S. 215-231 
    Details
    ISSN: 1573-7691
    Keywords: Transport models ; shallow water ; splitting methods ; stability
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Notes: Abstract We investigate the use of splitting methods for the numerical integration of three-dimensional transport-chemistry models. In particular, we investigate various possibilities for the time discretization that can take advantage of the parallelization and vectorization facilities offered by multi-processor vector computers. To suppress wiggles in the numerical solution, we use third-order, upwind-biased discretization of the advection terms, resulting in a five-point coupling in each direction. As an alternative to the usual splitting functions, such as co-ordinate splitting or operator splitting, we consider a splitting function that is based on a three-coloured hopscotch-type splitting in the horizontal direction, whereas full coupling is retained in the vertical direction. Advantages of this splitting function are the easy application of domain decomposition techniques and unconditional stability in the vertical, which is an important property for transport in shallow water. The splitting method is obtained by combining the hopscotch-type splitting function with various second-order splitting formulae from the literature. Although some of the resulting methods are highly accurate, their stability behaviour (due to horizontal advection) is quite poor. Therefore we also discuss several new splitting formulae with the aim to improve the stability characteristics. It turns out that this is possible indeed, but the price to pay is a reduction of the accuracy. Therefore, such methods are to be preferred if accuracy is less crucial than stability; such a situation is frequently encountered in solving transport problems. As part of the project TRUST (Transport and Reactions Unified by Splitting Techniques), preliminary versions of the schemes are implemented on the Cray C98 4256 computer and are available for benchmarking.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1025645326705
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  • 3
    Electronic Resource
    Electronic Resource
    On the numerical integration of second-order initial value problems with a periodic forcing function (1986)
    Springer
    Computing 37 (1986), S. 195-218 
    Details
    ISSN: 1436-5057
    Keywords: 65L05 ; G.1.7 ; G.1.8 ; Numerical analysis ; ordinary differential equations ; Runge-Kutta methods ; predictorcorrector methods ; periodic solutions
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Für die numerische Behandlung von Differentialgleichungen zweiter Ordnung, bei dennen die Lösung im wesentlichen durch die von einer äußeren periodischen Kraft erzwungenen Schwingung bestimmt wird, werden Diskretisierungsmethoden vom Runge-Kutta-Nyström Typ und spezielle Prädiktor-Korrektor Methoden konstruiert. Für eine Klasse expliziter und linear-impliziter Runge-Kutta-Nyström Methoden der Ordung zwei zeigen wir, daß die erzwungene Schwingung keinen Phasenfehler aufweist. Für eine Klasse von Prädiktor-Korrektor Methoden vierter Ordnung wird nachgewiesen, daß die Phasen- und Dissipationsfehlerordnung beliebig groß gemacht werden kann. Numerische Beispiele bestätigen die Wirksamkeit unserer Methoden mit reduziertem Phasenfehler.
    Notes: Abstract Runge-Kutta-Nyström type methods and special predictor-corrector methods are constructed for the accurate solution of second-order differential equations of which the solution is dominated by the forced oscillation originating from an external, periodic forcing term. For a family of second-order explicit and linearly implicit Runge-Kutta-Nyström methods it is shown that the forced oscillation is represented with zero phase lag. For a family of predictor-corrector methods of fourth-order, it is shown that both the phase lag order and the dissipation of the forced oscillation can be made arbitrarily high. Numerical examples illustrate the effectiveness of our reduced phase lag methods.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02252512
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  • 4
    Electronic Resource
    Electronic Resource
    Stability of collocation-based Runge-Kutta-Nyström methods (1991)
    Springer
    BIT 31 (1991), S. 469-481 
    Details
    ISSN: 1572-9125
    Keywords: 65M10 ; 65M20
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We analyse the attainable order and the stability of Runge-Kutta-Nyström (RKN) methods for special second-order initial-value problems derived by collocation techniques. Like collocation methods for first-order equations the step point order ofs-stage methods can be raised to 2s for alls. The attainable stage order is one higher and equalss+1. However, the stability results derived in this paper show that we have to pay a high price for the increased stage order.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01933263
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  • 5
    Electronic Resource
    Electronic Resource
    Parallelism across the steps in iterated Runge-Kutta methods for stiff initial value problems (1994)
    Springer
    Numerical algorithms 8 (1994), S. 293-312 
    Details
    ISSN: 1572-9265
    Keywords: Numerical analysis ; Runge-Kutta methods ; parallelism ; G.1.7.
