Effective highly accurate time integrators for linear Klein–Gordon equations across the scales
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Karolina Kropielnicka
,
Karolina Lademann
and
Katharina Schratz
Abstract
We propose an efficient approach for time integration of Klein–Gordon equations with highly oscillatory in time input terms. The new methods are highly accurate in the entire range, from slowly varying up to highly oscillatory regimes. Our approach is based on splitting methods tailored to the structure of the input term which allows us to resolve the oscillations in the system uniformly in all frequencies, while the error constant does not grow as the oscillations increase. Numerical experiments highlight our theoretical findings and demonstrate the efficiency of the new schemes.
Funding source: European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme
Award Identifier / Grant number: 850941
Funding source: National Science Centre (NCN)
Award Identifier / Grant number: 2019/34/E/ST1/00390
Funding source: Institute of Mathematics of the Polish Academy of Sciences for the years 2021–2023
Award Identifier / Grant number: 663281
Funding source: Academic Computer Center in Gdańsk
Acknowledgment
We are grateful to Arieh Iserles for his friendship, encouragement and invaluable editorial support. The authors with 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”, supported by EPSRC grant EP/R014604/1, where this work has been initiated.
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Research ethics: The local Institutional Review Board deemed the study exempt from review.
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Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
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Competing interests: Authors state no conflict of interest.
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Research funding: The work of Katharina Schratz in this project was funded by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No.850941). The work of Karolina Kropielnicka and Karolina Lademann in this project was funded by the National Science Centre (NCN) project no. 2019/34/E/ ST1/00390. Numerical simulations were carried out by Karolina Lademann at the Academic Computer Center in Gdańsk (CI TASK). This work was partially financed by Simons Foundation Award No. 663281 granted to the Institute of Mathematics of the Polish Academy of Sciences for the years 2021–2023.
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Data availability: Not applicable.
Appendix A: Simplification in Equations (2.4)–(2.7)
Let us recall that
In the following part we are calculating twofold and threefold nested commutators:
where
Analogously
where
For the threefold nested commutators we have
where
Likewise,
with
Finally,
where
and
where
To estimate term Θ4 we need to aggregate the matrices originating in individual commutators. The result is a matrix
and
where
Appendix B: Simplification in Strang splitting error in Section 3.2
Taking
we have
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Articles in the same Issue
- Frontmatter
- Effective highly accurate time integrators for linear Klein–Gordon equations across the scales
- Optimal complexity of goal-oriented adaptive FEM for nonsymmetric linear elliptic PDEs
- Stability and error analysis of a semi-implicit scheme for incompressible flows with variable density and viscosity
- Error analysis of virtual element method for the Poisson–Boltzmann equation
Articles in the same Issue
- Frontmatter
- Effective highly accurate time integrators for linear Klein–Gordon equations across the scales
- Optimal complexity of goal-oriented adaptive FEM for nonsymmetric linear elliptic PDEs
- Stability and error analysis of a semi-implicit scheme for incompressible flows with variable density and viscosity
- Error analysis of virtual element method for the Poisson–Boltzmann equation
