Conservative Three-Layer Schemes for Kirchhoff-Type Equations
Abstract
We study a Kirchhoff-type nonlinear integro-differential equation in two spatial dimensions whose coefficients are allowed to depend on time, and we construct conservative discretizations in time for the associated initial--boundary value problem. We consider two symmetric three-layer schemes of Crank--Nicolson type, a locally linear one and a genuinely nonlinear one, and we show that each of them preserves a discrete analogue of the total mechanical energy of the homogeneous problem with constant coefficients. For the nonlinear scheme we establish uniform apriori bounds on the discrete solution and on its discrete velocity by working directly with the discrete energies, without invoking a nonlinear discrete Grönwall inequality, the constants still grow exponentially in the final time, and we prove local second-order convergence in time, both for the solution and for the central-difference approximation of its first time derivative. The nonlinear system arising at each time level is solved by a fixed-point iteration: given iterates at the two preceding levels that satisfy the apriori bounds, that equation has exactly one solution and the iteration converges to it at a geometric rate once the time step is small enough. Since the apriori bounds are index-local, alternating them with that one-step solver constructs the trajectory stepwise, for time-dependent coefficients as well, on the local interval on which those bounds hold. Numerical experiments, carried out in a setting in which the spatial discretization contributes no error, exhibit the conservation of the discrete invariants, confirm the second order in time and verify the geometric convergence of the fixed-point iteration.
Related
- Conservative Finite Element Methods for Kirchhoff-Type Equations
- Linear ill-posedness of three-dimensional MHD boundary layer equations
- Least energy sign-changing solutions for asymptotically cubic Kirchhoff equations on locally finite graphs
- Systems of Kirchhoff quasilinear Schrodinger equations with Hardy potential and critical Choquard-type nonlinearity
- On structure-preserving and pointwise conservative continuous DG schemes for hyperbolic systems
BibTeX
@misc{krumbiegel2026conservative,
title = {{Conservative Three-Layer Schemes for Kirchhoff-Type Equations}},
author = {Felix Krumbiegel and Jemal Rogava and Andreas Rupp and Zurab Vashakidze},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27707},
eprint = {2609.27707},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27707},
}
RIS
TY - UNPB
TI - Conservative Three-Layer Schemes for Kirchhoff-Type Equations
AU - Felix Krumbiegel
AU - Jemal Rogava
AU - Andreas Rupp
AU - Zurab Vashakidze
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.27707
UR - https://arxiv.org/abs/2609.27707
AB - We study a Kirchhoff-type nonlinear integro-differential equation in two spatial dimensions whose coefficients are allowed to depend on time, and we construct conservative discretizations in time for the associated initial--boundary value problem. We consider two symmetric three-layer schemes of Crank--Nicolson type, a locally linear one and a genuinely nonlinear one, and we show that each of them preserves a discrete analogue of the total mechanical energy of the homogeneous problem with constant coefficients. For the nonlinear scheme we establish uniform apriori bounds on the discrete solution and on its discrete velocity by working directly with the discrete energies, without invoking a nonlinear discrete Grönwall inequality, the constants still grow exponentially in the final time, and we prove local second-order convergence in time, both for the solution and for the central-difference approximation of its first time derivative. The nonlinear system arising at each time level is solved by a fixed-point iteration: given iterates at the two preceding levels that satisfy the apriori bounds, that equation has exactly one solution and the iteration converges to it at a geometric rate once the time step is small enough. Since the apriori bounds are index-local, alternating them with that one-step solver constructs the trajectory stepwise, for time-dependent coefficients as well, on the local interval on which those bounds hold. Numerical experiments, carried out in a setting in which the spatial discretization contributes no error, exhibit the conservation of the discrete invariants, confirm the second order in time and verify the geometric convergence of the fixed-point iteration.
ER -