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract For the parallel integration of stiff initial value problems (IVPs) three main approaches can be distinguished: approaches based on “parallelism across the problem”, on “parallelism across the method” and on “parallelism across the steps”. The first type of parallelism does not require special integration methods can be exploited within any available IVP solver. The methodparallel approach received some attention in the case of Runge-Kutta based methods. For these methods, the required number of processors is roughly half the order of the generating Runge-Kutta method and the speed-up with respect to a good sequential IVP solver is about a factor 2. The third type of parallelism (step-parallelism) can be achieved in any IVP solver based on predictor-corrector iteration. Most step-parallel methods proposed so far employ a large number of processors, but lack the property of robustness, due to a poor convergence behaviour in the iteration process. Hence, the effective speed-up is rather poor. The step-parallel iteraction process proposed in the present paper is less massively parallel, but turns out to be sufficiently robust to solve the four-stage Radau IIA corrector used in our experiments within a few effective iterations per step and to achieve speed-up factors up to 10 with respect to the best sequential codes.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02142695
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  • 6
    Electronic Resource
    Electronic Resource
    Implementation and performance of the time integration of a 3D transport model (1995)
    Chichester : Wiley-Blackwell
    International Journal for Numerical Methods in Fluids 20 (1995), S. 213-231 
    Details
    ISSN: 0271-2091
    Keywords: Transport models ; 3D advection-diffusion equations ; Numerical time integration ; Vectorization ; Parallel processing ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
    Notes: The total solution of a three-dimensional model for computing the transport of salinity, pollutants, suspended material (such as sediment or mud), etc. in shallow seas involves many aspects, each of which has to be treated in an optimal way in order to cope with the tremendous computational task involved. In this paper we focus on one of these aspects, i.e. on the time integration, and discuss two numerical solution methods. The emphasis in this paper is on the performance of the methods when implemented on a vector/parallel, shared memory computer such as a Cray-type machine. The first method is an explicit time integrator and can straightforwardly be vectorized and parallelized. Although a stabilizing technique has been applied to this method, it still suffers from a severe time step restriction. The second method is partly implicit, resulting in much better stability characteristics; however, as a consequence of the implicitness, it requires in each step the solution of a large number of tridiagonal systems. When implemented in a standard way, the recursive nature would prevent vectorization, resulting in a very long solution time. Following a suggestion of Golub and Van Loan, this part of the algorithm has been tuned for use on the Cray C98/4256. On the basis of a large-scale test problem, performance results will be presented for various implementations.
    Additional Material: 2 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/fld.1650200303
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  • 7
    Electronic Resource
    Electronic Resource
    Implementation and performance of the time integration of a 3D numerical transport model (1995)
    Chichester : Wiley-Blackwell
    International Journal for Numerical Methods in Fluids 21 (1995), S. 349-367 
    Details
    ISSN: 0271-2091
    Keywords: transport models ; 3D advection-diffusion equations ; numerical time integration ; vectorization ; parallel processing ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
    Notes: The total solution of a three-dimensional model for computing the transport of salinity, pollutants, suspended material (such as sediment or mud), etc. in shallow seas involves many aspects, each of which has to be treated in an optimal way in order to cope with the tremendous computational task involved. In this paper we focus on one of these aspects, i.e. on the time integration, and discuss two numerical solution methods. The emphasis in this paper is on the performance of the methods when implemented on a vector/parallel, shared memory computer such as a Cray-type machine. The first method is an explicit time integrator and can straightforwardly be vectorized and parallelized. Although a stabilizing technique has been applied to this method, it still suffers from a severe time step restriction. The second method is partly implicit, resulting in much beter stability characteristics; however, as a consequence of the implicitness, it requires in each step the solution of a large number of tridiagonal systems. When implemented in a standard way, the recursive nature would prevent vectorization, resulting in a very long solution time. Following a suggestion of Golub and Van Loan, this part of the algorithm has been tuned for use on the Cray C98/4256. On the basis of a large-scale test problem, performance results will be presented for various implementations.
    Additional Material: 2 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/fld.1650210407
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    Paper (German National Licenses)
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  • 8
    Electronic Resource
    Electronic Resource
    On the treatment of time-dependent boundary conditions in splitting methods for parabolic differential equations (1981)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 17 (1981), S. 335-346 
    Details
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: Splitting methods for time-dependent partial differential equations usually exhibit a drop in accuracy if boundary conditions become time-dependent. This phenomenon is investigated for a class of splitting methods for two-space dimensional parabolic partial differential equations. A boundary-value correction discussed in a paper by Fairweather and Mitchell for the Laplace equation with Dirichlet conditions, is generalized for a wide class of initial boundary-value problems. A numerical comparison is made for the ADI method of Peaceman-Rachford and the LOD method of Yanenko applied to problems with Dirichlet boundary conditions and non-Dirichlet boundary conditions.
    Additional Material: 6 Tab.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nme.1620170304
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  • 9
    Electronic Resource
    Electronic Resource
    Iterated θ-method for hyperbolic equations (1990)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 30 (1990), S. 271-290 
    Details
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The iterated θ-methods employing residue smoothing for finding both steady state and time-accurate solutions of semidiscrete hyperbolic differential equations are analysed. By the technique of residue smoothing the stability condition is considerably relaxed, so that larger time steps are allowed which improves the efficiency of the method. The additional computational effort involved by the explicit smoothing technique used here is rather low when compared with its stabilizing effect. However, in the case where time-accurate solutions are desired, the overall accuracy may be decreased. This paper investigates the effect of residue smoothing on both the stability and accuracy, and presents a number of explicitly given methods based on the iterated implicit midpoint rule (θ = 1/2). Numerical examples confirm the theoretical results.
    Additional Material: 12 Tab.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nme.1620300205
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