Energy stability of explicit--implicit--null time discretizations for nonlinear degenerate parabolic equations with state-dependent coefficients
Abstract
We study nonlinear degenerate parabolic equations with state-dependent coefficients admitting a scalar energy potential. The explicit--implicit--null (EIN) method avoids severe time step restrictions of explicit time stepping and costly Newton--Raphson iterations required by a fully implicit treatment of the nonlinearity. Using orthogonal projection of the Kirchhoff transform, we prove nonlinear energy stability for first- to third-order EIN time stepping coupled with a wide range of spatial discretizations that admit self-adjoint positive semidefinite discrete operators. Under an upper bound for the diffusion coefficient on relevant stage segments, two IMEX coefficient conditions yield sufficient stabilization bounds without any time step restriction. The estimates dissipate the original convex energy and allow degenerate diffusion. We also extend the analysis to diffusion systems with a scalar energy potential in a fixed positive definite metric, and to consistent positive-weight quadrature discretizations. Applications include porous-medium flow, nonlinear heat transfer in plasma, and multicomponent diffusion.
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BibTeX
@misc{wu2026energy,
title = {{Energy stability of explicit--implicit--null time discretizations for nonlinear degenerate parabolic equations with state-dependent coefficients}},
author = {Yue Wu and Chi-Wang Shu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34539},
eprint = {2610.34539},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34539},
}
RIS
TY - UNPB
TI - Energy stability of explicit--implicit--null time discretizations for nonlinear degenerate parabolic equations with state-dependent coefficients
AU - Yue Wu
AU - Chi-Wang Shu
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2610.34539
UR - https://arxiv.org/abs/2610.34539
AB - We study nonlinear degenerate parabolic equations with state-dependent coefficients admitting a scalar energy potential. The explicit--implicit--null (EIN) method avoids severe time step restrictions of explicit time stepping and costly Newton--Raphson iterations required by a fully implicit treatment of the nonlinearity. Using orthogonal projection of the Kirchhoff transform, we prove nonlinear energy stability for first- to third-order EIN time stepping coupled with a wide range of spatial discretizations that admit self-adjoint positive semidefinite discrete operators. Under an upper bound for the diffusion coefficient on relevant stage segments, two IMEX coefficient conditions yield sufficient stabilization bounds without any time step restriction. The estimates dissipate the original convex energy and allow degenerate diffusion. We also extend the analysis to diffusion systems with a scalar energy potential in a fixed positive definite metric, and to consistent positive-weight quadrature discretizations. Applications include porous-medium flow, nonlinear heat transfer in plasma, and multicomponent diffusion.
ER -