A New Efficient Explicit Deferred Correction Framework: Analysis and Applications to Hyperbolic PDEs and Adaptivity

doi:10.1007/s42967-023-00294-6

Communications on Applied Mathematics and Computation ›› 2024, Vol. 6 ›› Issue (3): 1629-1664.doi: 10.1007/s42967-023-00294-6

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A New Efficient Explicit Deferred Correction Framework: Analysis and Applications to Hyperbolic PDEs and Adaptivity

Lorenzo Micalizzi¹, Davide Torlo²

1 Institute of Mathematics, University of Zurich, Winterthurerstrasse 190, Zurich 8057, Switzerland;
2 SISSA mathLab, SISSA, via Bonomea 265, Trieste 34136, Italy

收稿日期:2022-10-06 修回日期:2023-05-18 接受日期:2023-06-25 发布日期:2024-12-20
通讯作者: Lorenzo Micalizzi,lorenzo.micalizzi@math.uzh.ch;Davide Torlo,davide.torlo@sissa.it E-mail:lorenzo.micalizzi@math.uzh.ch;davide.torlo@sissa.it
基金资助:
L. Micalizzi has been funded by the SNF grant 200020_204917 “Structure preserving and fast methods for hyperbolic systems of conservation laws” and by the Forschungskredit grant FK-21- 098. D. Torlo has been funded by a SISSA Mathematical Fellowship. The authors warmly acknowledge Sixtine Michel for providing the code Parasol.

A New Efficient Explicit Deferred Correction Framework: Analysis and Applications to Hyperbolic PDEs and Adaptivity

Lorenzo Micalizzi¹, Davide Torlo²

1 Institute of Mathematics, University of Zurich, Winterthurerstrasse 190, Zurich 8057, Switzerland;
2 SISSA mathLab, SISSA, via Bonomea 265, Trieste 34136, Italy

Received:2022-10-06 Revised:2023-05-18 Accepted:2023-06-25 Published:2024-12-20
Contact: Lorenzo Micalizzi,lorenzo.micalizzi@math.uzh.ch;Davide Torlo,davide.torlo@sissa.it E-mail:lorenzo.micalizzi@math.uzh.ch;davide.torlo@sissa.it
Supported by:
L. Micalizzi has been funded by the SNF grant 200020_204917 “Structure preserving and fast methods for hyperbolic systems of conservation laws” and by the Forschungskredit grant FK-21- 098. D. Torlo has been funded by a SISSA Mathematical Fellowship. The authors warmly acknowledge Sixtine Michel for providing the code Parasol.

摘要/Abstract

摘要： The deferred correction (DeC) is an iterative procedure, characterized by increasing the accuracy at each iteration, which can be used to design numerical methods for systems of ODEs. The main advantage of such framework is the automatic way of getting arbitrarily high order methods, which can be put in the Runge-Kutta (RK) form. The drawback is the larger computational cost with respect to the most used RK methods. To reduce such cost, in an explicit setting, we propose an efficient modification: we introduce interpolation processes between the DeC iterations, decreasing the computational cost associated to the low order ones. We provide the Butcher tableaux of the new modified methods and we study their stability, showing that in some cases the computational advantage does not affect the stability. The flexibility of the novel modification allows nontrivial applications to PDEs and construction of adaptive methods. The good performances of the introduced methods are broadly tested on several benchmarks both in ODE and PDE contexts.

关键词: Efficient deferred correction (DeC), Arbitrary high order, Stability, Adaptive methods, Hyperbolic PDEs

Abstract: The deferred correction (DeC) is an iterative procedure, characterized by increasing the accuracy at each iteration, which can be used to design numerical methods for systems of ODEs. The main advantage of such framework is the automatic way of getting arbitrarily high order methods, which can be put in the Runge-Kutta (RK) form. The drawback is the larger computational cost with respect to the most used RK methods. To reduce such cost, in an explicit setting, we propose an efficient modification: we introduce interpolation processes between the DeC iterations, decreasing the computational cost associated to the low order ones. We provide the Butcher tableaux of the new modified methods and we study their stability, showing that in some cases the computational advantage does not affect the stability. The flexibility of the novel modification allows nontrivial applications to PDEs and construction of adaptive methods. The good performances of the introduced methods are broadly tested on several benchmarks both in ODE and PDE contexts.

Key words: Efficient deferred correction (DeC), Arbitrary high order, Stability, Adaptive methods, Hyperbolic PDEs

中图分类号:

65M12
65L20

Lorenzo Micalizzi, Davide Torlo. A New Efficient Explicit Deferred Correction Framework: Analysis and Applications to Hyperbolic PDEs and Adaptivity[J]. Communications on Applied Mathematics and Computation, 2024, 6(3): 1629-1664.

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A New Efficient Explicit Deferred Correction Framework: Analysis and Applications to Hyperbolic PDEs and Adaptivity

A New Efficient Explicit Deferred Correction Framework: Analysis and Applications to Hyperbolic PDEs and Adaptivity

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