2021
DOI: 10.1137/20m1349680
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A New Block Preconditioner for Implicit Runge--Kutta Methods for Parabolic PDE Problems

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Cited by 14 publications
(19 citation statements)
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“…Remark Clearly, the approach can be applied to Aq$$ {A}_q $$, which is tested in Reference 40. The variant š’«LD from that article would correspond to our approach when applied to Aq$$ {A}_q $$.…”
Section: Stageā€parallel Preconditioning Methods For the Radau Tensor ...mentioning
confidence: 96%
See 1 more Smart Citation
“…Remark Clearly, the approach can be applied to Aq$$ {A}_q $$, which is tested in Reference 40. The variant š’«LD from that article would correspond to our approach when applied to Aq$$ {A}_q $$.…”
Section: Stageā€parallel Preconditioning Methods For the Radau Tensor ...mentioning
confidence: 96%
“…A comparison study of a series of LDUā€based preconditioning approaches, applied on the nonā€transformed Equation (4) is recently presented in Reference 40. The authors test various combinations of the LDU$$ LDU $$ factorization of Aq$$ {A}_q $$.…”
Section: A Summary Of Some Earlier Presented Stageā€parallel Precondit...mentioning
confidence: 99%
“…Figure 4 shows that the error does not depend on N = 2 2 , 2 3 , ā€¢ ā€¢ ā€¢ , 2 15 (refer to + and ā€¢ in Figure 4 for the errors by implicit Runge-Kutta method, and to Ɨ and for those by Īø method). Regardless of the choice of N, it is confirmed in Figure 4 that the errors with āˆ†t = 2 āˆ’13 are smaller than those with āˆ†t = 2 āˆ’6 (note that in case of āˆ†t = 2 āˆ’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 āˆ†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%
“…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%