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PreprintCC BY 4.0arXiv

Finite Volume Convergence for a Heat Equation with a Measure-Valued Source

  • Matthias Gobbert
  • Martin Kruž\'ık
  • Andreas Meister
  • Michael Muscedere
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Abstract

Past work with Tom Seidman analyzed the spatial convergence order of the numerical finite element method applied to time-dependent diffusion-reaction equations with a measure-valued source term. For smooth source terms, the numerical method has second-order accuracy, but the non-smooth source terms introduce a limitation of the order that depends on the spatial dimension. This work in honor of Tom extends the analysis to the finite volume method in its clas- sical two-point flux form and shows that the regularity of the source term reduces the convergence order to 1 in two and 0.5 in three spatial dimensions which is reduced from the nominal second-order and depends on the space dimension. Computational tests with a point source modeled by a Dirac delta distribution exhibit the behavior predicted by the analytical results.

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Cite

BibTeXRIS
BibTeX
@misc{gobbert2026finite,
  title = {{Finite Volume Convergence for a Heat Equation with a Measure-Valued Source}},
  author = {Matthias Gobbert and Martin Kruž\'ık and Andreas Meister and Michael Muscedere},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.22063},
  eprint = {2610.22063},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.22063},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Finite Volume Convergence for a Heat Equation with a Measure-Valued Source
AU  - Matthias Gobbert
AU  - Martin Kruž\'ık
AU  - Andreas Meister
AU  - Michael Muscedere
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2610.22063
UR  - https://arxiv.org/abs/2610.22063
AB  - Past work with Tom Seidman analyzed the spatial convergence order of the numerical finite element method applied to time-dependent diffusion-reaction equations with a measure-valued source term. For smooth source terms, the numerical method has second-order accuracy, but the non-smooth source terms introduce a limitation of the order that depends on the spatial dimension. This work in honor of Tom extends the analysis to the finite volume method in its clas- sical two-point flux form and shows that the regularity of the source term reduces the convergence order to 1 in two and 0.5 in three spatial dimensions which is reduced from the nominal second-order and depends on the space dimension. Computational tests with a point source modeled by a Dirac delta distribution exhibit the behavior predicted by the analytical results.
ER  -