Implict-Explicit Runge-Kutta schemes for geometric flows
Abstract
We propose a one-step implicit-explicit (IMEX) Runge-Kutta framework for high-order time integration of geometric flows. Starting from the Dziuk and Deckelnick-Dziuk parametric finite element formulations of curve shortening flow, we recast the spatially semidiscrete equations as systems of ordinary differential equations and construct second- and third-order schemes. Each stage requires only a linear solve. The key ingredient is a reduced-coefficient structure that allows the variational problem at each stage to be posed on the geometry generated by the preceding stage. The Runge-Kutta stages thus supply the required prediction geometry without a separate prediction procedure. The resulting schemes require no additional starting values. The framework also extends directly to the Barrett-Garcke-Nürnberg formulations of mean curvature flow and surface diffusion for curves and surfaces. Numerical experiments support the designed orders of temporal convergence and indicate that suitable coefficient choices yield mesh quality comparable to that of the corresponding first-order methods.
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BibTeX
@misc{jiang2026implictexplicit,
title = {{Implict-Explicit Runge-Kutta schemes for geometric flows}},
author = {Wei Jiang and Chunmei Su and Kaike Tang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08019},
eprint = {2610.08019},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08019},
}
RIS
TY - UNPB
TI - Implict-Explicit Runge-Kutta schemes for geometric flows
AU - Wei Jiang
AU - Chunmei Su
AU - Kaike Tang
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08019
UR - https://arxiv.org/abs/2610.08019
AB - We propose a one-step implicit-explicit (IMEX) Runge-Kutta framework for high-order time integration of geometric flows. Starting from the Dziuk and Deckelnick-Dziuk parametric finite element formulations of curve shortening flow, we recast the spatially semidiscrete equations as systems of ordinary differential equations and construct second- and third-order schemes. Each stage requires only a linear solve. The key ingredient is a reduced-coefficient structure that allows the variational problem at each stage to be posed on the geometry generated by the preceding stage. The Runge-Kutta stages thus supply the required prediction geometry without a separate prediction procedure. The resulting schemes require no additional starting values. The framework also extends directly to the Barrett-Garcke-Nürnberg formulations of mean curvature flow and surface diffusion for curves and surfaces. Numerical experiments support the designed orders of temporal convergence and indicate that suitable coefficient choices yield mesh quality comparable to that of the corresponding first-order methods.
ER -