## Explicit stabilized multirate method for stiff differential equations

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Assyr Abdulle, Marcus J. Grote and Giacomo Rosilho de Souza
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## Abstract:

Stabilized Runge–Kutta methods are especially efficient for the numerical solution of large systems of stiff nonlinear differential equations because they are fully explicit. For semi-discrete parabolic problems, for instance, stabilized Runge–Kutta methods overcome the stringent stability condition of standard methods without sacrificing explicitness. However, when stiffness is only induced by a few components, as in the presence of spatially local mesh refinement, their efficiency deteriorates. To remove the crippling effect of a few severely stiff components on the entire system of differential equations, we derive a modified equation, whose stiffness solely depends on the remaining mildly stiff components. By applying stabilized Runge–Kutta methods to this modified equation, we then devise an explicit multirate Runge–Kutta–Chebyshev (mRKC) method whose stability conditions are independent of a few severely stiff components. Stability of the mRKC method is proved for a model problem, whereas its efficiency and usefulness are demonstrated through a series of numerical experiments.## References

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## Additional Information

**Assyr Abdulle**- Affiliation: ANMC, Institute of Mathematics, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland
- Email: giacomo.rosilhodesouza@usi.ch
**Marcus J. Grote**- Affiliation: Department of Mathematics and Computer Science, University of Basel, Spiegelgasse 1, 4051 Basel, Switzerland
- MR Author ID: 360720
- ORCID: 0000-0001-8129-0799
- Email: marcus.grote@unibas.ch
**Giacomo Rosilho de Souza**- MR Author ID: 1330077
- ORCID: 0000-0002-0176-8455
- Received by editor(s): December 8, 2020
- Received by editor(s) in revised form: September 17, 2021, and April 4, 2022
- Published electronically: July 19, 2022
- Additional Notes: This research was partially supported by the Swiss National Science Foundation, grant no. 20020_172710.

Our esteemed colleague, Assyr Abdulle, passed away on September 1, 2021 during the revision of this article. A wonderful mentor and friend, his enthusiasm for applied mathematics and for music will always remain dearly missed. - © Copyright 2022 American Mathematical Society
- Journal: Math. Comp.
**91**(2022), 2681-2714 - MSC (2020): Primary 65L04, 65L06, 65L20
- DOI: https://doi.org/10.1090/mcom/3753
- MathSciNet review: 4473100