Linearize or Discretize First? Error Estimates under Mesh-Uniform Regularity and Exponential Stability for the KdV Equation with Boundary Dissipation
Abstract
We study the Korteweg-de Vries equation on a bounded interval with boundary dissipation, and its spatial semidiscretization by a weighted cubic Hermite Galerkin method. A small nonlinear wave can be approximated by a linear discrete simulation in two orders: linearize first and then discretize, or discretize first and then linearize. Both orders give the same total error, but they divide it differently between linearization and discretization. The difference of the two divisions is the ordering error. We show that it is quadratic in the size of the initial datum and carries a positive power of the mesh size, for all time. For the continuous problem, the solution map is analytic, with a power series that converges for all time in exponentially weighted trajectory norms. The linearized semigroup has Crandall-Pazy regularity with the exact index 3/2. For the weighted scheme, exponential stability, Kato smoothing, Crandall-Pazy regularity and analyticity of the solution maps hold uniformly in the mesh size. These properties give error estimates in which the error and the datum are measured in the same norm: in the energy norm, in H01 and in H2. In exponentially weighted time-integrated norms the semidiscrete solutions converge uniformly on balls, at the rate h1/6 in L4(0,∞;L2) for data in L2 and at the rate h2/3 in L2(0,∞;H1) for data in H01. On every interval [t0,∞), t0>0, same-norm convergence holds uniformly on balls at algebraic rates. The ordering error is of order h1/6 times the square of the energy norm of the datum, uniformly for all t≥0. For the unweighted scheme, a grid wave packet of zero group velocity destroys mesh-uniform Kato smoothing and Crandall-Pazy regularity.
Related
- Global regularity for the unidirectional Euler-alignment system with supercritical dissipation
- A unified framework for original energy dissipation of equilibrium-preserving exponential Runge--Kutta methods for L2 gradient flows
- Regularity of the solution map for the Oldroyd--B model with fractional stress dissipation
- Explicit Exponential Sum Estimates and Approximate Functional Equations for the Zeta Function
- Regularity Estimates for Quasilinear Elliptic Equations with Measure Data and Matrix-Valued Weight
BibTeX
@misc{liu2026linearize,
title = {{Linearize or Discretize First? Error Estimates under Mesh-Uniform Regularity and Exponential Stability for the KdV Equation with Boundary Dissipation}},
author = {Kangsheng Liu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39269},
eprint = {2610.39269},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39269},
}
RIS
TY - UNPB
TI - Linearize or Discretize First? Error Estimates under Mesh-Uniform Regularity and Exponential Stability for the KdV Equation with Boundary Dissipation
AU - Kangsheng Liu
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39269
UR - https://arxiv.org/abs/2610.39269
AB - We study the Korteweg-de Vries equation on a bounded interval with boundary dissipation, and its spatial semidiscretization by a weighted cubic Hermite Galerkin method. A small nonlinear wave can be approximated by a linear discrete simulation in two orders: linearize first and then discretize, or discretize first and then linearize. Both orders give the same total error, but they divide it differently between linearization and discretization. The difference of the two divisions is the ordering error. We show that it is quadratic in the size of the initial datum and carries a positive power of the mesh size, for all time. For the continuous problem, the solution map is analytic, with a power series that converges for all time in exponentially weighted trajectory norms. The linearized semigroup has Crandall-Pazy regularity with the exact index 3/2. For the weighted scheme, exponential stability, Kato smoothing, Crandall-Pazy regularity and analyticity of the solution maps hold uniformly in the mesh size. These properties give error estimates in which the error and the datum are measured in the same norm: in the energy norm, in H_0^1 and in H^2. In exponentially weighted time-integrated norms the semidiscrete solutions converge uniformly on balls, at the rate h^1/6 in L^4(0,∞;L^2) for data in L^2 and at the rate h^2/3 in L^2(0,∞;H^1) for data in H_0^1. On every interval [t_0,∞), t_0>0, same-norm convergence holds uniformly on balls at algebraic rates. The ordering error is of order h^1/6 times the square of the energy norm of the datum, uniformly for all t≥0. For the unweighted scheme, a grid wave packet of zero group velocity destroys mesh-uniform Kato smoothing and Crandall-Pazy regularity.
ER -