A New Block Preconditioner for Implicit Runge--Kutta Methods for Parabolic PDE Problems

Rana, Md. Masud; Howle, Victoria E.; Long, Katharine; Meek, Ashley; Milestone, W.

doi:10.1137/20m1349680

Cited by 14 publications

(19 citation statements)

References 5 publications

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“…Remark Clearly, the approach can be applied to

{A}_q

, which is tested in Reference 40. The variant

𝒫_{L D}

from that article would correspond to our approach when applied to

{A}_q

.…”

Section: Stage‐parallel Preconditioning Methods For the Radau Tensor ...mentioning

confidence: 96%

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Stage‐parallel preconditioners for implicit Runge–Kutta methods of arbitrarily high order, linear problems

Axelsson,

Dravins,

Neytcheva

2023

Numerical Linear Algebra App

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Fully implicit Runge–Kutta methods offer the possibility to use high order accurate time discretization to match space discretization accuracy, an issue of significant importance for many large scale problems of current interest, where we may have fine space resolution with many millions of spatial degrees of freedom and long time intervals. In this work, we consider strongly A‐stable implicit Runge–Kutta methods of arbitrary order of accuracy, based on Radau quadratures. For the arising large algebraic systems we introduce efficient preconditioners, that (1) use only real arithmetic, (2) demonstrate robustness with respect to problem and discretization parameters, and (3) allow for fully stage‐parallel solution. The preconditioners are based on the observation that the lower‐triangular part of the coefficient matrices in the Butcher tableau has larger in magnitude values, compared to the corresponding strictly upper‐triangular part. We analyze the spectrum of the corresponding preconditioned systems and illustrate their performance with numerical experiments. Even though the observation has been made some time ago, its impact on constructing stage‐parallel preconditioners has not yet been done and its systematic study constitutes the novelty of this article.

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“…Remark Clearly, the approach can be applied to

{A}_q

, which is tested in Reference 40. The variant

𝒫_{L D}

from that article would correspond to our approach when applied to

{A}_q

.…”

Section: Stage‐parallel Preconditioning Methods For the Radau Tensor ...mentioning

confidence: 96%

“…A comparison study of a series of LDU‐based preconditioning approaches, applied on the non‐transformed Equation (4) is recently presented in Reference 40. The authors test various combinations of the

LDU

factorization of

{A}_q

.…”

Section: A Summary Of Some Earlier Presented Stage‐parallel Precondit...mentioning

confidence: 99%

Stage‐parallel preconditioners for implicit Runge–Kutta methods of arbitrarily high order, linear problems

Axelsson,

Dravins,

Neytcheva

2023

Numerical Linear Algebra App

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“…Figure 4 shows that the error does not depend on N = 2 2 , 2 3 , • • • , 2 15 (refer to + and • in Figure 4 for the errors by implicit Runge-Kutta method, and to × and for those by θ method). Regardless of the choice of N, it is confirmed in Figure 4 that the errors with ∆t = 2 −13 are smaller than those with ∆t = 2 −6 (note that in case of ∆t = 2 −6 , the numerical calculation does not converge and no numerical solution is obtained for large N).…”

Section: Accuracy Depending On Discretization Of Spatial Variablesmentioning

confidence: 99%

“…Therefore, a numerical scheme combining explicit and implicit methods has been studied: e.g., a method for weakening the restriction on ∆t with keeping the stability of calculations [12][13][14]. In this context unconditionally-stable fully implicit method has been studied recently [15][16][17]. When it comes to a fully implicit method, the problem is generally reduced to a nonlinear type self-consistent equation, and it is necessary to apply numerical iteration.…”

Section: Introductionmentioning

confidence: 99%

Numerical Scheme Based on the Implicit Runge-Kutta Method and Spectral Method for Calculating Nonlinear Hyperbolic Evolution Equations

Takei

Iwata

2022

Axioms

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A numerical scheme for nonlinear hyperbolic evolution equations is made based on the implicit Runge-Kutta method and the Fourier spectral method. The detailed discretization processes are discussed in the case of one-dimensional Klein-Gordon equations. In conclusion, a numerical scheme with third-order accuracy is presented. The order of total calculation cost is O(Nlog2N). As a benchmark, the relations between numerical accuracy and discretization unit size and that between the stability of calculation and discretization unit size are demonstrated for both linear and nonlinear cases.

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“…e.g., [1,2,3,4,8,9]. In addition to the above qualities, the IRK matrices, arising from Radau IIA quadratures, possess some special algebraic properties, used when constructing a preconditioner to A, which makes the preconditioner superior to other preconditioners of similar algebraic structure, see [10] and the discussion in [5].…”

Section: Linear Systems Arising In Irk and Preconditioningmentioning

confidence: 99%

Fine spectral analysis of preconditioned matrices and matrix-sequences arising from stage-parallel implicit Runge-Kutta methods of arbitrarily high order

Dravins¹,

Serra‐Capizzano²,

Neytcheva³

2023

Preprint

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The use of high order fully implicit Runge-Kutta methods is of significant importance in the context of the numerical solution of transient partial differential equations, in particular when solving large scale problems due to fine space resolution with many millions of spatial degrees of freedom and long time intervals. In this study we consider strongly A-stable implicit Runge-Kutta methods of arbitrary order of accuracy, based on Radau quadratures, for which efficient preconditioners have been introduced. A refined spectral analysis of the corresponding matrices and matrix-sequences is presented, both in terms of localization and asymptotic global distribution of the eigenvalues. Specific expressions of the eigenvectors are also obtained. The given study fully agrees with the numerically observed spectral behavior and substantially improves the theoretical studies done in this direction so far. Concluding remarks and open problems end the current work, with specific attention to the potential generalizations of the hereby suggested general approach.

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A New Block Preconditioner for Implicit Runge--Kutta Methods for Parabolic PDE Problems

Cited by 14 publications

References 5 publications

Stage‐parallel preconditioners for implicit Runge–Kutta methods of arbitrarily high order, linear problems

Stage‐parallel preconditioners for implicit Runge–Kutta methods of arbitrarily high order, linear problems

Numerical Scheme Based on the Implicit Runge-Kutta Method and Spectral Method for Calculating Nonlinear Hyperbolic Evolution Equations

Fine spectral analysis of preconditioned matrices and matrix-sequences arising from stage-parallel implicit Runge-Kutta methods of arbitrarily high order

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