Efficient Iterative Arbitrary High-Order Methods: an Adaptive Bridge Between Low and High Order

doi:10.1007/s42967-023-00290-w

Communications on Applied Mathematics and Computation ›› 2025, Vol. 7 ›› Issue (1): 40-77.doi: 10.1007/s42967-023-00290-w

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Efficient Iterative Arbitrary High-Order Methods: an Adaptive Bridge Between Low and High Order

Lorenzo Micalizzi¹, Davide Torlo², Walter Boscheri³

1 Institute of Mathematics, University of Zurich, Winterthurerstrasse 190, Zurich 8057, Switzerland;
2 SISSA mathLab, SISSA, via Bonomea 265, Trieste 34136, Italy;
3 Dipartimento di Matematica e Informatica, University of Ferrara, via Machiavelli 30, Ferrara 44121, Italy

收稿日期:2022-12-16 修回日期:2023-05-05 接受日期:2023-05-29 出版日期:2025-04-21 发布日期:2025-04-21
通讯作者: Walter Boscheri,walter.boscheri@unfe.it;Lorenzo Micalizzi,lorenzo.micalizzi@math.uzh.ch;Davide Torlo,davide.torlo@sissa.it E-mail:walter.boscheri@unfe.it;lorenzo.micalizzi@math.uzh.ch;davide.torlo@sissa.it
基金资助:
LM has been funded by the SNF under Grant 200020_204917 “Structure preserving and fast methods for hyperbolic systems of conservation laws”. DT has been funded by an SISSA Mathematical Fellowship. WB received financial support by Fondazione Cariplo and Fondazione CDP (Italy) under Grant No. 2022-1895.

Efficient Iterative Arbitrary High-Order Methods: an Adaptive Bridge Between Low and High Order

Lorenzo Micalizzi¹, Davide Torlo², Walter Boscheri³

1 Institute of Mathematics, University of Zurich, Winterthurerstrasse 190, Zurich 8057, Switzerland;
2 SISSA mathLab, SISSA, via Bonomea 265, Trieste 34136, Italy;
3 Dipartimento di Matematica e Informatica, University of Ferrara, via Machiavelli 30, Ferrara 44121, Italy

Received:2022-12-16 Revised:2023-05-05 Accepted:2023-05-29 Online:2025-04-21 Published:2025-04-21
Supported by:
LM has been funded by the SNF under Grant 200020_204917 “Structure preserving and fast methods for hyperbolic systems of conservation laws”. DT has been funded by an SISSA Mathematical Fellowship. WB received financial support by Fondazione Cariplo and Fondazione CDP (Italy) under Grant No. 2022-1895.

摘要/Abstract

摘要： We propose a new paradigm for designing efficient p-adaptive arbitrary high-order methods. We consider arbitrary high-order iterative schemes that gain one order of accuracy at each iteration and we modify them to match the accuracy achieved in a specific iteration with the discretization accuracy of the same iteration. Apart from the computational advantage, the newly modified methods allow to naturally perform the p-adaptivity, stopping the iterations when appropriate conditions are met. Moreover, the modification is very easy to be included in an existing implementation of an arbitrary high-order iterative scheme and it does not ruin the possibility of parallelization, if this was achievable by the original method. An application to the Arbitrary DERivative (ADER) method for hyperbolic Partial Differential Equations (PDEs) is presented here. We explain how such a framework can be interpreted as an arbitrary high-order iterative scheme, by recasting it as a Deferred Correction (DeC) method, and how to easily modify it to obtain a more efficient formulation, in which a local a posteriori limiter can be naturally integrated leading to the p-adaptivity and structure-preserving properties. Finally, the novel approach is extensively tested against classical benchmarks for compressible gas dynamics to show the robustness and the computational efficiency.

关键词: p-Adaptivity, Arbitrary DERivative (ADER), Arbitrary high order, Deferred Correction (DeC), Positivity preserving

Abstract: We propose a new paradigm for designing efficient p-adaptive arbitrary high-order methods. We consider arbitrary high-order iterative schemes that gain one order of accuracy at each iteration and we modify them to match the accuracy achieved in a specific iteration with the discretization accuracy of the same iteration. Apart from the computational advantage, the newly modified methods allow to naturally perform the p-adaptivity, stopping the iterations when appropriate conditions are met. Moreover, the modification is very easy to be included in an existing implementation of an arbitrary high-order iterative scheme and it does not ruin the possibility of parallelization, if this was achievable by the original method. An application to the Arbitrary DERivative (ADER) method for hyperbolic Partial Differential Equations (PDEs) is presented here. We explain how such a framework can be interpreted as an arbitrary high-order iterative scheme, by recasting it as a Deferred Correction (DeC) method, and how to easily modify it to obtain a more efficient formulation, in which a local a posteriori limiter can be naturally integrated leading to the p-adaptivity and structure-preserving properties. Finally, the novel approach is extensively tested against classical benchmarks for compressible gas dynamics to show the robustness and the computational efficiency.

Key words: p-Adaptivity, Arbitrary DERivative (ADER), Arbitrary high order, Deferred Correction (DeC), Positivity preserving

中图分类号:

Lorenzo Micalizzi, Davide Torlo, Walter Boscheri. Efficient Iterative Arbitrary High-Order Methods: an Adaptive Bridge Between Low and High Order[J]. Communications on Applied Mathematics and Computation, 2025, 7(1): 40-77.

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Efficient Iterative Arbitrary High-Order Methods: an Adaptive Bridge Between Low and High Order

Efficient Iterative Arbitrary High-Order Methods: an Adaptive Bridge Between Low and High Order

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