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PreprintCC BY 4.0arXiv

Operator splitting methods for non-autonomous evolution equations comprising three parts, with application to the Bloch-Torrey equation of MRI

  • Barbara Kaltenbacher
  • Pablo Muñoz
  • Mechthild Thalhammer
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Abstract

In this paper we devise and analyze Lie Trotter splitting schemes for non-autonomous evolution equations, following two approaches: 1) A fully non-autonomous scheme, motivated by the potential use of explicit solutions of some subproblems. 2) A splitting scheme relying on the concept of Magnus integrators, where we freeze the operators at the beginning of each time interval as this suffices for first order convergence of the Lie-Trotter integrator. We derive first order convergence in a general setting and apply the analysis to the Bloch-Torrey equation of magnetic resonance imaging, where a decomposition into three subproblems is well-motivated by the physical role of the underlying operators. Numerical tests, complementing the time stepping scheme by a finite element space discretization, illustrate the theoretical findings.

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Cite

BibTeXRIS
BibTeX
@misc{kaltenbacher2026operator,
  title = {{Operator splitting methods for non-autonomous evolution equations comprising three parts, with application to the Bloch-Torrey equation of MRI}},
  author = {Barbara Kaltenbacher and Pablo Muñoz and Mechthild Thalhammer},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.04355},
  eprint = {2610.04355},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.04355},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Operator splitting methods for non-autonomous evolution equations comprising three parts, with application to the Bloch-Torrey equation of MRI
AU  - Barbara Kaltenbacher
AU  - Pablo Muñoz
AU  - Mechthild Thalhammer
PY  - 2026
DA  - 2026/10/03
PB  - arXiv
DO  - 10.48550/arxiv.2610.04355
UR  - https://arxiv.org/abs/2610.04355
AB  - In this paper we devise and analyze Lie Trotter splitting schemes for non-autonomous evolution equations, following two approaches: 1) A fully non-autonomous scheme, motivated by the potential use of explicit solutions of some subproblems. 2) A splitting scheme relying on the concept of Magnus integrators, where we freeze the operators at the beginning of each time interval as this suffices for first order convergence of the Lie-Trotter integrator. We derive first order convergence in a general setting and apply the analysis to the Bloch-Torrey equation of magnetic resonance imaging, where a decomposition into three subproblems is well-motivated by the physical role of the underlying operators. Numerical tests, complementing the time stepping scheme by a finite element space discretization, illustrate the theoretical findings.
ER  -