Triangularly Implicit Iteration Methods for ODE-IVP Solvers

Houwen, P. J. van der; Swart, J.J.B. de

doi:10.1137/s1064827595287456

Cited by 44 publications

(46 citation statements)

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“…In [10] and [11] where the PDIRK and PTIRK methods were analysed, it turned out that strong damping of the stiff error components, that is, small amplification factors for error components corresponding to eigenvectors of J with eigenvalues λ of large magnitude, is crucial for a fast overall convergence. This leads us to require the matrix B to be such that ρ(Z(∞)) = ρ(I − B −1 A) vanishes.…”

Section: Convergence Region Of the Pilsrk Methodsmentioning

confidence: 99%

“…A poor initial convergence implies that no accurate predictor value is available for starting the iteration process at t n+1 . A substantial improvement in the initial phase of the convergence of the PILSRK method is obtained by employing the matrices L used in the Parallel Triangular-implicitly Iterated RK methods (PTIRK methods) constructed in [11] (like the PDIRK methods, the PTIRK methods are nonlinear system solvers). The PTIRK matrices L are defined by the lower triangular factor of the Crout decomposition LU of the RK matrix A.…”

Section: Iterative Solution Of the Linear Systemsmentioning

confidence: 99%

“…A preliminary parallel implementation of the Newton-PILSRK method based on the one-step 4-stage Radau IIA formula and using the PTIRK matrix showed on the four-processor Cray-C98/4256 speed-up factors ranging from at least 2.4 until at least 3.1 with respect to RADAU5 in one-processor mode (cf. [11]). …”

Section: Iterative Solution Of the Linear Systemsmentioning

confidence: 99%

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Houwen¹,

Swart²

1997

Advances in Computational Mathematics

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Link to publication General rightsIt is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons). Disclaimer/Complaints regulationsIf you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: http://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible. If the nonlinear systems arising in implicit Runge-Kutta methods like the Radau IIA methods are iterated by (modified) Newton, then we have to solve linear systems whose matrix of coefficients is of the form I − A ⊗ hJ with A the Runge-Kutta matrix and J an approximation to the Jacobian of the righthand side function of the system of differential equations. For larger systems of differential equations, the solution of these linear systems by a direct linear solver is very costly, mainly because of the LU-decompositions. We try to reduce these costs by solving the linear systems by a second (inner) iteration process. This inner iteration process is such that each inner iteration again requires the solution of a linear system. However, the matrix of coefficients in these new linear systems is of the form I − B ⊗ hJ where B is similar to a diagonal matrix with positive diagonal entries. Hence, after performing a similarity transformation, the linear systems are decoupled into s subsystems, so that the costs of the LU-decomposition are reduced to the costs of s LUdecompositions of dimension d. Since these LU-decompositions can be computed in parallel, the effective LU-costs on a parallel computer system are reduced by a factor s 3 . It will be shown that matrices B can be constructed such that the inner iterations converge whenever A and J have their eigenvalues in the positive and nonpositive halfplane, respectively. The theoretical results will be illustrated by a few numerical examples. A parallel implementation on the four-processor Cray-C98/4256 shows a speed-up ranging from at least 2.4 until at least 3.1 with respect to RADAU5 applied in one-processor mode.

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Section: Convergence Region Of the Pilsrk Methodsmentioning

confidence: 99%

Section: Iterative Solution Of the Linear Systemsmentioning

confidence: 99%

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Houwen¹,

Swart²

1997

Advances in Computational Mathematics

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“…[12,13]). In order to see the intrinsic parallelism of the inner iteration process, we proceed as in the preceding section.…”

Section: Iterative Solution Of the Newton Systemsmentioning

confidence: 99%

“…At present, the full method {(2.4), (2.5), (2.8)} is tested on a sequential computer system and only the case where ω = r = 1 and K = K * = I, J * = J (and hence q = 1) has been implemented on the fourprocessor Cray-C98/4256. The results reported in [12] show that with respect to the code RADAU5 of Hairer and Wanner [5], to be considered as one of the best sequential codes, the speed-ups are in the range [2.4, 3.1]. Implementation of the full method {(2.4), (2.5), (2.8)} on the Cray-C98/4256 will be subject of future research.…”

Section: Summary and Concluding Remarksmentioning

confidence: 99%

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Houwen¹,

Veen²

1997

Advances in Computational Mathematics

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Waveform relaxation methods for implicit differential equationsvan der Houwen, P.J.; van der Veen, W.A. General rightsIt is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons). Disclaimer/Complaints regulationsIf you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: http://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible. We apply a Runge-Kutta-based waveform relaxation method to initial-value problems for implicit differential equations. In the implementation of such methods, a sequence of nonlinear systems has to be solved iteratively in each step of the integration process. The size of these systems increases linearly with the number of stages of the underlying RungeKutta method, resulting in high linear algebra costs in the iterative process for high-order Runge-Kutta methods. In our earlier investigations of iterative solvers for implicit initialvalue problems, we designed an iteration method in which the linear algebra costs are almost independent of the number of stages when implemented on a parallel computer system. In this paper, we use this parallel iteration process in the Runge-Kutta waveform relaxation method. In particular, we analyse the convergence of the method. The theoretical results are illustrated by a few numerical examples.

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An efficient method for solving implicit and explicit stiff differential equations

Pašić

2000

Int. J. Numer. Meth. Engng.

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SUMMARYWhen the s-stage fully implicit Runge}Kutta (RK) method is used to solve a system of n ordinary di!erential equations (ODE) the resulting algebraic system has a dimension ns. Its solution by Gauss elimination is expensive and requires 2sn/3 operations. In this paper we present an e$cient algorithm, which di!ers from the traditional RK method. The formal procedure for uncoupling the algebraic system into a block-diagonal matrix with s blocks of size n is derived for any s. Its solution is s/2 times faster than the original, nondiagonalized system, for s even, and s/(s!1) for s odd in terms of number of multiplications, as well as s times in terms of number of additions/multiplications. In particular, for s"3 the method has the same precision and stability properties as the well-known RK-based RadauIIA quadrature of Ehle, implemented by Hairer and Wanner in RADAU5 algorithm. Unlike RADAU5, however, the method is applicable with any s and not only to the explicit ODEs My"f (x, y), where M"const., but also to the general implicit ODEs of the form f (x, y, y)"0. The block-diagonal form of the algebraic system allows parallel processing. The algorithm formally di!ers from the implicit RK methods in that the solution for y is assumed to have a form of the algebraic polynomial whose coe$cients are found by enforcing y to satisfy the di!erential equation at the collocation points. Locations of those points are found from the derived stability function such as to guarantee either A-or¸-stability properties as well as a superior precision of the algorithm. If constructed such as to be¸-stable the method is a good candidate for solving di!erential-algebraic equations (DAEs). Although not limited to any speci"c "eld, the application of the method is illustrated by its implementation in the multibody dynamics described by both ODEs and DAEs.

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Triangularly Implicit Iteration Methods for ODE-IVP Solvers

Cited by 44 publications

References 14 publications

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An efficient method for solving implicit and explicit stiff differential equations

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