Optimal parameters for numerical solvers of PDEs

Gianluca Frasca-Caccia; Pranav Singh

doi:10.1007/s10915-023-02324-0

Optimal parameters for numerical solvers of PDEs

Gianluca Frasca-Caccia, Pranav Singh

Università degli Studi di Salerno

Research output: Contribution to journal › Article › peer-review

1 Citation (SciVal)

Abstract

In this paper we introduce a procedure for identifying optimal methods in parametric families of numerical schemes for initial value problems in partial differential equations. The procedure maximizes accuracy by adaptively computing optimal parameters that minimize a defect-based estimate of the local error at each time-step. Viable refinements are proposed to reduce the computational overheads involved in the solution of the optimization problem, and to maintain conservation properties of the original methods. We apply the new strategy to recently introduced families of conservative schemes for the Korteweg-de Vries equation and for a nonlinear heat equation. Numerical tests demonstrate the improved efficiency of the new technique in comparison with existing methods.

Original language	English
Article number	11
Journal	Journal of Scientific Computing
Volume	97
Issue number	1
Early online date	7 Sept 2023
DOIs	https://doi.org/10.1007/s10915-023-02324-0
Publication status	Published - 31 Oct 2023

Bibliographical note

Funding Information:
The authors would like to thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the programme Geometry, Compatibility and Structure Preservation in Computational Differential Equations, when work on this paper was undertaken. This work was supported by EPSRC grant number EP/R014604/1. The first author is member of the INdAM Research group GNCS.

Funding Information:
Open access funding provided by Università degli Studi di Salerno within the CRUI-CARE Agreement.

Funding Information:
The authors would like to thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the programme Geometry, Compatibility and Structure Preservation in Computational Differential Equations, when work on this paper was undertaken. This work was supported by EPSRC grant number EP/R014604/1. The first author is member of the INdAM Research group GNCS.

Keywords

Conservation laws
Finite difference methods
KdV equation
Nonlinear diffusion equation
Parameter optimization

ASJC Scopus subject areas

Software
General Engineering
Computational Mathematics
Theoretical Computer Science
Applied Mathematics
Numerical Analysis
Computational Theory and Mathematics

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Cite this

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title = "Optimal parameters for numerical solvers of PDEs",

abstract = "In this paper we introduce a procedure for identifying optimal methods in parametric families of numerical schemes for initial value problems in partial differential equations. The procedure maximizes accuracy by adaptively computing optimal parameters that minimize a defect-based estimate of the local error at each time-step. Viable refinements are proposed to reduce the computational overheads involved in the solution of the optimization problem, and to maintain conservation properties of the original methods. We apply the new strategy to recently introduced families of conservative schemes for the Korteweg-de Vries equation and for a nonlinear heat equation. Numerical tests demonstrate the improved efficiency of the new technique in comparison with existing methods. ",

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author = "Gianluca Frasca-Caccia and Pranav Singh",

note = "Funding Information: The authors would like to thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the programme Geometry, Compatibility and Structure Preservation in Computational Differential Equations, when work on this paper was undertaken. This work was supported by EPSRC grant number EP/R014604/1. The first author is member of the INdAM Research group GNCS. Funding Information: Open access funding provided by Universit{\`a} degli Studi di Salerno within the CRUI-CARE Agreement. Funding Information: The authors would like to thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the programme Geometry, Compatibility and Structure Preservation in Computational Differential Equations, when work on this paper was undertaken. This work was supported by EPSRC grant number EP/R014604/1. The first author is member of the INdAM Research group GNCS. ",

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N2 - In this paper we introduce a procedure for identifying optimal methods in parametric families of numerical schemes for initial value problems in partial differential equations. The procedure maximizes accuracy by adaptively computing optimal parameters that minimize a defect-based estimate of the local error at each time-step. Viable refinements are proposed to reduce the computational overheads involved in the solution of the optimization problem, and to maintain conservation properties of the original methods. We apply the new strategy to recently introduced families of conservative schemes for the Korteweg-de Vries equation and for a nonlinear heat equation. Numerical tests demonstrate the improved efficiency of the new technique in comparison with existing methods.

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Optimal parameters for numerical solvers of PDEs

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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